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## G = C23×A5order 480 = 25·3·5

### Direct product of C23 and A5

Aliases: C23×A5, SmallGroup(480,1187)

Series: ChiefDerived Lower central Upper central

 Chief series C1 — C2 — C22 — C23 — C23×A5
 Derived series A5 — C23×A5
 Lower central A5 — C23×A5
 Upper central C1 — C23

Subgroups: 2974 in 312 conjugacy classes, 32 normal (4 characteristic)
C1, C2, C2, C3, C22, C22, C5, S3, C6, C23, C23, D5, C10, A4, D6, C2×C6, C24, D10, C2×C10, C2×A4, C22×S3, C22×C6, C25, C22×D5, C22×C10, C22×A4, S3×C23, A5, C23×D5, C23×A4, C2×A5, C22×A5, C23×A5
Quotients: C1, C2, C22, C23, A5, C2×A5, C22×A5, C23×A5

Smallest permutation representation of C23×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 40 7 36 5 32)(2 31 10 37 6 35)(3 38)(4 39)(8 33)(9 34)(11 27 17 23 15 29)(12 28 20 24 16 22)(13 25)(14 26)(18 30)(19 21)
(1 13 7 19 5 11 3 17 9 15)(2 14 10 16 8 12 4 20 6 18)(21 40 23 38 29 34 27 36 25 32)(22 31 30 37 26 35 28 33 24 39)

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

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

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

40 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3 5A 5B 6A ··· 6G 10A ··· 10N order 1 2 ··· 2 2 ··· 2 3 5 5 6 ··· 6 10 ··· 10 size 1 1 ··· 1 15 ··· 15 20 12 12 20 ··· 20 12 ··· 12

40 irreducible representations

 dim 1 1 3 3 4 4 5 5 type + + + + + + + + image C1 C2 A5 C2×A5 A5 C2×A5 A5 C2×A5 kernel C23×A5 C22×A5 C23 C22 C23 C22 C23 C22 # reps 1 7 2 14 1 7 1 7

Matrix representation of C23×A5 in GL6(𝔽31)

 30 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 22 25 28 0 0 0 15 22 6 0 0 0 30 0 0
,
 30 0 0 0 0 0 0 30 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 30 0 0 0 30 1 30
,
 30 0 0 0 0 0 0 30 0 0 0 0 0 0 30 0 0 0 0 0 0 25 15 22 0 0 0 22 25 28 0 0 0 0 1 0

G:=sub<GL(6,GF(31))| [30,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,22,15,30,0,0,0,25,22,0,0,0,0,28,6,0],[30,0,0,0,0,0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,30,0,0,0,0,0,1,0,0,0,0,30,30],[30,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,25,22,0,0,0,0,15,25,1,0,0,0,22,28,0] >;

C23×A5 in GAP, Magma, Sage, TeX

C_2^3\times A_5
% in TeX

G:=Group("C2^3xA5");
// GroupNames label

G:=SmallGroup(480,1187);
// by ID

G=gap.SmallGroup(480,1187);
# by ID

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