direct product, metabelian, soluble, monomial, A-group
Aliases: C3×C32⋊2C8, C33⋊3C8, C32⋊3C24, (C3×C6).2C12, C6.4(C32⋊C4), C3⋊Dic3.2C6, (C32×C6).1C4, C2.(C3×C32⋊C4), (C3×C3⋊Dic3).1C2, SmallGroup(216,117)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C32 — C3×C6 — C3⋊Dic3 — C3×C3⋊Dic3 — C3×C32⋊2C8 |
| C32 — C3×C32⋊2C8 |
Generators and relations for C3×C32⋊2C8
G = < a,b,c,d | a3=b3=c3=d8=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >
Permutation representations of C3×C32⋊2C8
►On 24 points - transitive group 24T554
Generators in S24
(1 9 21)(2 10 22)(3 11 23)(4 12 24)(5 13 17)(6 14 18)(7 15 19)(8 16 20)
(2 10 22)(4 24 12)(6 14 18)(8 20 16)
(1 9 21)(2 10 22)(3 23 11)(4 24 12)(5 13 17)(6 14 18)(7 19 15)(8 20 16)
(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)| (1,9,21)(2,10,22)(3,11,23)(4,12,24)(5,13,17)(6,14,18)(7,15,19)(8,16,20), (2,10,22)(4,24,12)(6,14,18)(8,20,16), (1,9,21)(2,10,22)(3,23,11)(4,24,12)(5,13,17)(6,14,18)(7,19,15)(8,20,16), (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( (1,9,21)(2,10,22)(3,11,23)(4,12,24)(5,13,17)(6,14,18)(7,15,19)(8,16,20), (2,10,22)(4,24,12)(6,14,18)(8,20,16), (1,9,21)(2,10,22)(3,23,11)(4,24,12)(5,13,17)(6,14,18)(7,19,15)(8,20,16), (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([[(1,9,21),(2,10,22),(3,11,23),(4,12,24),(5,13,17),(6,14,18),(7,15,19),(8,16,20)], [(2,10,22),(4,24,12),(6,14,18),(8,20,16)], [(1,9,21),(2,10,22),(3,23,11),(4,24,12),(5,13,17),(6,14,18),(7,19,15),(8,20,16)], [(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(24,554);
C3×C32⋊2C8 is a maximal subgroup of
C6.F9 C33⋊5(C2×C8) C33⋊M4(2) C33⋊2M4(2) C32⋊2D24 C33⋊8SD16 C33⋊3Q16
36 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3H | 4A | 4B | 6A | 6B | 6C | ··· | 6H | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | ··· | 24H |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 9 | 9 | 1 | 1 | 4 | ··· | 4 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | ··· | 9 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | ||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C24 | C32⋊C4 | C32⋊2C8 | C3×C32⋊C4 | C3×C32⋊2C8 |
| kernel | C3×C32⋊2C8 | C3×C3⋊Dic3 | C32⋊2C8 | C32×C6 | C3⋊Dic3 | C33 | C3×C6 | C32 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of C3×C32⋊2C8 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 5 | 1 | 5 | 6 |
| 5 | 0 | 3 | 2 |
| 4 | 4 | 3 | 3 |
| 0 | 5 | 6 | 4 |
| 6 | 4 | 6 | 3 |
| 3 | 6 | 1 | 5 |
| 4 | 6 | 3 | 1 |
| 0 | 4 | 3 | 0 |
| 4 | 0 | 4 | 3 |
| 2 | 4 | 1 | 0 |
| 3 | 2 | 5 | 3 |
| 3 | 2 | 6 | 1 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[5,5,4,0,1,0,4,5,5,3,3,6,6,2,3,4],[6,3,4,0,4,6,6,4,6,1,3,3,3,5,1,0],[4,2,3,3,0,4,2,2,4,1,5,6,3,0,3,1] >;
C3×C32⋊2C8 in GAP, Magma, Sage, TeX
C_3\times C_3^2\rtimes_2C_8
% in TeX
G:=Group("C3xC3^2:2C8"); // GroupNames label
G:=SmallGroup(216,117);
// by ID
G=gap.SmallGroup(216,117);
# by ID
G:=PCGroup([6,-2,-3,-2,-2,-3,3,36,50,5044,256,6917,881]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;
// generators/relations
Export