direct product, metabelian, soluble, monomial, A-group
Aliases: C8×A4, C22⋊C24, C23.2C12, (C22×C8)⋊C3, C4.4(C2×A4), C2.1(C4×A4), (C4×A4).4C2, (C2×A4).2C4, (C22×C4).2C6, SmallGroup(96,73)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — C8×A4 |
Generators and relations for C8×A4
G = < a,b,c,d | a8=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Permutation representations of C8×A4
►On 24 points - transitive group 24T85
Generators in S24
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)
(1 5)(2 6)(3 7)(4 8)(17 21)(18 22)(19 23)(20 24)
(1 15 19)(2 16 20)(3 9 21)(4 10 22)(5 11 23)(6 12 24)(7 13 17)(8 14 18)
G:=sub<Sym(24)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24), (1,5)(2,6)(3,7)(4,8)(17,21)(18,22)(19,23)(20,24), (1,15,19)(2,16,20)(3,9,21)(4,10,22)(5,11,23)(6,12,24)(7,13,17)(8,14,18)>;
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), (9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24), (1,5)(2,6)(3,7)(4,8)(17,21)(18,22)(19,23)(20,24), (1,15,19)(2,16,20)(3,9,21)(4,10,22)(5,11,23)(6,12,24)(7,13,17)(8,14,18) );
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)], [(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24)], [(1,5),(2,6),(3,7),(4,8),(17,21),(18,22),(19,23),(20,24)], [(1,15,19),(2,16,20),(3,9,21),(4,10,22),(5,11,23),(6,12,24),(7,13,17),(8,14,18)]])
G:=TransitiveGroup(24,85);
C8×A4 is a maximal subgroup of
A4⋊C16 A4⋊Q16 C8⋊S4 C8⋊2S4 A4⋊D8
C8×A4 is a maximal quotient of C16.A4
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 3 | 3 | 4 | 4 | 1 | 1 | 3 | 3 | 4 | 4 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | ||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C24 | A4 | C2×A4 | C4×A4 | C8×A4 |
| kernel | C8×A4 | C4×A4 | C22×C8 | C2×A4 | C22×C4 | A4 | C23 | C22 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | 1 | 2 | 4 |
Matrix representation of C8×A4 ►in GL3(𝔽73) generated by
| 22 | 0 | 0 |
| 0 | 22 | 0 |
| 0 | 0 | 22 |
| 1 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 0 | 72 |
| 72 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
G:=sub<GL(3,GF(73))| [22,0,0,0,22,0,0,0,22],[1,0,0,0,72,0,0,0,72],[72,0,0,0,72,0,0,0,1],[0,0,1,1,0,0,0,1,0] >;
C8×A4 in GAP, Magma, Sage, TeX
C_8\times A_4
% in TeX
G:=Group("C8xA4"); // GroupNames label
G:=SmallGroup(96,73);
// by ID
G=gap.SmallGroup(96,73);
# by ID
G:=PCGroup([6,-2,-3,-2,-2,-2,2,36,50,730,1307]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
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