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## G = C2×C4×A5order 480 = 25·3·5

### Direct product of C2×C4 and A5

Aliases: C2×C4×A5, C2.1(C22×A5), C22.3(C2×A5), (C2×A5).6C22, (C22×A5).2C2, SmallGroup(480,954)

Series: ChiefDerived Lower central Upper central

 Chief series C1 — C2 — C22 — C2×C4 — C2×C4×A5
 Derived series A5 — C2×C4×A5
 Lower central A5 — C2×C4×A5
 Upper central C1 — C2×C4

Subgroups: 1038 in 120 conjugacy classes, 16 normal (8 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C23, D5, C10, Dic3, C12, A4, D6, C2×C6, C22×C4, C24, Dic5, C20, D10, C2×C10, C4×S3, C2×Dic3, C2×C12, C2×A4, C22×S3, C23×C4, C4×D5, C2×Dic5, C2×C20, C22×D5, C4×A4, S3×C2×C4, C22×A4, A5, C2×C4×D5, C2×C4×A4, C2×A5, C2×A5, C4×A5, C22×A5, C2×C4×A5
Quotients: C1, C2, C4, C22, C2×C4, A5, C2×A5, C4×A5, C22×A5, C2×C4×A5

Smallest permutation representation of C2×C4×A5
On 40 points
Generators in S40
(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)
(1 33 15 27 17 25 19 39 13 21 11 23 5 37 7 35 9 29 3 31)(2 30 4 24 18 26 16 28 10 22 12 40 14 34 8 36 6 38 20 32)

G:=sub<Sym(40)| (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), (1,33,15,27,17,25,19,39,13,21,11,23,5,37,7,35,9,29,3,31)(2,30,4,24,18,26,16,28,10,22,12,40,14,34,8,36,6,38,20,32)>;

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), (1,33,15,27,17,25,19,39,13,21,11,23,5,37,7,35,9,29,3,31)(2,30,4,24,18,26,16,28,10,22,12,40,14,34,8,36,6,38,20,32) );

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)], [(1,33,15,27,17,25,19,39,13,21,11,23,5,37,7,35,9,29,3,31),(2,30,4,24,18,26,16,28,10,22,12,40,14,34,8,36,6,38,20,32)]])

40 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 6A 6B 6C 10A ··· 10F 12A 12B 12C 12D 20A ··· 20H order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 5 5 6 6 6 10 ··· 10 12 12 12 12 20 ··· 20 size 1 1 1 1 15 15 15 15 20 1 1 1 1 15 15 15 15 12 12 20 20 20 12 ··· 12 20 20 20 20 12 ··· 12

40 irreducible representations

 dim 1 1 1 1 3 3 3 3 4 4 4 4 5 5 5 5 type + + + + + + + + + + + + image C1 C2 C2 C4 A5 C2×A5 C2×A5 C4×A5 A5 C2×A5 C2×A5 C4×A5 A5 C2×A5 C2×A5 C4×A5 kernel C2×C4×A5 C4×A5 C22×A5 C2×A5 C2×C4 C4 C22 C2 C2×C4 C4 C22 C2 C2×C4 C4 C22 C2 # reps 1 2 1 4 2 4 2 8 1 2 1 4 1 2 1 4

Matrix representation of C2×C4×A5 in GL5(𝔽61)

 60 0 0 0 0 0 11 0 0 0 0 0 13 9 22 0 0 22 30 9 0 0 0 60 0
,
 1 0 0 0 0 0 11 0 0 0 0 0 39 13 9 0 0 60 0 0 0 0 30 9 39

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,11,0,0,0,0,0,13,22,0,0,0,9,30,60,0,0,22,9,0],[1,0,0,0,0,0,11,0,0,0,0,0,39,60,30,0,0,13,0,9,0,0,9,0,39] >;

C2×C4×A5 in GAP, Magma, Sage, TeX

C_2\times C_4\times A_5
% in TeX

G:=Group("C2xC4xA5");
// GroupNames label

G:=SmallGroup(480,954);
// by ID

G=gap.SmallGroup(480,954);
# by ID

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