non-abelian, soluble, monomial
Aliases: C62.1D4, C22.7S3≀C2, C6.D6⋊4C4, C3⋊S3.2C42, C32⋊1(C2.C42), C2.2(S32⋊C4), (C2×C32⋊C4)⋊1C4, (C2×C3⋊S3).8D4, (C2×C3⋊S3).2Q8, C3⋊S3.3(C4⋊C4), (C3×C6).3(C4⋊C4), C2.2(C3⋊S3.Q8), C3⋊S3.4(C22⋊C4), (C3×C6).7(C22⋊C4), (C2×C6.D6).6C2, (C22×C32⋊C4).1C2, (C22×C3⋊S3).1C22, (C2×C3⋊S3).9(C2×C4), SmallGroup(288,385)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.D4
G = < a,b,c,d | a6=b6=c4=1, d2=b3, ab=ba, cac-1=a3b4, dad-1=a-1, cbc-1=a2b3, bd=db, dcd-1=a3b3c-1 >
Subgroups: 648 in 130 conjugacy classes, 31 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C23, C32, Dic3, C12, D6, C2×C6, C22×C4, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C2.C42, C3×Dic3, C32⋊C4, C2×C3⋊S3, C2×C3⋊S3, C62, S3×C2×C4, C6.D6, C6.D6, C6×Dic3, C2×C32⋊C4, C2×C32⋊C4, C22×C3⋊S3, C2×C6.D6, C22×C32⋊C4, C62.D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2.C42, S3≀C2, S32⋊C4, C3⋊S3.Q8, C62.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 15 3 17 5 13)(2 16 4 18 6 14)(7 44 11 48 9 46)(8 45 12 43 10 47)(19 29 21 25 23 27)(20 30 22 26 24 28)(31 37 35 41 33 39)(32 38 36 42 34 40)
(1 7 17 48)(2 8 16 47)(3 9 15 46)(4 10 14 45)(5 11 13 44)(6 12 18 43)(19 38 29 32)(20 39 28 31)(21 40 27 36)(22 41 26 35)(23 42 25 34)(24 37 30 33)
(1 40 17 36)(2 39 18 35)(3 38 13 34)(4 37 14 33)(5 42 15 32)(6 41 16 31)(7 30 48 24)(8 29 43 23)(9 28 44 22)(10 27 45 21)(11 26 46 20)(12 25 47 19)
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), (1,15,3,17,5,13)(2,16,4,18,6,14)(7,44,11,48,9,46)(8,45,12,43,10,47)(19,29,21,25,23,27)(20,30,22,26,24,28)(31,37,35,41,33,39)(32,38,36,42,34,40), (1,7,17,48)(2,8,16,47)(3,9,15,46)(4,10,14,45)(5,11,13,44)(6,12,18,43)(19,38,29,32)(20,39,28,31)(21,40,27,36)(22,41,26,35)(23,42,25,34)(24,37,30,33), (1,40,17,36)(2,39,18,35)(3,38,13,34)(4,37,14,33)(5,42,15,32)(6,41,16,31)(7,30,48,24)(8,29,43,23)(9,28,44,22)(10,27,45,21)(11,26,46,20)(12,25,47,19)>;
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), (1,15,3,17,5,13)(2,16,4,18,6,14)(7,44,11,48,9,46)(8,45,12,43,10,47)(19,29,21,25,23,27)(20,30,22,26,24,28)(31,37,35,41,33,39)(32,38,36,42,34,40), (1,7,17,48)(2,8,16,47)(3,9,15,46)(4,10,14,45)(5,11,13,44)(6,12,18,43)(19,38,29,32)(20,39,28,31)(21,40,27,36)(22,41,26,35)(23,42,25,34)(24,37,30,33), (1,40,17,36)(2,39,18,35)(3,38,13,34)(4,37,14,33)(5,42,15,32)(6,41,16,31)(7,30,48,24)(8,29,43,23)(9,28,44,22)(10,27,45,21)(11,26,46,20)(12,25,47,19) );
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)], [(1,15,3,17,5,13),(2,16,4,18,6,14),(7,44,11,48,9,46),(8,45,12,43,10,47),(19,29,21,25,23,27),(20,30,22,26,24,28),(31,37,35,41,33,39),(32,38,36,42,34,40)], [(1,7,17,48),(2,8,16,47),(3,9,15,46),(4,10,14,45),(5,11,13,44),(6,12,18,43),(19,38,29,32),(20,39,28,31),(21,40,27,36),(22,41,26,35),(23,42,25,34),(24,37,30,33)], [(1,40,17,36),(2,39,18,35),(3,38,13,34),(4,37,14,33),(5,42,15,32),(6,41,16,31),(7,30,48,24),(8,29,43,23),(9,28,44,22),(10,27,45,21),(11,26,46,20),(12,25,47,19)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 6A | ··· | 6F | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 4 | 4 | 6 | ··· | 6 | 18 | 18 | 18 | 18 | 4 | ··· | 4 | 12 | ··· | 12 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | + | + | ||||
| image | C1 | C2 | C2 | C4 | C4 | D4 | Q8 | D4 | S3≀C2 | S32⋊C4 | S32⋊C4 | C3⋊S3.Q8 |
| kernel | C62.D4 | C2×C6.D6 | C22×C32⋊C4 | C6.D6 | C2×C32⋊C4 | C2×C3⋊S3 | C2×C3⋊S3 | C62 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 8 | 4 | 2 | 1 | 1 | 4 | 4 | 4 | 4 |
Matrix representation of C62.D4 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 6 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 8 | 5 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 4 | 3 | 0 | 0 | 0 | 0 |
| 3 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[1,6,0,0,0,0,0,12,0,0,0,0,0,0,0,0,8,0,0,0,0,0,5,5,0,0,5,0,0,0,0,0,0,5,0,0],[4,3,0,0,0,0,3,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
C62.D4 in GAP, Magma, Sage, TeX
C_6^2.D_4
% in TeX
G:=Group("C6^2.D4"); // GroupNames label
G:=SmallGroup(288,385);
// by ID
G=gap.SmallGroup(288,385);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,112,85,64,2693,2028,691,797,2372]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^4=1,d^2=b^3,a*b=b*a,c*a*c^-1=a^3*b^4,d*a*d^-1=a^-1,c*b*c^-1=a^2*b^3,b*d=d*b,d*c*d^-1=a^3*b^3*c^-1>;
// generators/relations