direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×C3⋊C8, C3⋊C24, C6.C12, C32⋊3C8, C12.2C6, C12.8S3, C6.4Dic3, C4.2(C3×S3), (C3×C6).2C4, C2.(C3×Dic3), (C3×C12).3C2, SmallGroup(72,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C3×C3⋊C8 |
Generators and relations for C3×C3⋊C8
G = < a,b,c | a3=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Permutation representations of C3×C3⋊C8
►On 24 points - transitive group 24T69
Generators in S24
(1 15 21)(2 16 22)(3 9 23)(4 10 24)(5 11 17)(6 12 18)(7 13 19)(8 14 20)
(1 15 21)(2 22 16)(3 9 23)(4 24 10)(5 11 17)(6 18 12)(7 13 19)(8 20 14)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
G:=sub<Sym(24)| (1,15,21)(2,16,22)(3,9,23)(4,10,24)(5,11,17)(6,12,18)(7,13,19)(8,14,20), (1,15,21)(2,22,16)(3,9,23)(4,24,10)(5,11,17)(6,18,12)(7,13,19)(8,20,14), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)>;
G:=Group( (1,15,21)(2,16,22)(3,9,23)(4,10,24)(5,11,17)(6,12,18)(7,13,19)(8,14,20), (1,15,21)(2,22,16)(3,9,23)(4,24,10)(5,11,17)(6,18,12)(7,13,19)(8,20,14), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24) );
G=PermutationGroup([[(1,15,21),(2,16,22),(3,9,23),(4,10,24),(5,11,17),(6,12,18),(7,13,19),(8,14,20)], [(1,15,21),(2,22,16),(3,9,23),(4,24,10),(5,11,17),(6,18,12),(7,13,19),(8,20,14)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)]])
G:=TransitiveGroup(24,69);
C3×C3⋊C8 is a maximal subgroup of
C12.29D6 D6.Dic3 C12.31D6 C3⋊D24 D12.S3 C32⋊5SD16 C32⋊3Q16 S3×C24 He3⋊3C8 C9⋊C24 He3⋊4C8 SL2(𝔽3).Dic3
C3×C3⋊C8 is a maximal quotient of
He3⋊3C8 C9⋊C24
36 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 24A | ··· | 24H |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C24 | S3 | Dic3 | C3×S3 | C3⋊C8 | C3×Dic3 | C3×C3⋊C8 |
| kernel | C3×C3⋊C8 | C3×C12 | C3⋊C8 | C3×C6 | C12 | C32 | C6 | C3 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of C3×C3⋊C8 ►in GL2(𝔽13) generated by
| 9 | 0 |
| 0 | 9 |
| 9 | 0 |
| 0 | 3 |
| 0 | 5 |
| 1 | 0 |
G:=sub<GL(2,GF(13))| [9,0,0,9],[9,0,0,3],[0,1,5,0] >;
C3×C3⋊C8 in GAP, Magma, Sage, TeX
C_3\times C_3\rtimes C_8
% in TeX
G:=Group("C3xC3:C8"); // GroupNames label
G:=SmallGroup(72,12);
// by ID
G=gap.SmallGroup(72,12);
# by ID
G:=PCGroup([5,-2,-3,-2,-2,-3,30,42,1204]);
// Polycyclic
G:=Group<a,b,c|a^3=b^3=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export