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## G = C7×A5order 420 = 22·3·5·7

### Direct product of C7 and A5

Aliases: C7×A5, SmallGroup(420,13)

Series: ChiefDerived Lower central Upper central

 Chief series C1 — C7 — C7×A5
 Derived series A5 — C7×A5
 Lower central A5 — C7×A5
 Upper central C1 — C7

15C2
10C3
6C5
5C22
10S3
6D5
15C14
10C21
6C35
5A4
10S3×C7

Smallest permutation representation of C7×A5
On 35 points
Generators in S35
(1 9 17 11 19 27 21 29 2 31 4 12 6 14 22 16 24 32 26 34 7)(3 18 33 13 28 8 23)(5 20 35 15 30 10 25)
(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)

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

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

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

35 conjugacy classes

 class 1 2 3 5A 5B 7A ··· 7F 14A ··· 14F 21A ··· 21F 35A ··· 35L order 1 2 3 5 5 7 ··· 7 14 ··· 14 21 ··· 21 35 ··· 35 size 1 15 20 12 12 1 ··· 1 15 ··· 15 20 ··· 20 12 ··· 12

35 irreducible representations

 dim 1 1 3 3 4 4 5 5 type + + + + image C1 C7 A5 C7×A5 A5 C7×A5 A5 C7×A5 kernel C7×A5 A5 C7 C1 C7 C1 C7 C1 # reps 1 6 2 12 1 6 1 6

Matrix representation of C7×A5 in GL3(𝔽211) generated by

 109 63 57 102 0 183 49 0 102
,
 80 29 87 37 2 148 132 128 35
G:=sub<GL(3,GF(211))| [109,102,49,63,0,0,57,183,102],[80,37,132,29,2,128,87,148,35] >;

C7×A5 in GAP, Magma, Sage, TeX

C_7\times A_5
% in TeX

G:=Group("C7xA5");
// GroupNames label

G:=SmallGroup(420,13);
// by ID

G=gap.SmallGroup(420,13);
# by ID

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