direct product, metabelian, supersoluble, monomial
Aliases: C3×C12.D4, (C6×D4).2C6, C12.8(C3×D4), (C6×D4).23S3, (C3×C12).46D4, C4.Dic3⋊3C6, (C2×C62).2C4, (C2×C12).222D6, (C22×C6).5C12, C62.97(C2×C4), (C6×C12).48C22, C32⋊7(C4.D4), (C22×C6).5Dic3, C22.2(C6×Dic3), C23.2(C3×Dic3), C12.100(C3⋊D4), C6.32(C6.D4), (D4×C3×C6).2C2, (C2×C4).3(S3×C6), (C2×D4).2(C3×S3), C3⋊2(C3×C4.D4), C4.13(C3×C3⋊D4), (C2×C6).38(C2×C12), (C2×C12).18(C2×C6), C6.14(C3×C22⋊C4), (C3×C4.Dic3)⋊19C2, (C2×C6).23(C2×Dic3), C2.4(C3×C6.D4), (C3×C6).65(C22⋊C4), SmallGroup(288,267)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C12.D4
G = < a,b,c,d | a3=b12=1, c4=b6, d2=b9, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b5, dcd-1=b3c3 >
Subgroups: 266 in 119 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C6, C6, C8, C2×C4, D4, C23, C32, C12, C12, C2×C6, C2×C6, M4(2), C2×D4, C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C4.D4, C3×C12, C62, C62, C4.Dic3, C3×M4(2), C6×D4, C6×D4, C3×C3⋊C8, C6×C12, D4×C32, C2×C62, C12.D4, C3×C4.D4, C3×C4.Dic3, D4×C3×C6, C3×C12.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, C4.D4, C3×Dic3, S3×C6, C6.D4, C3×C22⋊C4, C6×Dic3, C3×C3⋊D4, C12.D4, C3×C4.D4, C3×C6.D4, C3×C12.D4
►On 24 points - transitive group 24T590
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 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 17 10 20 7 23 4 14)(2 16 11 19 8 22 5 13)(3 15 12 18 9 21 6 24)
(1 20 10 17 7 14 4 23)(2 13 11 22 8 19 5 16)(3 18 12 15 9 24 6 21)
G:=sub<Sym(24)| (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,17,10,20,7,23,4,14)(2,16,11,19,8,22,5,13)(3,15,12,18,9,21,6,24), (1,20,10,17,7,14,4,23)(2,13,11,22,8,19,5,16)(3,18,12,15,9,24,6,21)>;
G:=Group( (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,17,10,20,7,23,4,14)(2,16,11,19,8,22,5,13)(3,15,12,18,9,21,6,24), (1,20,10,17,7,14,4,23)(2,13,11,22,8,19,5,16)(3,18,12,15,9,24,6,21) );
G=PermutationGroup([[(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,17,21),(14,18,22),(15,19,23),(16,20,24)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,17,10,20,7,23,4,14),(2,16,11,19,8,22,5,13),(3,15,12,18,9,21,6,24)], [(1,20,10,17,7,14,4,23),(2,13,11,22,8,19,5,16),(3,18,12,15,9,24,6,21)]])
G:=TransitiveGroup(24,590);
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | ··· | 6M | 6N | ··· | 6AC | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | ··· | 12 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | ||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | S3 | D4 | D6 | Dic3 | C3×S3 | C3⋊D4 | C3×D4 | S3×C6 | C3×Dic3 | C3×C3⋊D4 | C4.D4 | C12.D4 | C3×C4.D4 | C3×C12.D4 |
| kernel | C3×C12.D4 | C3×C4.Dic3 | D4×C3×C6 | C12.D4 | C2×C62 | C4.Dic3 | C6×D4 | C22×C6 | C6×D4 | C3×C12 | C2×C12 | C22×C6 | C2×D4 | C12 | C12 | C2×C4 | C23 | C4 | C32 | C3 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 2 | 4 | 8 | 1 | 2 | 2 | 4 |
Matrix representation of C3×C12.D4 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 3 | 1 | 0 | 2 |
| 0 | 5 | 4 | 3 |
| 6 | 6 | 6 | 2 |
| 1 | 6 | 3 | 0 |
| 6 | 4 | 3 | 5 |
| 6 | 6 | 3 | 0 |
| 2 | 5 | 6 | 6 |
| 2 | 2 | 5 | 3 |
| 4 | 0 | 3 | 0 |
| 4 | 5 | 0 | 5 |
| 5 | 2 | 1 | 1 |
| 5 | 5 | 1 | 4 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[3,0,6,1,1,5,6,6,0,4,6,3,2,3,2,0],[6,6,2,2,4,6,5,2,3,3,6,5,5,0,6,3],[4,4,5,5,0,5,2,5,3,0,1,1,0,5,1,4] >;
C3×C12.D4 in GAP, Magma, Sage, TeX
C_3\times C_{12}.D_4 % in TeX
G:=Group("C3xC12.D4"); // GroupNames label
G:=SmallGroup(288,267);
// by ID
G=gap.SmallGroup(288,267);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,365,850,136,2524,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^12=1,c^4=b^6,d^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^5,d*c*d^-1=b^3*c^3>;
// generators/relations