metabelian, supersoluble, monomial
Aliases: C62.110D4, (C2×C6).10D12, C62⋊5C4⋊2C2, C6.21(D6⋊C4), C32⋊7(C23⋊C4), C62.39(C2×C4), (C22×C6).56D6, C3⋊2(C23.6D6), (C2×C62).11C22, C22.2(C12⋊S3), C2.4(C6.11D12), C22.8(C32⋊7D4), (C22×C3⋊S3)⋊2C4, (C3×C22⋊C4)⋊2S3, (C2×C6).13(C4×S3), (C2×C3⋊Dic3)⋊2C4, C23.6(C2×C3⋊S3), C22.3(C4×C3⋊S3), C22⋊C4⋊1(C3⋊S3), (C2×C6).86(C3⋊D4), (C32×C22⋊C4)⋊2C2, (C2×C32⋊7D4).2C2, (C3×C6).52(C22⋊C4), SmallGroup(288,281)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.110D4
G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=dad=a-1b3, cbc-1=dbd=b-1, dcd=a3c-1 >
Subgroups: 732 in 156 conjugacy classes, 47 normal (19 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C2×D4, C3⋊S3, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C23⋊C4, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, C62, C6.D4, C3×C22⋊C4, C2×C3⋊D4, C2×C3⋊Dic3, C2×C3⋊Dic3, C32⋊7D4, C6×C12, C22×C3⋊S3, C2×C62, C23.6D6, C62⋊5C4, C32×C22⋊C4, C2×C32⋊7D4, C62.110D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C3⋊S3, C4×S3, D12, C3⋊D4, C23⋊C4, C2×C3⋊S3, D6⋊C4, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C23.6D6, C6.11D12, C62.110D4
►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 51 46 17 22 58)(2 52 47 18 23 59)(3 53 48 13 24 60)(4 54 43 14 19 55)(5 49 44 15 20 56)(6 50 45 16 21 57)(7 31 68 61 26 42)(8 32 69 62 27 37)(9 33 70 63 28 38)(10 34 71 64 29 39)(11 35 72 65 30 40)(12 36 67 66 25 41)
(2 16)(3 5)(4 14)(6 18)(7 36 61 25)(8 30 62 35)(9 34 63 29)(10 28 64 33)(11 32 65 27)(12 26 66 31)(13 15)(19 55)(20 48)(21 59)(22 46)(23 57)(24 44)(37 40 69 72)(38 71 70 39)(41 42 67 68)(43 54)(45 52)(47 50)(49 60)(51 58)(53 56)
(1 37)(2 68)(3 41)(4 72)(5 39)(6 70)(7 47)(8 58)(9 45)(10 56)(11 43)(12 60)(13 67)(14 40)(15 71)(16 38)(17 69)(18 42)(19 30)(20 34)(21 28)(22 32)(23 26)(24 36)(25 53)(27 51)(29 49)(31 52)(33 50)(35 54)(44 64)(46 62)(48 66)(55 65)(57 63)(59 61)
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), (1,51,46,17,22,58)(2,52,47,18,23,59)(3,53,48,13,24,60)(4,54,43,14,19,55)(5,49,44,15,20,56)(6,50,45,16,21,57)(7,31,68,61,26,42)(8,32,69,62,27,37)(9,33,70,63,28,38)(10,34,71,64,29,39)(11,35,72,65,30,40)(12,36,67,66,25,41), (2,16)(3,5)(4,14)(6,18)(7,36,61,25)(8,30,62,35)(9,34,63,29)(10,28,64,33)(11,32,65,27)(12,26,66,31)(13,15)(19,55)(20,48)(21,59)(22,46)(23,57)(24,44)(37,40,69,72)(38,71,70,39)(41,42,67,68)(43,54)(45,52)(47,50)(49,60)(51,58)(53,56), (1,37)(2,68)(3,41)(4,72)(5,39)(6,70)(7,47)(8,58)(9,45)(10,56)(11,43)(12,60)(13,67)(14,40)(15,71)(16,38)(17,69)(18,42)(19,30)(20,34)(21,28)(22,32)(23,26)(24,36)(25,53)(27,51)(29,49)(31,52)(33,50)(35,54)(44,64)(46,62)(48,66)(55,65)(57,63)(59,61)>;
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), (1,51,46,17,22,58)(2,52,47,18,23,59)(3,53,48,13,24,60)(4,54,43,14,19,55)(5,49,44,15,20,56)(6,50,45,16,21,57)(7,31,68,61,26,42)(8,32,69,62,27,37)(9,33,70,63,28,38)(10,34,71,64,29,39)(11,35,72,65,30,40)(12,36,67,66,25,41), (2,16)(3,5)(4,14)(6,18)(7,36,61,25)(8,30,62,35)(9,34,63,29)(10,28,64,33)(11,32,65,27)(12,26,66,31)(13,15)(19,55)(20,48)(21,59)(22,46)(23,57)(24,44)(37,40,69,72)(38,71,70,39)(41,42,67,68)(43,54)(45,52)(47,50)(49,60)(51,58)(53,56), (1,37)(2,68)(3,41)(4,72)(5,39)(6,70)(7,47)(8,58)(9,45)(10,56)(11,43)(12,60)(13,67)(14,40)(15,71)(16,38)(17,69)(18,42)(19,30)(20,34)(21,28)(22,32)(23,26)(24,36)(25,53)(27,51)(29,49)(31,52)(33,50)(35,54)(44,64)(46,62)(48,66)(55,65)(57,63)(59,61) );
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)], [(1,51,46,17,22,58),(2,52,47,18,23,59),(3,53,48,13,24,60),(4,54,43,14,19,55),(5,49,44,15,20,56),(6,50,45,16,21,57),(7,31,68,61,26,42),(8,32,69,62,27,37),(9,33,70,63,28,38),(10,34,71,64,29,39),(11,35,72,65,30,40),(12,36,67,66,25,41)], [(2,16),(3,5),(4,14),(6,18),(7,36,61,25),(8,30,62,35),(9,34,63,29),(10,28,64,33),(11,32,65,27),(12,26,66,31),(13,15),(19,55),(20,48),(21,59),(22,46),(23,57),(24,44),(37,40,69,72),(38,71,70,39),(41,42,67,68),(43,54),(45,52),(47,50),(49,60),(51,58),(53,56)], [(1,37),(2,68),(3,41),(4,72),(5,39),(6,70),(7,47),(8,58),(9,45),(10,56),(11,43),(12,60),(13,67),(14,40),(15,71),(16,38),(17,69),(18,42),(19,30),(20,34),(21,28),(22,32),(23,26),(24,36),(25,53),(27,51),(29,49),(31,52),(33,50),(35,54),(44,64),(46,62),(48,66),(55,65),(57,63),(59,61)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 6A | ··· | 6L | 6M | ··· | 6T | 12A | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 2 | 2 | 36 | 2 | 2 | 2 | 2 | 4 | 4 | 36 | 36 | 36 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D6 | C4×S3 | D12 | C3⋊D4 | C23⋊C4 | C23.6D6 |
| kernel | C62.110D4 | C62⋊5C4 | C32×C22⋊C4 | C2×C32⋊7D4 | C2×C3⋊Dic3 | C22×C3⋊S3 | C3×C22⋊C4 | C62 | C22×C6 | C2×C6 | C2×C6 | C2×C6 | C32 | C3 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 8 | 8 | 8 | 1 | 8 |
Matrix representation of C62.110D4 ►in GL6(𝔽13)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 4 | 2 | 7 |
| 0 | 0 | 9 | 11 | 7 | 4 |
| 0 | 0 | 0 | 0 | 11 | 4 |
| 0 | 0 | 0 | 0 | 9 | 2 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 7 | 5 |
| 0 | 0 | 12 | 0 | 5 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 8 | 5 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 6 | 9 |
| 0 | 0 | 0 | 12 | 9 | 10 |
| 0 | 0 | 0 | 0 | 9 | 2 |
| 0 | 0 | 0 | 0 | 11 | 4 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 12 | 4 |
| 0 | 0 | 2 | 4 | 8 | 3 |
| 0 | 0 | 12 | 1 | 7 | 10 |
| 0 | 0 | 1 | 2 | 6 | 10 |
G:=sub<GL(6,GF(13))| [0,12,0,0,0,0,1,1,0,0,0,0,0,0,2,9,0,0,0,0,4,11,0,0,0,0,2,7,11,9,0,0,7,4,4,2],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,7,5,0,12,0,0,5,1,1,1],[8,0,0,0,0,0,5,5,0,0,0,0,0,0,1,0,0,0,0,0,1,12,0,0,0,0,6,9,9,11,0,0,9,10,2,4],[12,0,0,0,0,0,1,1,0,0,0,0,0,0,5,2,12,1,0,0,5,4,1,2,0,0,12,8,7,6,0,0,4,3,10,10] >;
C62.110D4 in GAP, Magma, Sage, TeX
C_6^2._{110}D_4 % in TeX
G:=Group("C6^2.110D4"); // GroupNames label
G:=SmallGroup(288,281);
// by ID
G=gap.SmallGroup(288,281);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,422,346,2693,9414]);
// 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=a^3*c^-1>;
// generators/relations