metabelian, supersoluble, monomial
Aliases: C62:12D4, C62.226C23, (C2xC12):2D6, (C2xC6):6D12, (C6xC12):2C22, C6.52(C2xD12), C6.108(S3xD4), C32:9C22wrC2, C3:2(D6:D4), (C22xC6).88D6, C6.11D12:4C2, C22:3(C12:S3), (C2xC62).65C22, C2.7(D4xC3:S3), (C2xC3:S3):16D4, (C23xC3:S3):2C2, (C3xC22:C4):3S3, (C2xC12:S3):4C2, C22:C4:2(C3:S3), C2.7(C2xC12:S3), (C3xC6).192(C2xD4), C23.20(C2xC3:S3), (C2xC32:7D4):7C2, (C32xC22:C4):4C2, (C22xC3:S3):3C22, (C2xC3:Dic3):7C22, (C2xC6).243(C22xS3), C22.41(C22xC3:S3), (C2xC4):1(C2xC3:S3), SmallGroup(288,739)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62:12D4
G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=ab3, dad=a-1b3, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 1964 in 390 conjugacy classes, 81 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, C23, C23, C32, Dic3, C12, D6, C2xC6, C2xC6, C22:C4, C22:C4, C2xD4, C24, C3:S3, C3xC6, C3xC6, C3xC6, D12, C2xDic3, C3:D4, C2xC12, C22xS3, C22xC6, C22wrC2, C3:Dic3, C3xC12, C2xC3:S3, C2xC3:S3, C62, C62, C62, D6:C4, C3xC22:C4, C2xD12, C2xC3:D4, S3xC23, C12:S3, C2xC3:Dic3, C32:7D4, C6xC12, C22xC3:S3, C22xC3:S3, C22xC3:S3, C2xC62, D6:D4, C6.11D12, C32xC22:C4, C2xC12:S3, C2xC32:7D4, C23xC3:S3, C62:12D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C3:S3, D12, C22xS3, C22wrC2, C2xC3:S3, C2xD12, S3xD4, C12:S3, C22xC3:S3, D6:D4, C2xC12:S3, D4xC3:S3, C62:12D4
►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 17 10 23 13 19)(2 18 11 24 14 20)(3 16 12 22 15 21)(4 33 25 29 8 36)(5 31 26 30 9 34)(6 32 27 28 7 35)(37 46 69 40 43 72)(38 47 70 41 44 67)(39 48 71 42 45 68)(49 65 59 52 62 56)(50 66 60 53 63 57)(51 61 55 54 64 58)
(1 70 9 54)(2 68 7 52)(3 72 8 50)(4 60 12 46)(5 58 10 44)(6 56 11 48)(13 38 26 61)(14 42 27 65)(15 40 25 63)(16 37 36 66)(17 41 34 64)(18 39 35 62)(19 47 30 55)(20 45 28 59)(21 43 29 57)(22 69 33 53)(23 67 31 51)(24 71 32 49)
(1 51)(2 53)(3 49)(4 39)(5 41)(6 37)(7 69)(8 71)(9 67)(10 64)(11 66)(12 62)(13 55)(14 57)(15 59)(16 56)(17 58)(18 60)(19 61)(20 63)(21 65)(22 52)(23 54)(24 50)(25 45)(26 47)(27 43)(28 40)(29 42)(30 38)(31 70)(32 72)(33 68)(34 44)(35 46)(36 48)
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), (1,17,10,23,13,19)(2,18,11,24,14,20)(3,16,12,22,15,21)(4,33,25,29,8,36)(5,31,26,30,9,34)(6,32,27,28,7,35)(37,46,69,40,43,72)(38,47,70,41,44,67)(39,48,71,42,45,68)(49,65,59,52,62,56)(50,66,60,53,63,57)(51,61,55,54,64,58), (1,70,9,54)(2,68,7,52)(3,72,8,50)(4,60,12,46)(5,58,10,44)(6,56,11,48)(13,38,26,61)(14,42,27,65)(15,40,25,63)(16,37,36,66)(17,41,34,64)(18,39,35,62)(19,47,30,55)(20,45,28,59)(21,43,29,57)(22,69,33,53)(23,67,31,51)(24,71,32,49), (1,51)(2,53)(3,49)(4,39)(5,41)(6,37)(7,69)(8,71)(9,67)(10,64)(11,66)(12,62)(13,55)(14,57)(15,59)(16,56)(17,58)(18,60)(19,61)(20,63)(21,65)(22,52)(23,54)(24,50)(25,45)(26,47)(27,43)(28,40)(29,42)(30,38)(31,70)(32,72)(33,68)(34,44)(35,46)(36,48)>;
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), (1,17,10,23,13,19)(2,18,11,24,14,20)(3,16,12,22,15,21)(4,33,25,29,8,36)(5,31,26,30,9,34)(6,32,27,28,7,35)(37,46,69,40,43,72)(38,47,70,41,44,67)(39,48,71,42,45,68)(49,65,59,52,62,56)(50,66,60,53,63,57)(51,61,55,54,64,58), (1,70,9,54)(2,68,7,52)(3,72,8,50)(4,60,12,46)(5,58,10,44)(6,56,11,48)(13,38,26,61)(14,42,27,65)(15,40,25,63)(16,37,36,66)(17,41,34,64)(18,39,35,62)(19,47,30,55)(20,45,28,59)(21,43,29,57)(22,69,33,53)(23,67,31,51)(24,71,32,49), (1,51)(2,53)(3,49)(4,39)(5,41)(6,37)(7,69)(8,71)(9,67)(10,64)(11,66)(12,62)(13,55)(14,57)(15,59)(16,56)(17,58)(18,60)(19,61)(20,63)(21,65)(22,52)(23,54)(24,50)(25,45)(26,47)(27,43)(28,40)(29,42)(30,38)(31,70)(32,72)(33,68)(34,44)(35,46)(36,48) );
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)], [(1,17,10,23,13,19),(2,18,11,24,14,20),(3,16,12,22,15,21),(4,33,25,29,8,36),(5,31,26,30,9,34),(6,32,27,28,7,35),(37,46,69,40,43,72),(38,47,70,41,44,67),(39,48,71,42,45,68),(49,65,59,52,62,56),(50,66,60,53,63,57),(51,61,55,54,64,58)], [(1,70,9,54),(2,68,7,52),(3,72,8,50),(4,60,12,46),(5,58,10,44),(6,56,11,48),(13,38,26,61),(14,42,27,65),(15,40,25,63),(16,37,36,66),(17,41,34,64),(18,39,35,62),(19,47,30,55),(20,45,28,59),(21,43,29,57),(22,69,33,53),(23,67,31,51),(24,71,32,49)], [(1,51),(2,53),(3,49),(4,39),(5,41),(6,37),(7,69),(8,71),(9,67),(10,64),(11,66),(12,62),(13,55),(14,57),(15,59),(16,56),(17,58),(18,60),(19,61),(20,63),(21,65),(22,52),(23,54),(24,50),(25,45),(26,47),(27,43),(28,40),(29,42),(30,38),(31,70),(32,72),(33,68),(34,44),(35,46),(36,48)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 6A | ··· | 6L | 6M | ··· | 6T | 12A | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 18 | 18 | 18 | 18 | 36 | 2 | 2 | 2 | 2 | 4 | 4 | 36 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D12 | S3xD4 |
| kernel | C62:12D4 | C6.11D12 | C32xC22:C4 | C2xC12:S3 | C2xC32:7D4 | C23xC3:S3 | C3xC22:C4 | C2xC3:S3 | C62 | C2xC12 | C22xC6 | C2xC6 | C6 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 4 | 2 | 8 | 4 | 16 | 8 |
Matrix representation of C62:12D4 ►in GL6(F13)
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 3 | 6 | 0 | 0 | 0 | 0 |
| 7 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 6 | 0 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 11 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 3 | 6 | 0 | 0 | 0 | 0 |
| 3 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 10 | 0 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[3,7,0,0,0,0,6,10,0,0,0,0,0,0,3,7,0,0,0,0,6,10,0,0,0,0,0,0,12,1,0,0,0,0,11,1],[3,3,0,0,0,0,6,10,0,0,0,0,0,0,3,7,0,0,0,0,10,10,0,0,0,0,0,0,1,0,0,0,0,0,2,12] >;
C62:12D4 in GAP, Magma, Sage, TeX
C_6^2\rtimes_{12}D_4 % in TeX
G:=Group("C6^2:12D4"); // GroupNames label
G:=SmallGroup(288,739);
// by ID
G=gap.SmallGroup(288,739);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,219,58,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^4=d^2=1,a*b=b*a,c*a*c^-1=a*b^3,d*a*d=a^-1*b^3,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations