direct product, metabelian, supersoluble, monomial
Aliases: C6×Dic6, C12.63D6, C62.26C22, C6⋊(C3×Q8), (C3×C6)⋊3Q8, C3⋊1(C6×Q8), C32⋊5(C2×Q8), C4.11(S3×C6), (C2×C12).5C6, (C6×C12).9C2, (C2×C6).46D6, (C2×C12).20S3, C12.11(C2×C6), C22.8(S3×C6), C6.1(C22×C6), C6.40(C22×S3), (C3×C6).19C23, (C6×Dic3).7C2, (C2×Dic3).3C6, Dic3.1(C2×C6), (C3×C12).40C22, (C3×Dic3).10C22, C2.3(S3×C2×C6), (C2×C4).4(C3×S3), (C2×C6).11(C2×C6), SmallGroup(144,158)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×Dic6
G = < a,b,c | a6=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 136 in 84 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, Q8, C32, Dic3, C12, C12, C2×C6, C2×C6, C2×Q8, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Dic3, C3×C12, C62, C2×Dic6, C6×Q8, C3×Dic6, C6×Dic3, C6×C12, C6×Dic6
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, D6, C2×C6, C2×Q8, C3×S3, Dic6, C3×Q8, C22×S3, C22×C6, S3×C6, C2×Dic6, C6×Q8, C3×Dic6, S3×C2×C6, C6×Dic6
►On 48 points
(1 24 9 20 5 16)(2 13 10 21 6 17)(3 14 11 22 7 18)(4 15 12 23 8 19)(25 47 29 39 33 43)(26 48 30 40 34 44)(27 37 31 41 35 45)(28 38 32 42 36 46)
(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 32 7 26)(2 31 8 25)(3 30 9 36)(4 29 10 35)(5 28 11 34)(6 27 12 33)(13 41 19 47)(14 40 20 46)(15 39 21 45)(16 38 22 44)(17 37 23 43)(18 48 24 42)
G:=sub<Sym(48)| (1,24,9,20,5,16)(2,13,10,21,6,17)(3,14,11,22,7,18)(4,15,12,23,8,19)(25,47,29,39,33,43)(26,48,30,40,34,44)(27,37,31,41,35,45)(28,38,32,42,36,46), (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), (1,32,7,26)(2,31,8,25)(3,30,9,36)(4,29,10,35)(5,28,11,34)(6,27,12,33)(13,41,19,47)(14,40,20,46)(15,39,21,45)(16,38,22,44)(17,37,23,43)(18,48,24,42)>;
G:=Group( (1,24,9,20,5,16)(2,13,10,21,6,17)(3,14,11,22,7,18)(4,15,12,23,8,19)(25,47,29,39,33,43)(26,48,30,40,34,44)(27,37,31,41,35,45)(28,38,32,42,36,46), (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), (1,32,7,26)(2,31,8,25)(3,30,9,36)(4,29,10,35)(5,28,11,34)(6,27,12,33)(13,41,19,47)(14,40,20,46)(15,39,21,45)(16,38,22,44)(17,37,23,43)(18,48,24,42) );
G=PermutationGroup([[(1,24,9,20,5,16),(2,13,10,21,6,17),(3,14,11,22,7,18),(4,15,12,23,8,19),(25,47,29,39,33,43),(26,48,30,40,34,44),(27,37,31,41,35,45),(28,38,32,42,36,46)], [(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)], [(1,32,7,26),(2,31,8,25),(3,30,9,36),(4,29,10,35),(5,28,11,34),(6,27,12,33),(13,41,19,47),(14,40,20,46),(15,39,21,45),(16,38,22,44),(17,37,23,43),(18,48,24,42)]])
C6×Dic6 is a maximal subgroup of
C12.14D12 Dic6⋊Dic3 C6.Dic12 D12.32D6 Dic6.29D6 C62.9C23 C62.13C23 D6⋊6Dic6 C62.33C23 C12.28D12 Dic3⋊Dic6 C12.30D12 C62.43C23 D6⋊1Dic6 C62.58C23 C62.77C23 D12.33D6 S3×C6×Q8
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | ··· | 6O | 12A | ··· | 12P | 12Q | ··· | 12X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 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 | C2 | C3 | C6 | C6 | C6 | S3 | Q8 | D6 | D6 | C3×S3 | Dic6 | C3×Q8 | S3×C6 | S3×C6 | C3×Dic6 |
| kernel | C6×Dic6 | C3×Dic6 | C6×Dic3 | C6×C12 | C2×Dic6 | Dic6 | C2×Dic3 | C2×C12 | C2×C12 | C3×C6 | C12 | C2×C6 | C2×C4 | C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 2 | 1 | 2 | 8 | 4 | 2 | 1 | 2 | 2 | 1 | 2 | 4 | 4 | 4 | 2 | 8 |
Matrix representation of C6×Dic6 ►in GL4(𝔽13) generated by
| 10 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 10 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
G:=sub<GL(4,GF(13))| [10,0,0,0,0,10,0,0,0,0,4,0,0,0,0,4],[10,0,0,0,0,4,0,0,0,0,7,0,0,0,0,2],[0,1,0,0,1,0,0,0,0,0,0,12,0,0,1,0] >;
C6×Dic6 in GAP, Magma, Sage, TeX
C_6\times {\rm Dic}_6 % in TeX
G:=Group("C6xDic6"); // GroupNames label
G:=SmallGroup(144,158);
// by ID
G=gap.SmallGroup(144,158);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-3,144,506,122,3461]);
// Polycyclic
G:=Group<a,b,c|a^6=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations