C2xCSU(2,5) - GroupNames

G = C₂xCSU₂(F₅) order 480 = 2⁵·3·5

Direct product of C₂ and CSU₂(F₅)

direct product, non-abelian, not soluble

Aliases: C₂xCSU₂(F₅), C₂².₃S₅, SL₂(F₅).₃C₂², C₂.₈(C₂xS₅), (C₂xSL₂(F₅)).₂C₂, SmallGroup(480,949)

Series: Chief►Derived ►Lower central ►Upper central

Chief series

C₁ — C₂ — C₂² — C₂xSL₂(F₅) — C₂xCSU₂(F₅)

Derived series

SL₂(F₅) — C₂xCSU₂(F₅)

Lower central

SL₂(F₅) — C₂xCSU₂(F₅)

Upper central

C₁ — C₂²

Subgroups: 586 in 66 conjugacy classes, 10 normal (6 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₅, C₆, C₈, C₂xC₄, Q₈, C₁₀, Dic₃, C₁₂, C₂xC₆, C₂xC₈, Q₁₆, C₂xQ₈, Dic₅, C₂xC₁₀, SL₂(F₃), Dic₆, C₂xDic₃, C₂xC₁₂, C₂xQ₁₆, C₅:C₈, C₂xDic₅, CSU₂(F₃), C₂xSL₂(F₃), C₂xDic₆, C₂xC₅:C₈, C₂xCSU₂(F₃), SL₂(F₅), CSU₂(F₅), C₂xSL₂(F₅), C₂xCSU₂(F₅)
Quotients: C₁, C₂, C₂², S₅, CSU₂(F₅), C₂xS₅, C₂xCSU₂(F₅)

Character table of C₂xCSU₂(F₅)

class	1	2A	2B	2C	3	4A	4B	4C	4D	5	6A	6B	6C	8A	8B	8C	8D	10A	10B	10C	12A	12B	12C	12D
size	1	1	1	1	20	20	20	30	30	24	20	20	20	30	30	30	30	24	24	24	20	20	20	20
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	-1	1	1	-1	1	-1	1	1	-1	-1	-1	-1	1	1	1	-1	-1	1	1	-1	-1	linear of order 2
ρ₃	1	1	1	1	1	-1	-1	1	1	1	1	1	1	-1	-1	-1	-1	1	1	1	-1	-1	-1	-1	linear of order 2
ρ₄	1	1	-1	-1	1	-1	1	1	-1	1	1	-1	-1	1	1	-1	-1	1	-1	-1	-1	-1	1	1	linear of order 2
ρ₅	4	4	-4	-4	1	2	-2	0	0	-1	1	-1	-1	0	0	0	0	-1	1	1	-1	-1	1	1	orthogonal lifted from C₂xS₅
ρ₆	4	4	4	4	1	2	2	0	0	-1	1	1	1	0	0	0	0	-1	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₅
ρ₇	4	4	-4	-4	1	-2	2	0	0	-1	1	-1	-1	0	0	0	0	-1	1	1	1	1	-1	-1	orthogonal lifted from C₂xS₅
ρ₈	4	4	4	4	1	-2	-2	0	0	-1	1	1	1	0	0	0	0	-1	-1	-1	1	1	1	1	orthogonal lifted from S₅
ρ₉	4	-4	-4	4	-2	0	0	0	0	-1	2	2	-2	0	0	0	0	1	1	-1	0	0	0	0	symplectic lifted from CSU₂(F₅), Schur index 2
ρ₁₀	4	-4	4	-4	-2	0	0	0	0	-1	2	-2	2	0	0	0	0	1	-1	1	0	0	0	0	symplectic lifted from CSU₂(F₅), Schur index 2
ρ₁₁	4	-4	4	-4	1	0	0	0	0	-1	-1	1	-1	0	0	0	0	1	-1	1	-√3	√3	-√3	√3	symplectic lifted from CSU₂(F₅), Schur index 2
ρ₁₂	4	-4	4	-4	1	0	0	0	0	-1	-1	1	-1	0	0	0	0	1	-1	1	√3	-√3	√3	-√3	symplectic lifted from CSU₂(F₅), Schur index 2
ρ₁₃	4	-4	-4	4	1	0	0	0	0	-1	-1	-1	1	0	0	0	0	1	1	-1	-√3	√3	√3	-√3	symplectic lifted from CSU₂(F₅), Schur index 2
ρ₁₄	4	-4	-4	4	1	0	0	0	0	-1	-1	-1	1	0	0	0	0	1	1	-1	√3	-√3	-√3	√3	symplectic lifted from CSU₂(F₅), Schur index 2
ρ₁₅	5	5	5	5	-1	-1	-1	1	1	0	-1	-1	-1	1	1	1	1	0	0	0	-1	-1	-1	-1	orthogonal lifted from S₅
ρ₁₆	5	5	-5	-5	-1	-1	1	1	-1	0	-1	1	1	-1	-1	1	1	0	0	0	-1	-1	1	1	orthogonal lifted from C₂xS₅
ρ₁₇	5	5	5	5	-1	1	1	1	1	0	-1	-1	-1	-1	-1	-1	-1	0	0	0	1	1	1	1	orthogonal lifted from S₅
ρ₁₈	5	5	-5	-5	-1	1	-1	1	-1	0	-1	1	1	1	1	-1	-1	0	0	0	1	1	-1	-1	orthogonal lifted from C₂xS₅
ρ₁₉	6	6	-6	-6	0	0	0	-2	2	1	0	0	0	0	0	0	0	1	-1	-1	0	0	0	0	orthogonal lifted from C₂xS₅
ρ₂₀	6	6	6	6	0	0	0	-2	-2	1	0	0	0	0	0	0	0	1	1	1	0	0	0	0	orthogonal lifted from S₅
ρ₂₁	6	-6	6	-6	0	0	0	0	0	1	0	0	0	√2	-√2	-√2	√2	-1	1	-1	0	0	0	0	symplectic lifted from CSU₂(F₅), Schur index 2
ρ₂₂	6	-6	-6	6	0	0	0	0	0	1	0	0	0	√2	-√2	√2	-√2	-1	-1	1	0	0	0	0	symplectic lifted from CSU₂(F₅), Schur index 2
ρ₂₃	6	-6	6	-6	0	0	0	0	0	1	0	0	0	-√2	√2	√2	-√2	-1	1	-1	0	0	0	0	symplectic lifted from CSU₂(F₅), Schur index 2
ρ₂₄	6	-6	-6	6	0	0	0	0	0	1	0	0	0	-√2	√2	-√2	√2	-1	-1	1	0	0	0	0	symplectic lifted from CSU₂(F₅), Schur index 2

Smallest permutation representation of C₂xCSU₂(F₅)
►On 96 points

Generators in S₉₆

(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)

(1 21 3 66 7 15 9 72)(2 16 57 20 8 22 51 14)(4 74 34 65 10 80 28 71)(5 83 48 73 11 77 42 79)(6 61 30 82 12 67 36 76)(13 58 85 46 19 52 91 40)(17 44 95 56 23 38 89 50)(18 41 84 43 24 47 78 37)(25 93 27 75 31 87 33 81)(26 88 45 92 32 94 39 86)(29 68 54 70 35 62 60 64)(49 96 53 63 55 90 59 69)

G:=sub<Sym(96)| (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,21,3,66,7,15,9,72)(2,16,57,20,8,22,51,14)(4,74,34,65,10,80,28,71)(5,83,48,73,11,77,42,79)(6,61,30,82,12,67,36,76)(13,58,85,46,19,52,91,40)(17,44,95,56,23,38,89,50)(18,41,84,43,24,47,78,37)(25,93,27,75,31,87,33,81)(26,88,45,92,32,94,39,86)(29,68,54,70,35,62,60,64)(49,96,53,63,55,90,59,69)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,21,3,66,7,15,9,72)(2,16,57,20,8,22,51,14)(4,74,34,65,10,80,28,71)(5,83,48,73,11,77,42,79)(6,61,30,82,12,67,36,76)(13,58,85,46,19,52,91,40)(17,44,95,56,23,38,89,50)(18,41,84,43,24,47,78,37)(25,93,27,75,31,87,33,81)(26,88,45,92,32,94,39,86)(29,68,54,70,35,62,60,64)(49,96,53,63,55,90,59,69) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,21,3,66,7,15,9,72),(2,16,57,20,8,22,51,14),(4,74,34,65,10,80,28,71),(5,83,48,73,11,77,42,79),(6,61,30,82,12,67,36,76),(13,58,85,46,19,52,91,40),(17,44,95,56,23,38,89,50),(18,41,84,43,24,47,78,37),(25,93,27,75,31,87,33,81),(26,88,45,92,32,94,39,86),(29,68,54,70,35,62,60,64),(49,96,53,63,55,90,59,69)]])

Matrix representation of C₂xCSU₂(F₅) ►in GL₈(F₂₄₁)

240	0	0	0	0	0	0	0
240	0	0	1	0	0	0	0
240	1	0	0	0	0	0	0
240	0	1	0	0	0	0	0
0	0	0	0	158	215	166	26
0	0	0	0	80	94	41	37
0	0	0	0	113	140	228	206
0	0	0	0	65	158	174	2

0	0	0	1	0	0	0	0
240	0	0	1	0	0	0	0
0	0	240	1	0	0	0	0
0	240	0	1	0	0	0	0
0	0	0	0	183	223	28	49
0	0	0	0	221	193	10	60
0	0	0	0	41	74	220	9
0	0	0	0	218	196	145	127

G:=sub<GL(8,GF(241))| [240,240,240,240,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,158,80,113,65,0,0,0,0,215,94,140,158,0,0,0,0,166,41,228,174,0,0,0,0,26,37,206,2],[0,240,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0,183,221,41,218,0,0,0,0,223,193,74,196,0,0,0,0,28,10,220,145,0,0,0,0,49,60,9,127] >;

C₂xCSU₂(F₅) in GAP, Magma, Sage, TeX

C_2\times {\rm CSU}_2({\mathbb F}_5)

% in TeX

G:=Group("C2xCSU(2,5)");

// GroupNames label

G:=SmallGroup(480,949);

// by ID

G=gap.SmallGroup(480,949);

# by ID

Export

Character table of C₂xCSU₂(F₅) in TeX

240	0	0	0	0	0	0	0
240	0	0	1	0	0	0	0
240	1	0	0	0	0	0	0
240	0	1	0	0	0	0	0
0	0	0	0	158	215	166	26
0	0	0	0	80	94	41	37
0	0	0	0	113	140	228	206
0	0	0	0	65	158	174	2

0	0	0	1	0	0	0	0
240	0	0	1	0	0	0	0
0	0	240	1	0	0	0	0
0	240	0	1	0	0	0	0
0	0	0	0	183	223	28	49
0	0	0	0	221	193	10	60
0	0	0	0	41	74	220	9
0	0	0	0	218	196	145	127

240	0	0	0	0	0	0	0
240	0	0	1	0	0	0	0
240	1	0	0	0	0	0	0
240	0	1	0	0	0	0	0
0	0	0	0	158	215	166	26
0	0	0	0	80	94	41	37
0	0	0	0	113	140	228	206
0	0	0	0	65	158	174	2

0	0	0	1	0	0	0	0
240	0	0	1	0	0	0	0
0	0	240	1	0	0	0	0
0	240	0	1	0	0	0	0
0	0	0	0	183	223	28	49
0	0	0	0	221	193	10	60
0	0	0	0	41	74	220	9
0	0	0	0	218	196	145	127

G = C2xCSU2(F5) order 480 = 25·3·5

Direct product of C2 and CSU2(F5)

G = C₂xCSU₂(F₅) order 480 = 2⁵·3·5

Direct product of C₂ and CSU₂(F₅)

240	0	0	0	0	0	0	0
240	0	0	1	0	0	0	0
240	1	0	0	0	0	0	0
240	0	1	0	0	0	0	0
0	0	0	0	158	215	166	26
0	0	0	0	80	94	41	37
0	0	0	0	113	140	228	206
0	0	0	0	65	158	174	2

0	0	0	1	0	0	0	0
240	0	0	1	0	0	0	0
0	0	240	1	0	0	0	0
0	240	0	1	0	0	0	0
0	0	0	0	183	223	28	49
0	0	0	0	221	193	10	60
0	0	0	0	41	74	220	9
0	0	0	0	218	196	145	127