direct product, non-abelian, not soluble
Aliases: C2×C4.A5, SL2(𝔽5)⋊3C22, C4.6(C2×A5), (C2×C4).2A5, C22.4(C2×A5), C2.2(C22×A5), (C2×SL2(𝔽5))⋊4C2, SmallGroup(480,955)
Series: Chief►Derived ►Lower central ►Upper central
| SL2(𝔽5) — C2×C4.A5 |
| SL2(𝔽5) — C2×C4.A5 |
Subgroups: 900 in 100 conjugacy classes, 13 normal (7 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, D4, Q8, C23, D5, C10, Dic3, C12, D6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, D10, C2×C10, SL2(𝔽3), C4×S3, C2×Dic3, C2×C12, C22×S3, C2×C4○D4, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×SL2(𝔽3), C4.A4, S3×C2×C4, C2×C4×D5, C2×C4.A4, SL2(𝔽5), C4.A5, C2×SL2(𝔽5), C2×C4.A5
Quotients: C1, C2, C22, A5, C2×A5, C4.A5, C22×A5, C2×C4.A5
►On 48 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 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 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 30 | 30 | 20 | 1 | 1 | 1 | 1 | 30 | 30 | 12 | 12 | 20 | 20 | 20 | 12 | ··· | 12 | 20 | 20 | 20 | 20 | 12 | ··· | 12 |
36 irreducible representations
| dim | 1 | 1 | 1 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C4.A5 | A5 | C2×A5 | C2×A5 | A5 | C2×A5 | C2×A5 | C4.A5 | A5 | C2×A5 | C2×A5 | C4.A5 |
| kernel | C2×C4.A5 | C4.A5 | C2×SL2(𝔽5) | C2 | C2×C4 | C4 | C22 | C2×C4 | C4 | C22 | C2 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 8 | 2 | 4 | 2 | 1 | 2 | 1 | 4 | 1 | 2 | 1 | 4 |
Matrix representation of C2×C4.A5 ►in GL3(𝔽61) generated by
| 60 | 0 | 0 |
| 0 | 5 | 41 |
| 0 | 49 | 60 |
| 60 | 0 | 0 |
| 0 | 21 | 15 |
| 0 | 47 | 22 |
G:=sub<GL(3,GF(61))| [60,0,0,0,5,49,0,41,60],[60,0,0,0,21,47,0,15,22] >;
C2×C4.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