metabelian, supersoluble, monomial
Aliases: C62⋊24D4, C62.264C23, (C23×C6)⋊9S3, C24⋊5(C3⋊S3), C32⋊13C22≀C2, (C22×C62)⋊4C2, C62⋊5C4⋊21C2, C3⋊3(C24⋊4S3), (C22×C6).162D6, C22⋊4(C32⋊7D4), (C2×C62).114C22, (C2×C6)⋊17(C3⋊D4), (C3×C6).297(C2×D4), C6.138(C2×C3⋊D4), C23.31(C2×C3⋊S3), (C2×C32⋊7D4)⋊14C2, (C22×C3⋊S3)⋊4C22, (C2×C3⋊Dic3)⋊9C22, C2.26(C2×C32⋊7D4), (C2×C6).281(C22×S3), C22.66(C22×C3⋊S3), SmallGroup(288,810)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62⋊24D4
G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=dad=a-1b3, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 1260 in 390 conjugacy classes, 101 normal (8 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, D6, C2×C6, C2×C6, C22⋊C4, C2×D4, C24, C3⋊S3, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C22×C6, C22≀C2, C3⋊Dic3, C2×C3⋊S3, C62, C62, C62, C6.D4, C2×C3⋊D4, C23×C6, C2×C3⋊Dic3, C32⋊7D4, C22×C3⋊S3, C2×C62, C2×C62, C24⋊4S3, C62⋊5C4, C2×C32⋊7D4, C22×C62, C62⋊24D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, C3⋊D4, C22×S3, C22≀C2, C2×C3⋊S3, C2×C3⋊D4, C32⋊7D4, C22×C3⋊S3, C24⋊4S3, C2×C32⋊7D4, C62⋊24D4
►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 18 56 31 52 46)(2 13 57 32 53 47)(3 14 58 33 54 48)(4 15 59 34 49 43)(5 16 60 35 50 44)(6 17 55 36 51 45)(7 61 27 41 72 24)(8 62 28 42 67 19)(9 63 29 37 68 20)(10 64 30 38 69 21)(11 65 25 39 70 22)(12 66 26 40 71 23)
(1 61 4 69)(2 71 5 63)(3 65 6 67)(7 15 38 18)(8 54 39 51)(9 13 40 16)(10 52 41 49)(11 17 42 14)(12 50 37 53)(19 48 25 45)(20 57 26 60)(21 46 27 43)(22 55 28 58)(23 44 29 47)(24 59 30 56)(31 72 34 64)(32 66 35 68)(33 70 36 62)
(1 64)(2 68)(3 62)(4 72)(5 66)(6 70)(7 49)(8 14)(9 53)(10 18)(11 51)(12 16)(13 37)(15 41)(17 39)(19 58)(20 47)(21 56)(22 45)(23 60)(24 43)(25 55)(26 44)(27 59)(28 48)(29 57)(30 46)(31 69)(32 63)(33 67)(34 61)(35 71)(36 65)(38 52)(40 50)(42 54)
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), (1,18,56,31,52,46)(2,13,57,32,53,47)(3,14,58,33,54,48)(4,15,59,34,49,43)(5,16,60,35,50,44)(6,17,55,36,51,45)(7,61,27,41,72,24)(8,62,28,42,67,19)(9,63,29,37,68,20)(10,64,30,38,69,21)(11,65,25,39,70,22)(12,66,26,40,71,23), (1,61,4,69)(2,71,5,63)(3,65,6,67)(7,15,38,18)(8,54,39,51)(9,13,40,16)(10,52,41,49)(11,17,42,14)(12,50,37,53)(19,48,25,45)(20,57,26,60)(21,46,27,43)(22,55,28,58)(23,44,29,47)(24,59,30,56)(31,72,34,64)(32,66,35,68)(33,70,36,62), (1,64)(2,68)(3,62)(4,72)(5,66)(6,70)(7,49)(8,14)(9,53)(10,18)(11,51)(12,16)(13,37)(15,41)(17,39)(19,58)(20,47)(21,56)(22,45)(23,60)(24,43)(25,55)(26,44)(27,59)(28,48)(29,57)(30,46)(31,69)(32,63)(33,67)(34,61)(35,71)(36,65)(38,52)(40,50)(42,54)>;
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), (1,18,56,31,52,46)(2,13,57,32,53,47)(3,14,58,33,54,48)(4,15,59,34,49,43)(5,16,60,35,50,44)(6,17,55,36,51,45)(7,61,27,41,72,24)(8,62,28,42,67,19)(9,63,29,37,68,20)(10,64,30,38,69,21)(11,65,25,39,70,22)(12,66,26,40,71,23), (1,61,4,69)(2,71,5,63)(3,65,6,67)(7,15,38,18)(8,54,39,51)(9,13,40,16)(10,52,41,49)(11,17,42,14)(12,50,37,53)(19,48,25,45)(20,57,26,60)(21,46,27,43)(22,55,28,58)(23,44,29,47)(24,59,30,56)(31,72,34,64)(32,66,35,68)(33,70,36,62), (1,64)(2,68)(3,62)(4,72)(5,66)(6,70)(7,49)(8,14)(9,53)(10,18)(11,51)(12,16)(13,37)(15,41)(17,39)(19,58)(20,47)(21,56)(22,45)(23,60)(24,43)(25,55)(26,44)(27,59)(28,48)(29,57)(30,46)(31,69)(32,63)(33,67)(34,61)(35,71)(36,65)(38,52)(40,50)(42,54) );
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)], [(1,18,56,31,52,46),(2,13,57,32,53,47),(3,14,58,33,54,48),(4,15,59,34,49,43),(5,16,60,35,50,44),(6,17,55,36,51,45),(7,61,27,41,72,24),(8,62,28,42,67,19),(9,63,29,37,68,20),(10,64,30,38,69,21),(11,65,25,39,70,22),(12,66,26,40,71,23)], [(1,61,4,69),(2,71,5,63),(3,65,6,67),(7,15,38,18),(8,54,39,51),(9,13,40,16),(10,52,41,49),(11,17,42,14),(12,50,37,53),(19,48,25,45),(20,57,26,60),(21,46,27,43),(22,55,28,58),(23,44,29,47),(24,59,30,56),(31,72,34,64),(32,66,35,68),(33,70,36,62)], [(1,64),(2,68),(3,62),(4,72),(5,66),(6,70),(7,49),(8,14),(9,53),(10,18),(11,51),(12,16),(13,37),(15,41),(17,39),(19,58),(20,47),(21,56),(22,45),(23,60),(24,43),(25,55),(26,44),(27,59),(28,48),(29,57),(30,46),(31,69),(32,63),(33,67),(34,61),(35,71),(36,65),(38,52),(40,50),(42,54)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 6A | ··· | 6BH |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | ··· | 6 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 36 | 2 | 2 | 2 | 2 | 36 | 36 | 36 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | C3⋊D4 |
| kernel | C62⋊24D4 | C62⋊5C4 | C2×C32⋊7D4 | C22×C62 | C23×C6 | C62 | C22×C6 | C2×C6 |
| # reps | 1 | 3 | 3 | 1 | 4 | 6 | 12 | 48 |
Matrix representation of C62⋊24D4 ►in GL4(𝔽13) generated by
| 4 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 9 | 4 |
| 10 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 6 | 5 |
| 0 | 0 | 6 | 7 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 7 | 8 |
| 0 | 0 | 7 | 6 |
G:=sub<GL(4,GF(13))| [4,0,0,0,0,3,0,0,0,0,3,9,0,0,0,4],[10,0,0,0,0,4,0,0,0,0,12,0,0,0,0,12],[0,1,0,0,12,0,0,0,0,0,6,6,0,0,5,7],[0,1,0,0,1,0,0,0,0,0,7,7,0,0,8,6] >;
C62⋊24D4 in GAP, Magma, Sage, TeX
C_6^2\rtimes_{24}D_4 % in TeX
G:=Group("C6^2:24D4"); // GroupNames label
G:=SmallGroup(288,810);
// by ID
G=gap.SmallGroup(288,810);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,254,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^4=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^3,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations