direct product, metabelian, soluble, monomial
Aliases: C3×C62⋊C4, C62⋊8C12, (C3×C62)⋊1C4, C33⋊6(C22⋊C4), (C6×C3⋊S3)⋊5C4, (C2×C3⋊S3)⋊5C12, (C6×C32⋊C4)⋊4C2, (C2×C32⋊C4)⋊2C6, C3⋊S3.5(C3×D4), C2.7(C6×C32⋊C4), (C2×C6)⋊1(C32⋊C4), (C3×C3⋊S3).16D4, C6.25(C2×C32⋊C4), C22⋊2(C3×C32⋊C4), (C3×C6).16(C2×C12), (C22×C3⋊S3).6C6, C32⋊4(C3×C22⋊C4), (C6×C3⋊S3).40C22, (C32×C6).14(C2×C4), (C2×C6×C3⋊S3).5C2, (C2×C3⋊S3).15(C2×C6), SmallGroup(432,634)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C62⋊C4
G = < a,b,c,d | a3=b6=c6=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c, dcd-1=b4c >
Subgroups: 764 in 152 conjugacy classes, 28 normal (20 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C23, C32, C32, C12, D6, C2×C6, C2×C6, C22⋊C4, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C2×C12, C22×S3, C22×C6, C33, C32⋊C4, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C3×C22⋊C4, C3×C3⋊S3, C3×C3⋊S3, C32×C6, C32×C6, C2×C32⋊C4, S3×C2×C6, C22×C3⋊S3, C3×C32⋊C4, C6×C3⋊S3, C6×C3⋊S3, C3×C62, C62⋊C4, C6×C32⋊C4, C2×C6×C3⋊S3, C3×C62⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, C2×C12, C3×D4, C32⋊C4, C3×C22⋊C4, C2×C32⋊C4, C3×C32⋊C4, C62⋊C4, C6×C32⋊C4, C3×C62⋊C4
►On 24 points - transitive group 24T1288
(1 6 7)(2 5 8)(3 10 11)(4 9 12)(13 15 14)(16 17 18)(19 21 20)(22 23 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)
(1 8 6 2 7 5)(3 9 11 4 10 12)(13 21 15 20 14 19)(16 23 18 22 17 24)
(1 21 12 16)(2 14 11 22)(3 23 5 13)(4 17 6 20)(7 19 9 18)(8 15 10 24)
G:=sub<Sym(24)| (1,6,7)(2,5,8)(3,10,11)(4,9,12)(13,15,14)(16,17,18)(19,21,20)(22,23,24), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,8,6,2,7,5)(3,9,11,4,10,12)(13,21,15,20,14,19)(16,23,18,22,17,24), (1,21,12,16)(2,14,11,22)(3,23,5,13)(4,17,6,20)(7,19,9,18)(8,15,10,24)>;
G:=Group( (1,6,7)(2,5,8)(3,10,11)(4,9,12)(13,15,14)(16,17,18)(19,21,20)(22,23,24), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,8,6,2,7,5)(3,9,11,4,10,12)(13,21,15,20,14,19)(16,23,18,22,17,24), (1,21,12,16)(2,14,11,22)(3,23,5,13)(4,17,6,20)(7,19,9,18)(8,15,10,24) );
G=PermutationGroup([[(1,6,7),(2,5,8),(3,10,11),(4,9,12),(13,15,14),(16,17,18),(19,21,20),(22,23,24)], [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,8,6,2,7,5),(3,9,11,4,10,12),(13,21,15,20,14,19),(16,23,18,22,17,24)], [(1,21,12,16),(2,14,11,22),(3,23,5,13),(4,17,6,20),(7,19,9,18),(8,15,10,24)]])
G:=TransitiveGroup(24,1288);
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | ··· | 3H | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | ··· | 6V | 6W | 6X | 6Y | 6Z | 6AA | 6AB | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 9 | 9 | 18 | 1 | 1 | 4 | ··· | 4 | 18 | 18 | 18 | 18 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 9 | 9 | 9 | 9 | 18 | 18 | 18 | ··· | 18 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | D4 | C3×D4 | C32⋊C4 | C2×C32⋊C4 | C3×C32⋊C4 | C62⋊C4 | C6×C32⋊C4 | C3×C62⋊C4 |
| kernel | C3×C62⋊C4 | C6×C32⋊C4 | C2×C6×C3⋊S3 | C62⋊C4 | C6×C3⋊S3 | C3×C62 | C2×C32⋊C4 | C22×C3⋊S3 | C2×C3⋊S3 | C62 | C3×C3⋊S3 | C3⋊S3 | C2×C6 | C6 | C22 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C3×C62⋊C4 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 5 | 4 | 6 | 1 |
| 0 | 4 | 5 | 6 |
| 3 | 0 | 3 | 4 |
| 6 | 6 | 6 | 2 |
| 1 | 3 | 0 | 1 |
| 5 | 6 | 4 | 6 |
| 1 | 1 | 3 | 4 |
| 1 | 5 | 0 | 3 |
| 5 | 1 | 4 | 1 |
| 6 | 1 | 2 | 1 |
| 5 | 0 | 3 | 0 |
| 1 | 6 | 3 | 5 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[5,0,3,6,4,4,0,6,6,5,3,6,1,6,4,2],[1,5,1,1,3,6,1,5,0,4,3,0,1,6,4,3],[5,6,5,1,1,1,0,6,4,2,3,3,1,1,0,5] >;
C3×C62⋊C4 in GAP, Magma, Sage, TeX
C_3\times C_6^2\rtimes C_4
% in TeX
G:=Group("C3xC6^2:C4"); // GroupNames label
G:=SmallGroup(432,634);
// by ID
G=gap.SmallGroup(432,634);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,84,365,14117,362,18822,1203]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^6=c^6=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c,d*c*d^-1=b^4*c>;
// generators/relations