--------------------------------------- Character Table of Group E of Order 108 ---------------------------------------------------------------- Class | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Size | 1 9 1 1 12 12 9 9 9 9 9 9 9 9 Order | 1 2 3 3 3 3 4 4 6 6 12 12 12 12 ---------------------------------------------------------------- p = 2 1 1 4 3 5 6 2 2 3 4 10 9 9 10 p = 3 1 2 1 1 1 1 8 7 2 2 7 8 7 8 ---------------------------------------------------------------- X.1 + 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 + 1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 X.3 0 1 -1 1 1 1 1 I -I -1 -1 -I I -I I X.4 0 1 -1 1 1 1 1 -I I -1 -1 I -I I -I X.5 0 3 -1 3*J-3-3*J 0 0 -1 -1 1+J -J 1+J -J -J 1+J X.6 0 3 -1-3-3*J 3*J 0 0 -1 -1 -J 1+J -J 1+J 1+J -J X.7 0 3 -1-3-3*J 3*J 0 0 1 1 -J 1+J J -1-J -1-J J X.8 0 3 -1 3*J-3-3*J 0 0 1 1 1+J -J -1-J J J -1-J X.9 0 3 1-3-3*J 3*J 0 0 -I I J-1-J Z1-Z1#5 Z1#5 -Z1 X.10 0 3 1 3*J-3-3*J 0 0 -I I-1-J J Z1#5 -Z1 Z1-Z1#5 X.11 0 3 1-3-3*J 3*J 0 0 I -I J-1-J -Z1 Z1#5-Z1#5 Z1 X.12 0 3 1 3*J-3-3*J 0 0 I -I-1-J J-Z1#5 Z1 -Z1 Z1#5 X.13 + 4 0 4 4 1 -2 0 0 0 0 0 0 0 0 X.14 + 4 0 4 4 -2 1 0 0 0 0 0 0 0 0 Explanation of Character Value Symbols -------------------------------------- J = RootOfUnity(3) I = RootOfUnity(4) Z1 = (CyclotomicField(12: Sparse := true)) ! [ RationalField() | 0, 0, 0, 1] --------------------------------------- Character Table of Group F of Order 216 ---------------------------------------------------------------------- Class | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Size | 1 9 1 1 24 18 18 18 9 9 18 18 18 18 18 18 Order | 1 2 3 3 3 4 4 4 6 6 12 12 12 12 12 12 ---------------------------------------------------------------------- p = 2 1 1 4 3 5 2 2 2 4 3 9 9 10 10 10 9 p = 3 1 2 1 1 1 6 7 8 2 2 6 7 7 6 8 8 ---------------------------------------------------------------------- X.1 + 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 + 1 1 1 1 1 -1 1 -1 1 1 -1 1 1 -1 -1 -1 X.3 + 1 1 1 1 1 1 -1 -1 1 1 1 -1 -1 1 -1 -1 X.4 + 1 1 1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 1 1 X.5 - 2 -2 2 2 2 0 0 0 -2 -2 0 0 0 0 0 0 X.6 0 3 -1-3-3*J 3*J 0 1 1 1 1+J -J J J-1-J-1-J-1-J J X.7 0 3 -1-3-3*J 3*J 0 1 -1 -1 1+J -J J -J 1+J-1-J 1+J -J X.8 0 3 -1-3-3*J 3*J 0 -1 1 -1 1+J -J -J J-1-J 1+J 1+J -J X.9 0 3 -1-3-3*J 3*J 0 -1 -1 1 1+J -J -J -J 1+J 1+J-1-J J X.10 0 3 -1 3*J-3-3*J 0 1 1 1 -J 1+J-1-J-1-J J J J-1-J X.11 0 3 -1 3*J-3-3*J 0 1 -1 -1 -J 1+J-1-J 1+J -J J -J 1+J X.12 0 3 -1 3*J-3-3*J 0 -1 -1 1 -J 1+J 1+J 1+J -J -J J-1-J X.13 0 3 -1 3*J-3-3*J 0 -1 1 -1 -J 1+J 1+J-1-J J -J -J 1+J X.14 0 6 2-6-6*J 6*J 0 0 0 0-2-2*J 2*J 0 0 0 0 0 0 X.15 0 6 2 6*J-6-6*J 0 0 0 0 2*J-2-2*J 0 0 0 0 0 0 X.16 + 8 0 8 8 -1 0 0 0 0 0 0 0 0 0 0 0 Explanation of Character Value Symbols -------------------------------------- J = RootOfUnity(3) ----------------------------------------------- Character Table of Hessian Group G of Order 648 --------------------------------------------------------------------------- Class | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Size | 1 9 1 1 24 72 72 54 9 9 12 12 12 12 12 Order | 1 2 3 3 3 3 3 4 6 6 9 9 9 9 9 --------------------------------------------------------------------------- p = 2 1 1 4 3 5 7 6 2 4 3 12 15 14 16 13 p = 3 1 2 1 1 1 1 1 8 2 2 3 4 4 3 3 --------------------------------------------------------------------------- X.1 + 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 0 1 1 1 1 1 J-1-J 1 1 1 J -1-J -1-J J J X.3 0 1 1 1 1 1-1-J J 1 1 1 -1-J J J -1-J -1-J X.4 - 2 -2 2 2 2 -1 -1 0 -2 -2 -1 -1 -1 -1 -1 X.5 0 2 -2 2 2 2 -J 1+J 0 -2 -2 -J 1+J 1+J -J -J X.6 0 2 -2 2 2 2 1+J -J 0 -2 -2 1+J -J -J 1+J 1+J X.7 + 3 3 3 3 3 0 0 -1 3 3 0 0 0 0 0 X.8 0 3 -1-3-3*J 3*J 0 0 0 1 1+J -J Z1 Z1#2 Z1#8 Z1#7 Z1#4 X.9 0 3 -1-3-3*J 3*J 0 0 0 1 1+J -J Z1#7 Z1#5 Z1#2 Z1#4 Z1 X.10 0 3 -1 3*J-3-3*J 0 0 0 1 -J 1+J Z1#2 Z1#4 Z1#7 Z1#5 Z1#8 X.11 0 3 -1 3*J-3-3*J 0 0 0 1 -J 1+J Z1#5 Z1 Z1#4 Z1#8 Z1#2 X.12 0 3 -1-3-3*J 3*J 0 0 0 1 1+J -J Z1#4 Z1#8 Z1#5 Z1 Z1#7 X.13 0 3 -1 3*J-3-3*J 0 0 0 1 -J 1+J Z1#8 Z1#7 Z1 Z1#2 Z1#5 X.14 0 6 2 6*J-6-6*J 0 0 0 0 2*J-2-2*J-Z1#5 -Z1-Z1#4-Z1#8-Z1#2 X.15 0 6 2-6-6*J 6*J 0 0 0 0-2-2*J 2*J-Z1#4-Z1#8-Z1#5 -Z1-Z1#7 X.16 0 6 2 6*J-6-6*J 0 0 0 0 2*J-2-2*J-Z1#8-Z1#7 -Z1-Z1#2-Z1#5 X.17 0 6 2 6*J-6-6*J 0 0 0 0 2*J-2-2*J-Z1#2-Z1#4-Z1#7-Z1#5-Z1#8 X.18 0 6 2-6-6*J 6*J 0 0 0 0-2-2*J 2*J -Z1-Z1#2-Z1#8-Z1#7-Z1#4 X.19 0 6 2-6-6*J 6*J 0 0 0 0-2-2*J 2*J-Z1#7-Z1#5-Z1#2-Z1#4 -Z1 X.20 + 8 0 8 8 -1 -1 -1 0 0 0 2 2 2 2 2 X.21 0 8 0 8 8 -1 1+J -J 0 0 0-2-2*J 2*J 2*J-2-2*J-2-2*J X.22 0 8 0 8 8 -1 -J 1+J 0 0 0 2*J-2-2*J-2-2*J 2*J 2*J X.23 0 9 -3-9-9*J 9*J 0 0 0 -1 3+3*J -3*J 0 0 0 0 0 X.24 0 9 -3 9*J-9-9*J 0 0 0 -1 -3*J 3+3*J 0 0 0 0 0 ---------------------------------------------------- Class | 16 17 18 19 20 21 22 23 24 Size | 12 54 54 36 36 36 36 36 36 Order | 9 12 12 18 18 18 18 18 18 ---------------------------------------------------- p = 2 11 10 9 12 11 15 16 13 14 p = 3 4 8 8 9 10 10 9 9 10 ---------------------------------------------------- X.1 + 1 1 1 1 1 1 1 1 1 X.2 0 -1-J 1 1 J -1-J -1-J J J -1-J X.3 0 J 1 1 -1-J J J -1-J -1-J J X.4 - -1 0 0 1 1 1 1 1 1 X.5 0 1+J 0 0 J -1-J -1-J J J -1-J X.6 0 -J 0 0 -1-J J J -1-J -1-J J X.7 + 0 -1 -1 0 0 0 0 0 0 X.8 0 Z1#5-1-J J Z2 Z2#5 Z2#2 Z2#7 Z2#4 Z2#8 X.9 0 Z1#8-1-J J Z2#7 Z2#8 Z2#5 Z2#4 Z2 Z2#2 X.10 0 Z1 J-1-J Z2#2 Z2 Z2#4 Z2#5 Z2#8 Z2#7 X.11 0 Z1#7 J-1-J Z2#5 Z2#7 Z2 Z2#8 Z2#2 Z2#4 X.12 0 Z1#2-1-J J Z2#4 Z2#2 Z2#8 Z2 Z2#7 Z2#5 X.13 0 Z1#4 J-1-J Z2#8 Z2#4 Z2#7 Z2#2 Z2#5 Z2 X.14 0 -Z1#7 0 0 Z2#5 Z2#7 Z2 Z2#8 Z2#2 Z2#4 X.15 0 -Z1#2 0 0 Z2#4 Z2#2 Z2#8 Z2 Z2#7 Z2#5 X.16 0 -Z1#4 0 0 Z2#8 Z2#4 Z2#7 Z2#2 Z2#5 Z2 X.17 0 -Z1 0 0 Z2#2 Z2 Z2#4 Z2#5 Z2#8 Z2#7 X.18 0 -Z1#5 0 0 Z2 Z2#5 Z2#2 Z2#7 Z2#4 Z2#8 X.19 0 -Z1#8 0 0 Z2#7 Z2#8 Z2#5 Z2#4 Z2 Z2#2 X.20 + 2 0 0 0 0 0 0 0 0 X.21 0 2*J 0 0 0 0 0 0 0 0 X.22 0 -2-2*J 0 0 0 0 0 0 0 0 X.23 0 0 1+J -J 0 0 0 0 0 0 X.24 0 0 -J 1+J 0 0 0 0 0 0 Explanation of Character Value Symbols -------------------------------------- J = RootOfUnity(3) Z1 = (CyclotomicField(9: Sparse := true)) ! [ RationalField() | 0, 0, -1, 0, 0, -2 ] Z2 = (CyclotomicField(9: Sparse := true)) ! [ RationalField() | 0, 0, -1, 0, 0, 0 ] --------------------------------------------- Character Table of Simple Group H of Order 60 --------------------------------------------- Class | 1 2 3 4 5 Size | 1 15 20 12 12 Order | 1 2 3 5 5 --------------------------- p = 2 1 1 3 5 4 p = 3 1 2 1 5 4 p = 5 1 2 3 1 1 --------------------------- X.1 + 1 1 1 1 1 X.2 + 3 -1 0 Z1 Z1#2 X.3 + 3 -1 0 Z1#2 Z1 X.4 + 4 0 1 -1 -1 X.5 + 5 1 -1 0 0 Explanation of Character Value Symbols -------------------------------------- Z1 = (CyclotomicField(5: Sparse := true)) ! [ RationalField() | 0, 0, -1, -1] ---------------------------------------------- Character Table of Simple Group I of Order 168 ---------------------------------------------- Class | 1 2 3 4 5 6 Size | 1 21 56 42 24 24 Order | 1 2 3 4 7 7 ------------------------------ p = 2 1 1 3 2 5 6 p = 3 1 2 1 4 6 5 p = 7 1 2 3 4 1 1 ------------------------------ X.1 + 1 1 1 1 1 1 X.2 0 3 -1 0 1 Z1 Z1#3 X.3 0 3 -1 0 1 Z1#3 Z1 X.4 + 6 2 0 0 -1 -1 X.5 + 7 -1 1 -1 0 0 X.6 + 8 0 -1 0 1 1 Explanation of Character Value Symbols -------------------------------------- Z1 = (CyclotomicField(7: Sparse := true)) ! [ RationalField() | 0, 1, 1, 0, 1, 0 ] --------------------------------------- Character Table of Group J of Order 180 ----------------------------------------------------------------------------- Class | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Size | 1 15 1 1 20 20 20 12 12 15 15 12 12 12 12 Order | 1 2 3 3 3 3 3 5 5 6 6 15 15 15 15 ----------------------------------------------------------------------------- p = 2 1 1 4 3 6 5 7 9 8 4 3 15 14 13 12 p = 3 1 2 1 1 1 1 1 9 8 2 2 8 9 8 9 p = 5 1 2 4 3 6 5 7 1 1 11 10 3 3 4 4 ----------------------------------------------------------------------------- X.1 + 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 0 1 1 -1-J J J-1-J 1 1 1-1-J J J J -1-J -1-J X.3 0 1 1 J -1-J-1-J J 1 1 1 J-1-J -1-J -1-J J J X.4 + 3 -1 3 3 0 0 0 Z1 Z1#2 -1 -1 Z1#2 Z1 Z1#2 Z1 X.5 + 3 -1 3 3 0 0 0 Z1#2 Z1 -1 -1 Z1 Z1#2 Z1 Z1#2 X.6 0 3 -1-3-3*J 3*J 0 0 0 Z1 Z1#2 1+J -J Z2 Z2#7 Z2#11 Z2#2 X.7 0 3 -1 3*J-3-3*J 0 0 0 Z1#2 Z1 -J 1+J Z2#2 Z2#11 Z2#7 Z2 X.8 0 3 -1 3*J-3-3*J 0 0 0 Z1 Z1#2 -J 1+J Z2#11 Z2#2 Z2 Z2#7 X.9 0 3 -1-3-3*J 3*J 0 0 0 Z1#2 Z1 1+J -J Z2#7 Z2 Z2#2 Z2#11 X.10 + 4 0 4 4 1 1 1 -1 -1 0 0 -1 -1 -1 -1 X.11 0 4 0-4-4*J 4*J J-1-J 1 -1 -1 0 0 -J -J 1+J 1+J X.12 0 4 0 4*J-4-4*J-1-J J 1 -1 -1 0 0 1+J 1+J -J -J X.13 + 5 1 5 5 -1 -1 -1 0 0 1 1 0 0 0 0 X.14 0 5 1-5-5*J 5*J -J 1+J -1 0 0-1-J J 0 0 0 0 X.15 0 5 1 5*J-5-5*J 1+J -J -1 0 0 J-1-J 0 0 0 0 Explanation of Character Value Symbols -------------------------------------- J = RootOfUnity(3) Z1 = (CyclotomicField(5: Sparse := true)) ! [ RationalField() | 0, 0, -1, -1] Z2 = (CyclotomicField(15: Sparse := true)) ! [ RationalField() | 0, 1, 0, 0, 0, 1, 0, 1 ] --------------------------------------- Character Table of Group K of Order 504 ------------------------------------------------------------------------------ Class | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Size | 1 21 1 1 56 56 56 42 21 21 24 24 42 42 24 24 Order | 1 2 3 3 3 3 3 4 6 6 7 7 12 12 21 21 ------------------------------------------------------------------------------ p = 2 1 1 4 3 7 6 5 2 3 4 11 12 10 9 17 18 p = 3 1 2 1 1 1 1 1 8 2 2 12 11 8 8 12 11 p = 7 1 2 3 4 5 6 7 8 9 10 1 1 13 14 4 4 ------------------------------------------------------------------------------ X.1 + 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 0 1 1 J -1-J J 1-1-J 1 -1-J J 1 1-1-J J -1-J -1-J X.3 0 1 1 -1-J J-1-J 1 J 1 J -1-J 1 1 J-1-J J J X.4 0 3 -1 3 3 0 0 0 1 -1 -1 Z1 Z1#3 1 1 Z1 Z1#3 X.5 0 3 -1 3 3 0 0 0 1 -1 -1 Z1#3 Z1 1 1 Z1#3 Z1 X.6 0 3 -1-3-3*J 3*J 0 0 0 1 -J 1+J Z1 Z1#3 J-1-J Z2 Z2#10 X.7 0 3 -1 3*J-3-3*J 0 0 0 1 1+J -J Z1 Z1#3-1-J J Z2#2 Z2#5 X.8 0 3 -1-3-3*J 3*J 0 0 0 1 -J 1+J Z1#3 Z1 J-1-J Z2#10 Z2 X.9 0 3 -1 3*J-3-3*J 0 0 0 1 1+J -J Z1#3 Z1-1-J J Z2#5 Z2#2 X.10 + 6 2 6 6 0 0 0 0 2 2 -1 -1 0 0 -1 -1 X.11 0 6 2 6*J-6-6*J 0 0 0 0-2-2*J 2*J -1 -1 0 0 1+J 1+J X.12 0 6 2-6-6*J 6*J 0 0 0 0 2*J-2-2*J -1 -1 0 0 -J -J X.13 + 7 -1 7 7 1 1 1 -1 -1 -1 0 0 -1 -1 0 0 X.14 0 7 -1-7-7*J 7*J-1-J 1 J -1 -J 1+J 0 0 -J 1+J 0 0 X.15 0 7 -1 7*J-7-7*J J 1-1-J -1 1+J -J 0 0 1+J -J 0 0 X.16 + 8 0 8 8 -1 -1 -1 0 0 0 1 1 0 0 1 1 X.17 0 8 0-8-8*J 8*J 1+J -1 -J 0 0 0 1 1 0 0 J J X.18 0 8 0 8*J-8-8*J -J -1 1+J 0 0 0 1 1 0 0 -1-J -1-J --------------------- Class | 17 18 Size | 24 24 Order | 21 21 --------------------- p = 2 15 16 p = 3 12 11 p = 7 3 3 --------------------- X.1 + 1 1 X.2 0 J J X.3 0 -1-J -1-J X.4 0 Z1 Z1#3 X.5 0 Z1#3 Z1 X.6 0 Z2#2 Z2#5 X.7 0 Z2 Z2#10 X.8 0 Z2#5 Z2#2 X.9 0 Z2#10 Z2 X.10 + -1 -1 X.11 0 -J -J X.12 0 1+J 1+J X.13 + 0 0 X.14 0 0 0 X.15 0 0 0 X.16 + 1 1 X.17 0 -1-J -1-J X.18 0 J J Explanation of Character Value Symbols -------------------------------------- J = RootOfUnity(3) Z1 = (CyclotomicField(7: Sparse := true)) ! [ RationalField() | 0, 1, 1, 0, 1, 0 ] Z2 = (CyclotomicField(21: Sparse := true)) ! [ RationalField() | 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0 ] ---------------------------------------- Character Table of Group L of Order 1080 -------------------------------------------------------------------------- Class | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Size | 1 45 1 1 120 120 90 72 72 45 45 90 90 72 Order | 1 2 3 3 3 3 4 5 5 6 6 12 12 15 -------------------------------------------------------------------------- p = 2 1 1 4 3 5 6 2 9 8 4 3 10 11 17 p = 3 1 2 1 1 1 1 7 9 8 2 2 7 7 8 p = 5 1 2 4 3 5 6 7 1 1 11 10 13 12 3 -------------------------------------------------------------------------- X.1 + 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 0 3 -1 -3-3*J 3*J 0 0 1 Z1 Z1#2 1+J -J J-1-J Z2 X.3 0 3 -1 3*J -3-3*J 0 0 1 Z1 Z1#2 -J 1+J-1-J J Z2#11 X.4 0 3 -1 -3-3*J 3*J 0 0 1 Z1#2 Z1 1+J -J J-1-J Z2#7 X.5 0 3 -1 3*J -3-3*J 0 0 1 Z1#2 Z1 -J 1+J-1-J J Z2#2 X.6 + 5 1 5 5 -1 2 -1 0 0 1 1 -1 -1 0 X.7 + 5 1 5 5 2 -1 -1 0 0 1 1 -1 -1 0 X.8 0 6 2 6*J -6-6*J 0 0 0 1 1 2*J-2-2*J 0 0 -1-J X.9 0 6 2 -6-6*J 6*J 0 0 0 1 1-2-2*J 2*J 0 0 J X.10 + 8 0 8 8 -1 -1 0 Z1#2 Z1 0 0 0 0 Z1 X.11 + 8 0 8 8 -1 -1 0 Z1 Z1#2 0 0 0 0 Z1#2 X.12 + 9 1 9 9 0 0 1 -1 -1 1 1 1 1 -1 X.13 0 9 1 -9-9*J 9*J 0 0 1 -1 -1 -1-J J J-1-J -J X.14 0 9 1 9*J -9-9*J 0 0 1 -1 -1 J -1-J-1-J J 1+J X.15 + 10 -2 10 10 1 1 0 0 0 -2 -2 0 0 0 X.16 0 15 -1 15*J-15-15*J 0 0 -1 0 0 -J 1+J 1+J -J 0 X.17 0 15 -1-15-15*J 15*J 0 0 -1 0 0 1+J -J -J 1+J 0 --------------------------- Class | 15 16 17 Size | 72 72 72 Order | 15 15 15 --------------------------- p = 2 16 15 14 p = 3 9 8 9 p = 5 3 4 4 --------------------------- X.1 + 1 1 1 X.2 0 Z2#7 Z2#11 Z2#2 X.3 0 Z2#2 Z2 Z2#7 X.4 0 Z2 Z2#2 Z2#11 X.5 0 Z2#11 Z2#7 Z2 X.6 + 0 0 0 X.7 + 0 0 0 X.8 0 -1-J J J X.9 0 J -1-J -1-J X.10 + Z1#2 Z1 Z1#2 X.11 + Z1 Z1#2 Z1 X.12 + -1 -1 -1 X.13 0 -J 1+J 1+J X.14 0 1+J -J -J X.15 + 0 0 0 X.16 0 0 0 0 X.17 0 0 0 0 Explanation of Character Value Symbols -------------------------------------- J = RootOfUnity(3) Z1 = (CyclotomicField(5: Sparse := true)) ! [ RationalField() | 1, 0, 1, 1 ] Z2 = (CyclotomicField(15: Sparse := true)) ! [ RationalField() | 0, 0, 0, 0, 0, -1, 0, -1 ]