metabelian, supersoluble, monomial
Aliases: C62.37D4, C32⋊11C4≀C2, C12⋊S3⋊9C4, C12.29(C4×S3), (C2×C12).92D6, (C2×C6).11D12, C32⋊4Q8⋊9C4, C6.29(D6⋊C4), (C3×C12).171D4, C3⋊2(D12⋊C4), M4(2)⋊4(C3⋊S3), (C3×M4(2))⋊10S3, (C6×C12).59C22, C12.132(C3⋊D4), C12.59D6.3C2, C4.29(C32⋊7D4), C22.3(C12⋊S3), (C32×M4(2))⋊14C2, C2.11(C6.11D12), C4.3(C4×C3⋊S3), (C4×C3⋊Dic3)⋊2C2, (C3×C12).51(C2×C4), (C3×C6).60(C22⋊C4), (C2×C4).39(C2×C3⋊S3), SmallGroup(288,300)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.37D4
G = < a,b,c,d | a6=b6=1, c4=b3, d2=a3, ab=ba, cac-1=ab3, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=a3b3c3 >
Subgroups: 572 in 132 conjugacy classes, 47 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, D6, C2×C6, C42, M4(2), C4○D4, C3⋊S3, C3×C6, C3×C6, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C4≀C2, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, C4×Dic3, C3×M4(2), C4○D12, C3×C24, C32⋊4Q8, C4×C3⋊S3, C12⋊S3, C2×C3⋊Dic3, C32⋊7D4, C6×C12, D12⋊C4, C4×C3⋊Dic3, C32×M4(2), C12.59D6, C62.37D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C3⋊S3, C4×S3, D12, C3⋊D4, C4≀C2, C2×C3⋊S3, D6⋊C4, C4×C3⋊S3, C12⋊S3, C32⋊7D4, D12⋊C4, C6.11D12, C62.37D4
►On 72 points
(1 52 68 5 56 72)(2 49 69)(3 54 70 7 50 66)(4 51 71)(6 53 65)(8 55 67)(9 37 57 13 33 61)(10 34 58)(11 39 59 15 35 63)(12 36 60)(14 38 62)(16 40 64)(17 31 41 21 27 45)(18 28 42)(19 25 43 23 29 47)(20 30 44)(22 32 46)(24 26 48)
(1 45 15 5 41 11)(2 46 16 6 42 12)(3 47 9 7 43 13)(4 48 10 8 44 14)(17 35 56 21 39 52)(18 36 49 22 40 53)(19 37 50 23 33 54)(20 38 51 24 34 55)(25 57 66 29 61 70)(26 58 67 30 62 71)(27 59 68 31 63 72)(28 60 69 32 64 65)
(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 7 5 3)(2 6)(4 8)(9 41 13 45)(10 48)(11 43 15 47)(12 42)(14 44)(16 46)(17 61 21 57)(18 60)(19 63 23 59)(20 62)(22 64)(24 58)(25 35 29 39)(26 34)(27 37 31 33)(28 36)(30 38)(32 40)(49 65)(50 68 54 72)(51 67)(52 70 56 66)(53 69)(55 71)
G:=sub<Sym(72)| (1,52,68,5,56,72)(2,49,69)(3,54,70,7,50,66)(4,51,71)(6,53,65)(8,55,67)(9,37,57,13,33,61)(10,34,58)(11,39,59,15,35,63)(12,36,60)(14,38,62)(16,40,64)(17,31,41,21,27,45)(18,28,42)(19,25,43,23,29,47)(20,30,44)(22,32,46)(24,26,48), (1,45,15,5,41,11)(2,46,16,6,42,12)(3,47,9,7,43,13)(4,48,10,8,44,14)(17,35,56,21,39,52)(18,36,49,22,40,53)(19,37,50,23,33,54)(20,38,51,24,34,55)(25,57,66,29,61,70)(26,58,67,30,62,71)(27,59,68,31,63,72)(28,60,69,32,64,65), (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), (1,7,5,3)(2,6)(4,8)(9,41,13,45)(10,48)(11,43,15,47)(12,42)(14,44)(16,46)(17,61,21,57)(18,60)(19,63,23,59)(20,62)(22,64)(24,58)(25,35,29,39)(26,34)(27,37,31,33)(28,36)(30,38)(32,40)(49,65)(50,68,54,72)(51,67)(52,70,56,66)(53,69)(55,71)>;
G:=Group( (1,52,68,5,56,72)(2,49,69)(3,54,70,7,50,66)(4,51,71)(6,53,65)(8,55,67)(9,37,57,13,33,61)(10,34,58)(11,39,59,15,35,63)(12,36,60)(14,38,62)(16,40,64)(17,31,41,21,27,45)(18,28,42)(19,25,43,23,29,47)(20,30,44)(22,32,46)(24,26,48), (1,45,15,5,41,11)(2,46,16,6,42,12)(3,47,9,7,43,13)(4,48,10,8,44,14)(17,35,56,21,39,52)(18,36,49,22,40,53)(19,37,50,23,33,54)(20,38,51,24,34,55)(25,57,66,29,61,70)(26,58,67,30,62,71)(27,59,68,31,63,72)(28,60,69,32,64,65), (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), (1,7,5,3)(2,6)(4,8)(9,41,13,45)(10,48)(11,43,15,47)(12,42)(14,44)(16,46)(17,61,21,57)(18,60)(19,63,23,59)(20,62)(22,64)(24,58)(25,35,29,39)(26,34)(27,37,31,33)(28,36)(30,38)(32,40)(49,65)(50,68,54,72)(51,67)(52,70,56,66)(53,69)(55,71) );
G=PermutationGroup([[(1,52,68,5,56,72),(2,49,69),(3,54,70,7,50,66),(4,51,71),(6,53,65),(8,55,67),(9,37,57,13,33,61),(10,34,58),(11,39,59,15,35,63),(12,36,60),(14,38,62),(16,40,64),(17,31,41,21,27,45),(18,28,42),(19,25,43,23,29,47),(20,30,44),(22,32,46),(24,26,48)], [(1,45,15,5,41,11),(2,46,16,6,42,12),(3,47,9,7,43,13),(4,48,10,8,44,14),(17,35,56,21,39,52),(18,36,49,22,40,53),(19,37,50,23,33,54),(20,38,51,24,34,55),(25,57,66,29,61,70),(26,58,67,30,62,71),(27,59,68,31,63,72),(28,60,69,32,64,65)], [(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)], [(1,7,5,3),(2,6),(4,8),(9,41,13,45),(10,48),(11,43,15,47),(12,42),(14,44),(16,46),(17,61,21,57),(18,60),(19,63,23,59),(20,62),(22,64),(24,58),(25,35,29,39),(26,34),(27,37,31,33),(28,36),(30,38),(32,40),(49,65),(50,68,54,72),(51,67),(52,70,56,66),(53,69),(55,71)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | 8B | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 36 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 18 | 18 | 18 | 18 | 36 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | D6 | C4×S3 | C3⋊D4 | D12 | C4≀C2 | D12⋊C4 |
| kernel | C62.37D4 | C4×C3⋊Dic3 | C32×M4(2) | C12.59D6 | C32⋊4Q8 | C12⋊S3 | C3×M4(2) | C3×C12 | C62 | C2×C12 | C12 | C12 | C2×C6 | C32 | C3 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 4 | 8 | 8 | 8 | 4 | 8 |
Matrix representation of C62.37D4 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 7 | 59 | 0 | 0 | 0 | 0 |
| 14 | 66 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 46 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[7,14,0,0,0,0,59,66,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,46,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,27,0,0,0,0,0,0,72] >;
C62.37D4 in GAP, Magma, Sage, TeX
C_6^2._{37}D_4 % in TeX
G:=Group("C6^2.37D4"); // GroupNames label
G:=SmallGroup(288,300);
// by ID
G=gap.SmallGroup(288,300);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,100,675,346,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=1,c^4=b^3,d^2=a^3,a*b=b*a,c*a*c^-1=a*b^3,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=a^3*b^3*c^3>;
// generators/relations