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G = C2xC4.A5order 480 = 25·3·5

Direct product of C2 and C4.A5

direct product, non-abelian, not soluble

Aliases: C2xC4.A5, SL2(F5):3C22, C4.6(C2xA5), (C2xC4).2A5, C22.4(C2xA5), C2.2(C22xA5), (C2xSL2(F5)):4C2, SmallGroup(480,955)

Series: ChiefDerived Lower central Upper central

C1C2C22C2xC4 — C2xC4.A5
SL2(F5) — C2xC4.A5
SL2(F5) — C2xC4.A5
C1C2xC4

Subgroups: 900 in 100 conjugacy classes, 13 normal (7 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2xC4, C2xC4, D4, Q8, C23, D5, C10, Dic3, C12, D6, C2xC6, C22xC4, C2xD4, C2xQ8, C4oD4, Dic5, C20, D10, C2xC10, SL2(F3), C4xS3, C2xDic3, C2xC12, C22xS3, C2xC4oD4, C4xD5, C2xDic5, C2xC20, C22xD5, C2xSL2(F3), C4.A4, S3xC2xC4, C2xC4xD5, C2xC4.A4, SL2(F5), C4.A5, C2xSL2(F5), C2xC4.A5
Quotients: C1, C2, C22, A5, C2xA5, C4.A5, C22xA5, C2xC4.A5

Smallest permutation representation of C2xC4.A5
On 48 points
Generators in S48
(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)
(1 31 23 38 20 8 14 40 21 37)(2 36 28 43 25 5 19 45 26 42)(3 41 13 48 10 6 24 30 11 47)(4 46 18 33 15 7 9 35 16 32)(12 29)(17 34)(22 39)(27 44)

G:=sub<Sym(48)| (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), (1,31,23,38,20,8,14,40,21,37)(2,36,28,43,25,5,19,45,26,42)(3,41,13,48,10,6,24,30,11,47)(4,46,18,33,15,7,9,35,16,32)(12,29)(17,34)(22,39)(27,44)>;

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), (1,31,23,38,20,8,14,40,21,37)(2,36,28,43,25,5,19,45,26,42)(3,41,13,48,10,6,24,30,11,47)(4,46,18,33,15,7,9,35,16,32)(12,29)(17,34)(22,39)(27,44) );

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)], [(1,31,23,38,20,8,14,40,21,37),(2,36,28,43,25,5,19,45,26,42),(3,41,13,48,10,6,24,30,11,47),(4,46,18,33,15,7,9,35,16,32),(12,29),(17,34),(22,39),(27,44)]])

36 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F5A5B6A6B6C10A···10F12A12B12C12D20A···20H
order12222234444445566610···101212121220···20
size111130302011113030121220202012···122020202012···12

36 irreducible representations

dim111233344445556
type++++++++++++
imageC1C2C2C4.A5A5C2xA5C2xA5A5C2xA5C2xA5C4.A5A5C2xA5C2xA5C4.A5
kernelC2xC4.A5C4.A5C2xSL2(F5)C2C2xC4C4C22C2xC4C4C22C2C2xC4C4C22C2
# reps121824212141214

Matrix representation of C2xC4.A5 in GL3(F61) generated by

6000
0541
04960
,
6000
02115
04722
G:=sub<GL(3,GF(61))| [60,0,0,0,5,49,0,41,60],[60,0,0,0,21,47,0,15,22] >;

C2xC4.A5 in GAP, Magma, Sage, TeX

C_2\times C_4.A_5
% in TeX

G:=Group("C2xC4.A5");
// GroupNames label

G:=SmallGroup(480,955);
// by ID

G=gap.SmallGroup(480,955);
# by ID

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