direct product, metabelian, supersoluble, monomial
Aliases: C3×C6.D4, C62⋊4C4, C62.20C22, (C2×C6)⋊4C12, C6.9(C2×C12), (C2×C6)⋊3Dic3, (C3×C6).32D4, (C2×C6).45D6, C6.11(C3×D4), (C6×Dic3)⋊4C2, (C2×Dic3)⋊2C6, (C22×C6).5S3, (C2×C62).2C2, C23.3(C3×S3), C22.7(S3×C6), (C22×C6).8C6, C2.5(C6×Dic3), C6.33(C3⋊D4), C32⋊7(C22⋊C4), C6.21(C2×Dic3), C22⋊3(C3×Dic3), C3⋊2(C3×C22⋊C4), C2.3(C3×C3⋊D4), (C3×C6).30(C2×C4), (C2×C6).10(C2×C6), SmallGroup(144,84)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C6.D4
G = < a,b,c,d | a3=b6=c4=1, d2=b3, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b3c-1 >
Subgroups: 152 in 84 conjugacy classes, 38 normal (18 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C32, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C22×C6, C3×Dic3, C62, C62, C62, C6.D4, C3×C22⋊C4, C6×Dic3, C2×C62, C3×C6.D4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Dic3, C12, D6, C2×C6, C22⋊C4, C3×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Dic3, S3×C6, C6.D4, C3×C22⋊C4, C6×Dic3, C3×C3⋊D4, C3×C6.D4
►On 24 points - transitive group 24T247
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 21 23)(20 22 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 19 17 7)(2 24 18 12)(3 23 13 11)(4 22 14 10)(5 21 15 9)(6 20 16 8)
(1 22 4 19)(2 21 5 24)(3 20 6 23)(7 17 10 14)(8 16 11 13)(9 15 12 18)
G:=sub<Sym(24)| (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,21,23)(20,22,24), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,19,17,7)(2,24,18,12)(3,23,13,11)(4,22,14,10)(5,21,15,9)(6,20,16,8), (1,22,4,19)(2,21,5,24)(3,20,6,23)(7,17,10,14)(8,16,11,13)(9,15,12,18)>;
G:=Group( (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,21,23)(20,22,24), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,19,17,7)(2,24,18,12)(3,23,13,11)(4,22,14,10)(5,21,15,9)(6,20,16,8), (1,22,4,19)(2,21,5,24)(3,20,6,23)(7,17,10,14)(8,16,11,13)(9,15,12,18) );
G=PermutationGroup([[(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,17,15),(14,18,16),(19,21,23),(20,22,24)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,19,17,7),(2,24,18,12),(3,23,13,11),(4,22,14,10),(5,21,15,9),(6,20,16,8)], [(1,22,4,19),(2,21,5,24),(3,20,6,23),(7,17,10,14),(8,16,11,13),(9,15,12,18)]])
G:=TransitiveGroup(24,247);
C3×C6.D4 is a maximal subgroup of
C62.31D4 C62.32D4 C62.94C23 C62.95C23 C62.97C23 C62.98C23 C62.99C23 C62.100C23 C62.101C23 C62.57D4 C62⋊3Q8 C62.60D4 C62.111C23 C62.113C23 C62.115C23 C62.116C23 C62.117C23 C62⋊5D4 C62⋊8D4 C62⋊4Q8 C3×S3×C22⋊C4 C12×C3⋊D4 C3×D4×Dic3 C62⋊3C12 C62.27D6 C62⋊4Dic3 C62.Dic3 C62⋊5Dic3 C62⋊6Dic3
C3×C6.D4 is a maximal quotient of
C62⋊3C12 C62.27D6
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | ··· | 6AE | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | |||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | S3 | D4 | Dic3 | D6 | C3×S3 | C3⋊D4 | C3×D4 | C3×Dic3 | S3×C6 | C3×C3⋊D4 |
| kernel | C3×C6.D4 | C6×Dic3 | C2×C62 | C6.D4 | C62 | C2×Dic3 | C22×C6 | C2×C6 | C22×C6 | C3×C6 | C2×C6 | C2×C6 | C23 | C6 | C6 | C22 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 1 | 2 | 2 | 1 | 2 | 4 | 4 | 4 | 2 | 8 |
Matrix representation of C3×C6.D4 ►in GL3(𝔽13) generated by
| 3 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 12 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 10 |
| 5 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 5 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 12 | 0 |
G:=sub<GL(3,GF(13))| [3,0,0,0,3,0,0,0,3],[12,0,0,0,4,0,0,0,10],[5,0,0,0,0,1,0,1,0],[5,0,0,0,0,12,0,1,0] >;
C3×C6.D4 in GAP, Magma, Sage, TeX
C_3\times C_6.D_4
% in TeX
G:=Group("C3xC6.D4"); // GroupNames label
G:=SmallGroup(144,84);
// by ID
G=gap.SmallGroup(144,84);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-3,72,313,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^6=c^4=1,d^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^3*c^-1>;
// generators/relations