metabelian, supersoluble, monomial
Aliases: C12.24D6, C62.14C22, (C3xD4):3S3, (C2xC6).6D6, D4:2(C3:S3), (D4xC32):6C2, C32:7D4:4C2, C3:5(D4:2S3), C32:4Q8:6C2, C32:10(C4oD4), C6.35(C22xS3), (C3xC6).34C23, (C3xC12).24C22, C3:Dic3.18C22, (C4xC3:S3):4C2, C4.5(C2xC3:S3), (C2xC3:Dic3):7C2, C22.1(C2xC3:S3), C2.7(C22xC3:S3), (C2xC3:S3).18C22, SmallGroup(144,173)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.D6
G = < a,b,c | a12=b6=1, c2=a6, bab-1=a7, cac-1=a-1, cbc-1=a6b-1 >
Subgroups: 346 in 120 conjugacy classes, 47 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2xC4, D4, D4, Q8, C32, Dic3, C12, D6, C2xC6, C4oD4, C3:S3, C3xC6, C3xC6, Dic6, C4xS3, C2xDic3, C3:D4, C3xD4, C3:Dic3, C3:Dic3, C3xC12, C2xC3:S3, C62, D4:2S3, C32:4Q8, C4xC3:S3, C2xC3:Dic3, C32:7D4, D4xC32, C12.D6
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, C3:S3, C22xS3, C2xC3:S3, D4:2S3, C22xC3:S3, C12.D6
Character table of C12.D6
class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 12A | 12B | 12C | 12D | |
size | 1 | 1 | 2 | 2 | 18 | 2 | 2 | 2 | 2 | 2 | 9 | 9 | 18 | 18 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
ρ2 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ3 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ4 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
ρ5 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
ρ6 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ7 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ8 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
ρ9 | 2 | 2 | -2 | -2 | 0 | -1 | -1 | 2 | -1 | 2 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -2 | 1 | -2 | 1 | 2 | -1 | -1 | -1 | orthogonal lifted from D6 |
ρ10 | 2 | 2 | 2 | -2 | 0 | -1 | 2 | -1 | -1 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | 2 | 1 | -2 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -2 | orthogonal lifted from D6 |
ρ11 | 2 | 2 | -2 | 2 | 0 | -1 | 2 | -1 | -1 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 1 | -2 | -1 | 2 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -2 | orthogonal lifted from D6 |
ρ12 | 2 | 2 | 2 | 2 | 0 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | 2 | -1 | -1 | orthogonal lifted from S3 |
ρ13 | 2 | 2 | -2 | 2 | 0 | 2 | -1 | -1 | -1 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | -2 | 1 | 2 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -2 | 1 | orthogonal lifted from D6 |
ρ14 | 2 | 2 | -2 | -2 | 0 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | -1 | 1 | 1 | 1 | 1 | 1 | -2 | 1 | -2 | -1 | 2 | -1 | -1 | orthogonal lifted from D6 |
ρ15 | 2 | 2 | 2 | -2 | 0 | 2 | -1 | -1 | -1 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | -1 | -2 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -2 | 1 | orthogonal lifted from D6 |
ρ16 | 2 | 2 | -2 | 2 | 0 | -1 | -1 | 2 | -1 | -2 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 2 | -1 | -2 | 1 | -2 | 1 | 1 | 1 | orthogonal lifted from D6 |
ρ17 | 2 | 2 | 2 | 2 | 0 | -1 | -1 | 2 | -1 | 2 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
ρ18 | 2 | 2 | -2 | -2 | 0 | 2 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | -2 | 1 | -2 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 2 | -1 | orthogonal lifted from D6 |
ρ19 | 2 | 2 | -2 | 2 | 0 | -1 | -1 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | -1 | 1 | 1 | -1 | -1 | -1 | 2 | 1 | -2 | 1 | -2 | 1 | 1 | orthogonal lifted from D6 |
ρ20 | 2 | 2 | 2 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | orthogonal lifted from S3 |
ρ21 | 2 | 2 | 2 | -2 | 0 | -1 | -1 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | -1 | -1 | -1 | 1 | 1 | 1 | -2 | -1 | 2 | 1 | -2 | 1 | 1 | orthogonal lifted from D6 |
ρ22 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | orthogonal lifted from S3 |
ρ23 | 2 | 2 | 2 | -2 | 0 | -1 | -1 | 2 | -1 | -2 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -2 | 1 | 2 | -1 | -2 | 1 | 1 | 1 | orthogonal lifted from D6 |
ρ24 | 2 | 2 | -2 | -2 | 0 | -1 | 2 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 1 | -2 | 1 | -2 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 2 | orthogonal lifted from D6 |
ρ25 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | -2i | 2i | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
ρ26 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 2i | -2i | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
ρ27 | 4 | -4 | 0 | 0 | 0 | -2 | -2 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | -4 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4:2S3, Schur index 2 |
ρ28 | 4 | -4 | 0 | 0 | 0 | 4 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4:2S3, Schur index 2 |
ρ29 | 4 | -4 | 0 | 0 | 0 | -2 | 4 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4:2S3, Schur index 2 |
ρ30 | 4 | -4 | 0 | 0 | 0 | -2 | -2 | -2 | 4 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4:2S3, Schur index 2 |
(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)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)
(1 26 52 48 17 64)(2 33 53 43 18 71)(3 28 54 38 19 66)(4 35 55 45 20 61)(5 30 56 40 21 68)(6 25 57 47 22 63)(7 32 58 42 23 70)(8 27 59 37 24 65)(9 34 60 44 13 72)(10 29 49 39 14 67)(11 36 50 46 15 62)(12 31 51 41 16 69)
(1 64 7 70)(2 63 8 69)(3 62 9 68)(4 61 10 67)(5 72 11 66)(6 71 12 65)(13 30 19 36)(14 29 20 35)(15 28 21 34)(16 27 22 33)(17 26 23 32)(18 25 24 31)(37 57 43 51)(38 56 44 50)(39 55 45 49)(40 54 46 60)(41 53 47 59)(42 52 48 58)
G:=sub<Sym(72)| (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,26,52,48,17,64)(2,33,53,43,18,71)(3,28,54,38,19,66)(4,35,55,45,20,61)(5,30,56,40,21,68)(6,25,57,47,22,63)(7,32,58,42,23,70)(8,27,59,37,24,65)(9,34,60,44,13,72)(10,29,49,39,14,67)(11,36,50,46,15,62)(12,31,51,41,16,69), (1,64,7,70)(2,63,8,69)(3,62,9,68)(4,61,10,67)(5,72,11,66)(6,71,12,65)(13,30,19,36)(14,29,20,35)(15,28,21,34)(16,27,22,33)(17,26,23,32)(18,25,24,31)(37,57,43,51)(38,56,44,50)(39,55,45,49)(40,54,46,60)(41,53,47,59)(42,52,48,58)>;
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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,26,52,48,17,64)(2,33,53,43,18,71)(3,28,54,38,19,66)(4,35,55,45,20,61)(5,30,56,40,21,68)(6,25,57,47,22,63)(7,32,58,42,23,70)(8,27,59,37,24,65)(9,34,60,44,13,72)(10,29,49,39,14,67)(11,36,50,46,15,62)(12,31,51,41,16,69), (1,64,7,70)(2,63,8,69)(3,62,9,68)(4,61,10,67)(5,72,11,66)(6,71,12,65)(13,30,19,36)(14,29,20,35)(15,28,21,34)(16,27,22,33)(17,26,23,32)(18,25,24,31)(37,57,43,51)(38,56,44,50)(39,55,45,49)(40,54,46,60)(41,53,47,59)(42,52,48,58) );
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),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72)], [(1,26,52,48,17,64),(2,33,53,43,18,71),(3,28,54,38,19,66),(4,35,55,45,20,61),(5,30,56,40,21,68),(6,25,57,47,22,63),(7,32,58,42,23,70),(8,27,59,37,24,65),(9,34,60,44,13,72),(10,29,49,39,14,67),(11,36,50,46,15,62),(12,31,51,41,16,69)], [(1,64,7,70),(2,63,8,69),(3,62,9,68),(4,61,10,67),(5,72,11,66),(6,71,12,65),(13,30,19,36),(14,29,20,35),(15,28,21,34),(16,27,22,33),(17,26,23,32),(18,25,24,31),(37,57,43,51),(38,56,44,50),(39,55,45,49),(40,54,46,60),(41,53,47,59),(42,52,48,58)]])
C12.D6 is a maximal subgroup of
C32:6C4wrC2 D12.D6 Dic6.D6 D12.8D6 C24:8D6 C24.26D6 C24.32D6 C24.40D6 C62.(C2xC4) Dic6.24D6 S3xD4:2S3 D12:12D6 C32:82+ 1+4 C4oD4xC3:S3 C32:92- 1+4 C62.13D6 C36.27D6 (C3xD12):S3 D12:(C3:S3) C62.90D6 C62.91D6 C62.100D6
C12.D6 is a maximal quotient of
C62.221C23 C62:6Q8 C62.223C23 C62.225C23 C62.227C23 C62.229C23 C62.69D4 C62.231C23 C62.233C23 C62.234C23 C62.236C23 C12.31D12 C62.242C23 D4xC3:Dic3 C62.72D4 C62.254C23 C62.256C23 C62:14D4 C36.27D6 C62.16D6 (C3xD12):S3 D12:(C3:S3) C62.90D6 C62.91D6 C62.100D6
Matrix representation of C12.D6 ►in GL6(F13)
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 | 5 | 0 |
0 | 0 | 0 | 0 | 0 | 8 |
12 | 12 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 1 | 0 | 0 |
0 | 0 | 12 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 1 | 0 |
12 | 12 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 12 |
0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,1,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0] >;
C12.D6 in GAP, Magma, Sage, TeX
C_{12}.D_6
% in TeX
G:=Group("C12.D6");
// GroupNames label
G:=SmallGroup(144,173);
// by ID
G=gap.SmallGroup(144,173);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,55,218,116,964,3461]);
// Polycyclic
G:=Group<a,b,c|a^12=b^6=1,c^2=a^6,b*a*b^-1=a^7,c*a*c^-1=a^-1,c*b*c^-1=a^6*b^-1>;
// generators/relations
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