direct product, metabelian, supersoluble, monomial
Aliases: C6×C6.D4, C62.127D4, C62.210C23, (C2×C62)⋊8C4, C6.62(C6×D4), C62⋊23(C2×C4), (C22×C6)⋊6C12, C24.3(C3×S3), (C23×C6).7S3, (C23×C6).12C6, C23.42(S3×C6), C22⋊4(C6×Dic3), C23⋊4(C3×Dic3), (C22×C6)⋊5Dic3, C6.28(C22×C12), (C22×C6).130D6, (C22×C62).2C2, (C6×Dic3)⋊25C22, (C22×Dic3)⋊10C6, C6.48(C22×Dic3), (C2×C62).102C22, C6⋊2(C3×C22⋊C4), C3⋊3(C6×C22⋊C4), C2.4(C6×C3⋊D4), (C2×C6)⋊11(C2×C12), C2.9(Dic3×C2×C6), (Dic3×C2×C6)⋊10C2, (C2×C6).53(C3×D4), C22.27(S3×C2×C6), (C3×C6)⋊7(C22⋊C4), (C2×Dic3)⋊7(C2×C6), (C2×C6)⋊10(C2×Dic3), (C3×C6).270(C2×D4), C6.163(C2×C3⋊D4), C32⋊13(C2×C22⋊C4), (C22×C6).66(C2×C6), (C2×C6).65(C22×C6), C22.25(C3×C3⋊D4), (C2×C6).118(C3⋊D4), (C3×C6).119(C22×C4), (C2×C6).343(C22×S3), SmallGroup(288,723)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×C6.D4
G = < a,b,c,d | a6=b6=c4=1, d2=b3, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b3c-1 >
Subgroups: 618 in 331 conjugacy classes, 130 normal (22 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C23, C23, C32, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C22×C4, C24, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C3×Dic3, C62, C62, C62, C6.D4, C3×C22⋊C4, C22×Dic3, C22×C12, C23×C6, C23×C6, C6×Dic3, C6×Dic3, C2×C62, C2×C62, C2×C62, C2×C6.D4, C6×C22⋊C4, C3×C6.D4, Dic3×C2×C6, C22×C62, C6×C6.D4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, C23, Dic3, C12, D6, C2×C6, C22⋊C4, C22×C4, C2×D4, C3×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C2×C22⋊C4, C3×Dic3, S3×C6, C6.D4, C3×C22⋊C4, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, C6×Dic3, C3×C3⋊D4, S3×C2×C6, C2×C6.D4, C6×C22⋊C4, C3×C6.D4, Dic3×C2×C6, C6×C3⋊D4, C6×C6.D4
►On 48 points
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)(25 26 27 28 29 30)(31 32 33 34 35 36)(37 38 39 40 41 42)(43 44 45 46 47 48)
(1 21 5 19 3 23)(2 22 6 20 4 24)(7 29 11 27 9 25)(8 30 12 28 10 26)(13 39 15 41 17 37)(14 40 16 42 18 38)(31 43 33 45 35 47)(32 44 34 46 36 48)
(1 18 26 43)(2 13 27 44)(3 14 28 45)(4 15 29 46)(5 16 30 47)(6 17 25 48)(7 36 20 41)(8 31 21 42)(9 32 22 37)(10 33 23 38)(11 34 24 39)(12 35 19 40)
(1 40 19 18)(2 41 20 13)(3 42 21 14)(4 37 22 15)(5 38 23 16)(6 39 24 17)(7 44 27 36)(8 45 28 31)(9 46 29 32)(10 47 30 33)(11 48 25 34)(12 43 26 35)
G:=sub<Sym(48)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,21,5,19,3,23)(2,22,6,20,4,24)(7,29,11,27,9,25)(8,30,12,28,10,26)(13,39,15,41,17,37)(14,40,16,42,18,38)(31,43,33,45,35,47)(32,44,34,46,36,48), (1,18,26,43)(2,13,27,44)(3,14,28,45)(4,15,29,46)(5,16,30,47)(6,17,25,48)(7,36,20,41)(8,31,21,42)(9,32,22,37)(10,33,23,38)(11,34,24,39)(12,35,19,40), (1,40,19,18)(2,41,20,13)(3,42,21,14)(4,37,22,15)(5,38,23,16)(6,39,24,17)(7,44,27,36)(8,45,28,31)(9,46,29,32)(10,47,30,33)(11,48,25,34)(12,43,26,35)>;
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)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,21,5,19,3,23)(2,22,6,20,4,24)(7,29,11,27,9,25)(8,30,12,28,10,26)(13,39,15,41,17,37)(14,40,16,42,18,38)(31,43,33,45,35,47)(32,44,34,46,36,48), (1,18,26,43)(2,13,27,44)(3,14,28,45)(4,15,29,46)(5,16,30,47)(6,17,25,48)(7,36,20,41)(8,31,21,42)(9,32,22,37)(10,33,23,38)(11,34,24,39)(12,35,19,40), (1,40,19,18)(2,41,20,13)(3,42,21,14)(4,37,22,15)(5,38,23,16)(6,39,24,17)(7,44,27,36)(8,45,28,31)(9,46,29,32)(10,47,30,33)(11,48,25,34)(12,43,26,35) );
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),(25,26,27,28,29,30),(31,32,33,34,35,36),(37,38,39,40,41,42),(43,44,45,46,47,48)], [(1,21,5,19,3,23),(2,22,6,20,4,24),(7,29,11,27,9,25),(8,30,12,28,10,26),(13,39,15,41,17,37),(14,40,16,42,18,38),(31,43,33,45,35,47),(32,44,34,46,36,48)], [(1,18,26,43),(2,13,27,44),(3,14,28,45),(4,15,29,46),(5,16,30,47),(6,17,25,48),(7,36,20,41),(8,31,21,42),(9,32,22,37),(10,33,23,38),(11,34,24,39),(12,35,19,40)], [(1,40,19,18),(2,41,20,13),(3,42,21,14),(4,37,22,15),(5,38,23,16),(6,39,24,17),(7,44,27,36),(8,45,28,31),(9,46,29,32),(10,47,30,33),(11,48,25,34),(12,43,26,35)]])
108 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3A | 3B | 3C | 3D | 3E | 4A | ··· | 4H | 6A | ··· | 6N | 6O | ··· | 6BO | 12A | ··· | 12P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 6 | ··· | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | S3 | D4 | Dic3 | D6 | C3×S3 | C3⋊D4 | C3×D4 | C3×Dic3 | S3×C6 | C3×C3⋊D4 |
| kernel | C6×C6.D4 | C3×C6.D4 | Dic3×C2×C6 | C22×C62 | C2×C6.D4 | C2×C62 | C6.D4 | C22×Dic3 | C23×C6 | C22×C6 | C23×C6 | C62 | C22×C6 | C22×C6 | C24 | C2×C6 | C2×C6 | C23 | C23 | C22 |
| # reps | 1 | 4 | 2 | 1 | 2 | 8 | 8 | 4 | 2 | 16 | 1 | 4 | 4 | 3 | 2 | 8 | 8 | 8 | 6 | 16 |
Matrix representation of C6×C6.D4 ►in GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 10 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 8 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,3,0,0,0,0,10,0,0,0,0,10],[12,0,0,0,0,12,0,0,0,0,4,0,0,0,0,10],[8,0,0,0,0,8,0,0,0,0,0,1,0,0,1,0],[8,0,0,0,0,5,0,0,0,0,0,12,0,0,1,0] >;
C6×C6.D4 in GAP, Magma, Sage, TeX
C_6\times C_6.D_4
% in TeX
G:=Group("C6xC6.D4"); // GroupNames label
G:=SmallGroup(288,723);
// by ID
G=gap.SmallGroup(288,723);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,1094,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=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