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## G = C2×C2.S5order 480 = 25·3·5

### Direct product of C2 and C2.S5

Aliases: C2×C2.S5, C22.4S5, SL2(𝔽5)⋊2C22, C2.9(C2×S5), (C2×SL2(𝔽5))⋊2C2, SmallGroup(480,950)

Series: ChiefDerived Lower central Upper central

 Chief series C1 — C2 — C22 — C2×SL2(𝔽5) — C2×C2.S5
 Derived series SL2(𝔽5) — C2×C2.S5
 Lower central SL2(𝔽5) — C2×C2.S5
 Upper central C1 — C22

Subgroups: 826 in 78 conjugacy classes, 10 normal (6 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C8, C2×C4, D4, Q8, C23, C10, Dic3, D6, C2×C6, C2×C8, SD16, C2×D4, C2×Q8, Dic5, C2×C10, SL2(𝔽3), C2×Dic3, C3⋊D4, C22×S3, C22×C6, C2×SD16, C5⋊C8, C2×Dic5, GL2(𝔽3), C2×SL2(𝔽3), C2×C3⋊D4, C2×C5⋊C8, C2×GL2(𝔽3), SL2(𝔽5), C2.S5, C2×SL2(𝔽5), C2×C2.S5
Quotients: C1, C2, C22, S5, C2.S5, C2×S5, C2×C2.S5

Character table of C2×C2.S5

 class 1 2A 2B 2C 2D 2E 3 4A 4B 5 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 10A 10B 10C size 1 1 1 1 20 20 20 30 30 24 20 20 20 20 20 20 20 30 30 30 30 24 24 24 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 -2 -2 1 0 0 -1 1 1 1 1 1 1 1 0 0 0 0 -1 -1 -1 orthogonal lifted from S5 ρ6 4 -4 4 -4 -2 2 1 0 0 -1 -1 1 1 -1 -1 -1 1 0 0 0 0 1 1 -1 orthogonal lifted from C2×S5 ρ7 4 4 4 4 2 2 1 0 0 -1 1 -1 -1 -1 -1 1 1 0 0 0 0 -1 -1 -1 orthogonal lifted from S5 ρ8 4 -4 4 -4 2 -2 1 0 0 -1 -1 -1 -1 1 1 -1 1 0 0 0 0 1 1 -1 orthogonal lifted from C2×S5 ρ9 4 -4 -4 4 0 0 -2 0 0 -1 2 0 0 0 0 -2 2 0 0 0 0 -1 1 1 symplectic lifted from C2.S5, Schur index 2 ρ10 4 4 -4 -4 0 0 -2 0 0 -1 -2 0 0 0 0 2 2 0 0 0 0 1 -1 1 symplectic lifted from C2.S5, Schur index 2 ρ11 4 -4 -4 4 0 0 1 0 0 -1 -1 √-3 -√-3 -√-3 √-3 1 -1 0 0 0 0 -1 1 1 complex lifted from C2.S5 ρ12 4 -4 -4 4 0 0 1 0 0 -1 -1 -√-3 √-3 √-3 -√-3 1 -1 0 0 0 0 -1 1 1 complex lifted from C2.S5 ρ13 4 4 -4 -4 0 0 1 0 0 -1 1 √-3 -√-3 √-3 -√-3 -1 -1 0 0 0 0 1 -1 1 complex lifted from C2.S5 ρ14 4 4 -4 -4 0 0 1 0 0 -1 1 -√-3 √-3 -√-3 √-3 -1 -1 0 0 0 0 1 -1 1 complex lifted from C2.S5 ρ15 5 5 5 5 -1 -1 -1 1 1 0 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 0 0 0 orthogonal lifted from S5 ρ16 5 -5 5 -5 -1 1 -1 1 -1 0 1 -1 -1 1 1 1 -1 -1 -1 1 1 0 0 0 orthogonal lifted from C2×S5 ρ17 5 -5 5 -5 1 -1 -1 1 -1 0 1 1 1 -1 -1 1 -1 1 1 -1 -1 0 0 0 orthogonal lifted from C2×S5 ρ18 5 5 5 5 1 1 -1 1 1 0 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 0 0 0 orthogonal lifted from S5 ρ19 6 6 6 6 0 0 0 -2 -2 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 orthogonal lifted from S5 ρ20 6 -6 6 -6 0 0 0 -2 2 1 0 0 0 0 0 0 0 0 0 0 0 -1 -1 1 orthogonal lifted from C2×S5 ρ21 6 -6 -6 6 0 0 0 0 0 1 0 0 0 0 0 0 0 √-2 -√-2 √-2 -√-2 1 -1 -1 complex lifted from C2.S5 ρ22 6 6 -6 -6 0 0 0 0 0 1 0 0 0 0 0 0 0 -√-2 √-2 √-2 -√-2 -1 1 -1 complex lifted from C2.S5 ρ23 6 -6 -6 6 0 0 0 0 0 1 0 0 0 0 0 0 0 -√-2 √-2 -√-2 √-2 1 -1 -1 complex lifted from C2.S5 ρ24 6 6 -6 -6 0 0 0 0 0 1 0 0 0 0 0 0 0 √-2 -√-2 -√-2 √-2 -1 1 -1 complex lifted from C2.S5

Smallest permutation representation of C2×C2.S5
On 80 points
Generators in S80
(1 47)(2 72 74 48 61 34)(3 18 21 41 50 53)(4 77 71 42 37 60)(5 43)(6 68 78 44 57 38)(7 22 17 45 54 49)(8 73 67 46 33 64)(9 29)(10 66 24 30 63 56)(11 76 75 31 36 35)(12 23 69 32 55 58)(13 25)(14 70 20 26 59 52)(15 80 79 27 40 39)(16 19 65 28 51 62)
(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)

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

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

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

Matrix representation of C2×C2.S5 in GL5(𝔽241)

 240 0 0 0 0 0 1 178 211 181 0 0 183 3 5 0 0 3 32 33 0 0 90 238 26
,
 1 0 0 0 0 0 0 240 1 0 0 1 240 0 0 0 0 240 0 240 0 0 2 0 1

G:=sub<GL(5,GF(241))| [240,0,0,0,0,0,1,0,0,0,0,178,183,3,90,0,211,3,32,238,0,181,5,33,26],[1,0,0,0,0,0,0,1,0,0,0,240,240,240,2,0,1,0,0,0,0,0,0,240,1] >;

C2×C2.S5 in GAP, Magma, Sage, TeX

C_2\times C_2.S_5
% in TeX

G:=Group("C2xC2.S5");
// GroupNames label

G:=SmallGroup(480,950);
// by ID

G=gap.SmallGroup(480,950);
# by ID

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