metabelian, supersoluble, monomial
Aliases: C12.24D6, C62.14C22, (C3×D4)⋊3S3, (C2×C6).6D6, D4⋊2(C3⋊S3), (D4×C32)⋊6C2, C32⋊7D4⋊4C2, C3⋊5(D4⋊2S3), C32⋊4Q8⋊6C2, C32⋊10(C4○D4), C6.35(C22×S3), (C3×C6).34C23, (C3×C12).24C22, C3⋊Dic3.18C22, (C4×C3⋊S3)⋊4C2, C4.5(C2×C3⋊S3), (C2×C3⋊Dic3)⋊7C2, C22.1(C2×C3⋊S3), C2.7(C22×C3⋊S3), (C2×C3⋊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, C2×C4, D4, D4, Q8, C32, Dic3, C12, D6, C2×C6, C4○D4, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C3⋊Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, D4⋊2S3, C32⋊4Q8, C4×C3⋊S3, C2×C3⋊Dic3, C32⋊7D4, D4×C32, C12.D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C3⋊S3, C22×S3, C2×C3⋊S3, D4⋊2S3, C22×C3⋊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 C4○D4 |
| 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 C4○D4 |
| 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 |
►On 72 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)(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⋊6C4≀C2 D12.D6 Dic6.D6 D12.8D6 C24⋊8D6 C24.26D6 C24.32D6 C24.40D6 C62.(C2×C4) Dic6.24D6 S3×D4⋊2S3 D12⋊12D6 C32⋊82+ 1+4 C4○D4×C3⋊S3 C32⋊92- 1+4 C62.13D6 C36.27D6 (C3×D12)⋊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 D4×C3⋊Dic3 C62.72D4 C62.254C23 C62.256C23 C62⋊14D4 C36.27D6 C62.16D6 (C3×D12)⋊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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