direct product, non-abelian, not soluble, A-group
Aliases: C2×C4×A5, C2.1(C22×A5), C22.3(C2×A5), (C2×A5).6C22, (C22×A5).2C2, SmallGroup(480,954)
Series: Chief►Derived ►Lower central ►Upper central
| A5 — C2×C4×A5 |
| A5 — C2×C4×A5 |
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
►On 40 points
(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