direct product, metabelian, supersoluble, monomial, A-group
Aliases: C2×C6.D6, Dic3⋊6D6, C62.9C22, C6⋊1(C4×S3), C22.9S32, (C2×C6).14D6, (C6×Dic3)⋊8C2, (C2×Dic3)⋊5S3, C32⋊4(C22×C4), (C3×C6).13C23, C6.13(C22×S3), (C3×Dic3)⋊7C22, C3⋊2(S3×C2×C4), C2.3(C2×S32), (C2×C3⋊S3)⋊4C4, C3⋊S3⋊2(C2×C4), (C3×C6)⋊3(C2×C4), (C22×C3⋊S3).4C2, (C2×C3⋊S3).16C22, SmallGroup(144,149)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C2×C6.D6 |
Generators and relations for C2×C6.D6
G = < a,b,c,d | a2=b6=d2=1, c6=b3, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c5 >
Subgroups: 400 in 124 conjugacy classes, 48 normal (8 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×2], C3, C4 [×4], C22, C22 [×6], S3 [×12], C6 [×6], C6 [×3], C2×C4 [×6], C23, C32, Dic3 [×4], C12 [×4], D6 [×18], C2×C6 [×2], C2×C6, C22×C4, C3⋊S3 [×4], C3×C6, C3×C6 [×2], C4×S3 [×8], C2×Dic3 [×2], C2×C12 [×2], C22×S3 [×3], C3×Dic3 [×4], C2×C3⋊S3 [×6], C62, S3×C2×C4 [×2], C6.D6 [×4], C6×Dic3 [×2], C22×C3⋊S3, C2×C6.D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], C23, D6 [×6], C22×C4, C4×S3 [×4], C22×S3 [×2], S32, S3×C2×C4 [×2], C6.D6 [×2], C2×S32, C2×C6.D6
►On 24 points - transitive group 24T225
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 13)(11 14)(12 15)
(1 11 9 7 5 3)(2 4 6 8 10 12)(13 15 17 19 21 23)(14 24 22 20 18 16)
(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 9)(3 7)(4 12)(6 10)(13 21)(15 19)(16 24)(18 22)
G:=sub<Sym(24)| (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,13)(11,14)(12,15), (1,11,9,7,5,3)(2,4,6,8,10,12)(13,15,17,19,21,23)(14,24,22,20,18,16), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,9)(3,7)(4,12)(6,10)(13,21)(15,19)(16,24)(18,22)>;
G:=Group( (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,13)(11,14)(12,15), (1,11,9,7,5,3)(2,4,6,8,10,12)(13,15,17,19,21,23)(14,24,22,20,18,16), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,9)(3,7)(4,12)(6,10)(13,21)(15,19)(16,24)(18,22) );
G=PermutationGroup([(1,16),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(10,13),(11,14),(12,15)], [(1,11,9,7,5,3),(2,4,6,8,10,12),(13,15,17,19,21,23),(14,24,22,20,18,16)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,9),(3,7),(4,12),(6,10),(13,21),(15,19),(16,24),(18,22)])
G:=TransitiveGroup(24,225);
C2×C6.D6 is a maximal subgroup of
C62.D4 C62.6C23 C62.24C23 Dic3⋊4D12 C62.53C23 C62.58C23 Dic3⋊5D12 C62.65C23 C62.67C23 C62.70C23 C62.74C23 Dic3⋊3D12 C62.91C23 C62.94C23 C62.100C23 C62.116C23 C62.117C23 C62.9D4 S32×C2×C4 Dic6⋊12D6
C2×C6.D6 is a maximal quotient of
C3⋊C8.22D6 C3⋊C8⋊20D6 Dic3⋊6Dic6 C62.19C23 C62.44C23 Dic3⋊5D12 C62.70C23 C62.94C23 C2×Dic32 C62.99C23 C62.116C23
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 4A | ··· | 4H | 6A | ··· | 6F | 6G | 6H | 6I | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 2 | 2 | 4 | 3 | ··· | 3 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | ··· | 6 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D6 | D6 | C4×S3 | S32 | C6.D6 | C2×S32 |
| kernel | C2×C6.D6 | C6.D6 | C6×Dic3 | C22×C3⋊S3 | C2×C3⋊S3 | C2×Dic3 | Dic3 | C2×C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 4 | 2 | 1 | 8 | 2 | 4 | 2 | 8 | 1 | 2 | 1 |
Matrix representation of C2×C6.D6 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,8,0,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C2×C6.D6 in GAP, Magma, Sage, TeX
C_2\times C_6.D_6
% in TeX
G:=Group("C2xC6.D6"); // GroupNames label
G:=SmallGroup(144,149);
// by ID
G=gap.SmallGroup(144,149);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,55,490,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=d^2=1,c^6=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^5>;
// generators/relations