direct product, metabelian, soluble, monomial, A-group
Aliases: C6×C32⋊C4, C3⋊S3⋊3C12, (C3×C6)⋊2C12, C33⋊2(C2×C4), (C32×C6)⋊1C4, C32⋊3(C2×C12), (C3×C3⋊S3)⋊2C4, (C6×C3⋊S3).4C2, (C2×C3⋊S3).3C6, C3⋊S3.3(C2×C6), (C3×C3⋊S3).7C22, SmallGroup(216,168)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C6×C32⋊C4 |
Generators and relations for C6×C32⋊C4
G = < a,b,c,d | a6=b3=c3=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >
Subgroups: 248 in 60 conjugacy classes, 20 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, C32, C32, C12, D6, C2×C6, C3×S3, C3⋊S3, C3×C6, C3×C6, C2×C12, C33, C32⋊C4, S3×C6, C2×C3⋊S3, C3×C3⋊S3, C32×C6, C2×C32⋊C4, C3×C32⋊C4, C6×C3⋊S3, C6×C32⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C12, C2×C6, C2×C12, C32⋊C4, C2×C32⋊C4, C3×C32⋊C4, C6×C32⋊C4
►On 24 points - transitive group 24T555
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(7 9 11)(8 10 12)(19 23 21)(20 24 22)
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 23 21)(20 24 22)
(1 8 16 24)(2 9 17 19)(3 10 18 20)(4 11 13 21)(5 12 14 22)(6 7 15 23)
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), (7,9,11)(8,10,12)(19,23,21)(20,24,22), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22), (1,8,16,24)(2,9,17,19)(3,10,18,20)(4,11,13,21)(5,12,14,22)(6,7,15,23)>;
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), (7,9,11)(8,10,12)(19,23,21)(20,24,22), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22), (1,8,16,24)(2,9,17,19)(3,10,18,20)(4,11,13,21)(5,12,14,22)(6,7,15,23) );
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)], [(7,9,11),(8,10,12),(19,23,21),(20,24,22)], [(1,3,5),(2,4,6),(7,9,11),(8,10,12),(13,17,15),(14,18,16),(19,23,21),(20,24,22)], [(1,8,16,24),(2,9,17,19),(3,10,18,20),(4,11,13,21),(5,12,14,22),(6,7,15,23)]])
G:=TransitiveGroup(24,555);
C6×C32⋊C4 is a maximal subgroup of
D6⋊(C32⋊C4) C33⋊(C4⋊C4) (C3×C6).8D12 (C3×C6).9D12 C6.PSU3(𝔽2) C6.2PSU3(𝔽2)
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3H | 4A | 4B | 4C | 4D | 6A | 6B | 6C | ··· | 6H | 6I | 6J | 6K | 6L | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 9 | 9 | 1 | 1 | 4 | ··· | 4 | 9 | 9 | 9 | 9 | 1 | 1 | 4 | ··· | 4 | 9 | 9 | 9 | 9 | 9 | ··· | 9 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | C32⋊C4 | C2×C32⋊C4 | C3×C32⋊C4 | C6×C32⋊C4 |
| kernel | C6×C32⋊C4 | C3×C32⋊C4 | C6×C3⋊S3 | C2×C32⋊C4 | C3×C3⋊S3 | C32×C6 | C32⋊C4 | C2×C3⋊S3 | C3⋊S3 | C3×C6 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 2 | 2 | 4 | 4 |
Matrix representation of C6×C32⋊C4 ►in GL4(𝔽7) generated by
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 4 | 6 |
| 2 | 3 | 1 | 0 |
| 6 | 1 | 2 | 2 |
| 4 | 4 | 5 | 0 |
| 4 | 2 | 5 | 0 |
| 2 | 3 | 5 | 2 |
| 2 | 2 | 6 | 2 |
| 2 | 4 | 3 | 2 |
| 5 | 0 | 5 | 3 |
| 6 | 4 | 6 | 4 |
| 5 | 4 | 6 | 6 |
| 1 | 6 | 3 | 6 |
G:=sub<GL(4,GF(7))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[0,2,6,4,0,3,1,4,4,1,2,5,6,0,2,0],[4,2,2,2,2,3,2,4,5,5,6,3,0,2,2,2],[5,6,5,1,0,4,4,6,5,6,6,3,3,4,6,6] >;
C6×C32⋊C4 in GAP, Magma, Sage, TeX
C_6\times C_3^2\rtimes C_4
% in TeX
G:=Group("C6xC3^2:C4"); // GroupNames label
G:=SmallGroup(216,168);
// by ID
G=gap.SmallGroup(216,168);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,3,72,5044,142,6917,455]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^3=c^3=d^4=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