direct product, non-abelian, not soluble, A-group
Aliases: C23×A5, SmallGroup(480,1187)
Series: Chief►Derived ►Lower central ►Upper central
| A5 — C23×A5 |
| A5 — C23×A5 |
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
►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