direct product, metabelian, supersoluble, monomial, A-group
Aliases: C2xC6.D6, Dic3:6D6, C62.9C22, C6:1(C4xS3), C22.9S32, (C2xC6).14D6, (C6xDic3):8C2, (C2xDic3):5S3, C32:4(C22xC4), (C3xC6).13C23, C6.13(C22xS3), (C3xDic3):7C22, C3:2(S3xC2xC4), C2.3(C2xS32), (C2xC3:S3):4C4, C3:S3:2(C2xC4), (C3xC6):3(C2xC4), (C22xC3:S3).4C2, (C2xC3:S3).16C22, SmallGroup(144,149)
Series: Derived ►Chief ►Lower central ►Upper central
C32 — C2xC6.D6 |
Generators and relations for C2xC6.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, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, C23, C32, Dic3, C12, D6, C2xC6, C2xC6, C22xC4, C3:S3, C3xC6, C3xC6, C4xS3, C2xDic3, C2xC12, C22xS3, C3xDic3, C2xC3:S3, C62, S3xC2xC4, C6.D6, C6xDic3, C22xC3:S3, C2xC6.D6
Quotients: C1, C2, C4, C22, S3, C2xC4, C23, D6, C22xC4, C4xS3, C22xS3, S32, S3xC2xC4, C6.D6, C2xS32, C2xC6.D6
(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);
C2xC6.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 S32xC2xC4 Dic6:12D6
C2xC6.D6 is a maximal quotient of
C3:C8.22D6 C3:C8:20D6 Dic3:6Dic6 C62.19C23 C62.44C23 Dic3:5D12 C62.70C23 C62.94C23 C2xDic32 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 | C4xS3 | S32 | C6.D6 | C2xS32 |
kernel | C2xC6.D6 | C6.D6 | C6xDic3 | C22xC3:S3 | C2xC3:S3 | C2xDic3 | Dic3 | C2xC6 | C6 | C22 | C2 | C2 |
# reps | 1 | 4 | 2 | 1 | 8 | 2 | 4 | 2 | 8 | 1 | 2 | 1 |
Matrix representation of C2xC6.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] >;
C2xC6.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