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 [×7], C2 [×8], C3, C22 [×7], C22 [×50], C5, S3 [×8], C6 [×7], C23, C23 [×56], D5 [×8], C10 [×7], A4, D6 [×28], C2×C6 [×7], C24 [×15], D10 [×28], C2×C10 [×7], C2×A4 [×7], C22×S3 [×14], C22×C6, C25, C22×D5 [×14], C22×C10, C22×A4 [×7], S3×C23, A5, C23×D5, C23×A4, C2×A5 [×7], C22×A5 [×7], C23×A5
Quotients:
C1, C2 [×7], C22 [×7], C23, A5, C2×A5 [×7], C22×A5 [×7], 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 11 7 17 5 13)(2 12 10 18 6 16)(3 19)(4 20)(8 14)(9 15)(21 35 27 31 25 37)(22 36 30 32 26 40)(23 33)(24 34)(28 38)(29 39)
(1 23 7 29 5 21 3 27 9 25)(2 24 10 26 8 22 4 30 6 28)(11 31 19 37 15 35 17 33 13 39)(12 38 18 34 16 36 14 32 20 40)
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), (1,11,7,17,5,13)(2,12,10,18,6,16)(3,19)(4,20)(8,14)(9,15)(21,35,27,31,25,37)(22,36,30,32,26,40)(23,33)(24,34)(28,38)(29,39), (1,23,7,29,5,21,3,27,9,25)(2,24,10,26,8,22,4,30,6,28)(11,31,19,37,15,35,17,33,13,39)(12,38,18,34,16,36,14,32,20,40)>;
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), (1,11,7,17,5,13)(2,12,10,18,6,16)(3,19)(4,20)(8,14)(9,15)(21,35,27,31,25,37)(22,36,30,32,26,40)(23,33)(24,34)(28,38)(29,39), (1,23,7,29,5,21,3,27,9,25)(2,24,10,26,8,22,4,30,6,28)(11,31,19,37,15,35,17,33,13,39)(12,38,18,34,16,36,14,32,20,40) );
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)], [(1,11,7,17,5,13),(2,12,10,18,6,16),(3,19),(4,20),(8,14),(9,15),(21,35,27,31,25,37),(22,36,30,32,26,40),(23,33),(24,34),(28,38),(29,39)], [(1,23,7,29,5,21,3,27,9,25),(2,24,10,26,8,22,4,30,6,28),(11,31,19,37,15,35,17,33,13,39),(12,38,18,34,16,36,14,32,20,40)])
Matrix representation ►G ⊆ 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] >;
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 |
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
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