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## G = C8×A5order 480 = 25·3·5

### Direct product of C8 and A5

Aliases: C8×A5, C2.1(C4×A5), C4.4(C2×A5), (C2×A5).2C4, (C4×A5).4C2, SmallGroup(480,220)

Series: ChiefDerived Lower central Upper central

 Chief series C1 — C2 — C4 — C8 — C8×A5
 Derived series A5 — C8×A5
 Lower central A5 — C8×A5
 Upper central C1 — C8

15C2
15C2
10C3
6C5
5C22
15C22
15C4
15C22
10C6
10S3
10S3
6D5
6D5
6C10
5C23
15C2×C4
15C2×C4
15C8
5A4
10C12
10D6
10Dic3
6Dic5
6C20
6D10
15C2×C8
15C2×C8
10C24
10C4×S3
10C3⋊C8
6C40
10S3×C8

Smallest permutation representation of C8×A5
On 40 points
Generators in S40
(1 33 6 38 11 23 16 28)(2 26 7 31 12 36 17 21)(3 35 8 40 13 25 18 30)(4 32 9 37 14 22 19 27)(5 29 10 34 15 39 20 24)
(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)

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

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

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

40 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 5A 5B 6 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 12A 12B 20A 20B 20C 20D 24A 24B 24C 24D 40A ··· 40H order 1 2 2 2 3 4 4 4 4 5 5 6 8 8 8 8 8 8 8 8 10 10 12 12 20 20 20 20 24 24 24 24 40 ··· 40 size 1 1 15 15 20 1 1 15 15 12 12 20 1 1 1 1 15 15 15 15 12 12 20 20 12 12 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 C4 C8 A5 C2×A5 C4×A5 C8×A5 A5 C2×A5 C4×A5 C8×A5 A5 C2×A5 C4×A5 C8×A5 kernel C8×A5 C4×A5 C2×A5 A5 C8 C4 C2 C1 C8 C4 C2 C1 C8 C4 C2 C1 # reps 1 1 2 4 2 2 4 8 1 1 2 4 1 1 2 4

Matrix representation of C8×A5 in GL4(𝔽241) generated by

 211 0 0 0 0 64 0 0 0 188 34 10 0 182 53 207
,
 177 0 0 0 0 182 210 97 0 233 0 0 0 97 231 7
G:=sub<GL(4,GF(241))| [211,0,0,0,0,64,188,182,0,0,34,53,0,0,10,207],[177,0,0,0,0,182,233,97,0,210,0,231,0,97,0,7] >;

C8×A5 in GAP, Magma, Sage, TeX

C_8\times A_5
% in TeX

G:=Group("C8xA5");
// GroupNames label

G:=SmallGroup(480,220);
// by ID

G=gap.SmallGroup(480,220);
# by ID

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