metabelian, supersoluble, monomial
Aliases: C62.93D6, (C6×Dic3)⋊6S3, C32⋊7D4⋊9S3, C33⋊8D4⋊2C2, C33⋊25(C4○D4), C33⋊15D4⋊2C2, C3⋊Dic3.39D6, C33⋊4Q8⋊10C2, C3⋊5(D6.3D6), (C3×Dic3).44D6, C3⋊4(C12.59D6), C32⋊17(C4○D12), (C32×C6).56C23, (C3×C62).27C22, C32⋊24(D4⋊2S3), C33⋊5C4.12C22, (C32×Dic3).30C22, C6.66(C2×S32), (C2×C6).40S32, (Dic3×C3×C6)⋊8C2, C33⋊8(C2×C4)⋊8C2, (C2×C3⋊S3).37D6, C22.1(S3×C3⋊S3), (Dic3×C3⋊S3)⋊10C2, C6.19(C22×C3⋊S3), (C3×C32⋊7D4)⋊2C2, Dic3.9(C2×C3⋊S3), (C2×Dic3)⋊3(C3⋊S3), (C6×C3⋊S3).29C22, (C3×C6).111(C22×S3), (C3×C3⋊Dic3).22C22, (C2×C33⋊C2).10C22, C2.20(C2×S3×C3⋊S3), (C2×C6).7(C2×C3⋊S3), SmallGroup(432,678)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.93D6
G = < a,b,c,d | a6=b6=1, c6=d2=b3, ab=ba, ac=ca, dad-1=a-1b3, cbc-1=dbd-1=b-1, dcd-1=c5 >
Subgroups: 1792 in 304 conjugacy classes, 68 normal (32 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, Q8, C32, C32, C32, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, C4○D12, D4⋊2S3, C3×C3⋊S3, C33⋊C2, C32×C6, C32×C6, S3×Dic3, C6.D6, C3⋊D12, C32⋊2Q8, C6×Dic3, C3×C3⋊D4, C32⋊4Q8, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C32⋊7D4, C6×C12, C32×Dic3, C3×C3⋊Dic3, C33⋊5C4, C6×C3⋊S3, C2×C33⋊C2, C3×C62, D6.3D6, C12.59D6, Dic3×C3⋊S3, C33⋊8(C2×C4), C33⋊8D4, C33⋊4Q8, Dic3×C3×C6, C3×C32⋊7D4, C33⋊15D4, C62.93D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C3⋊S3, C22×S3, S32, C2×C3⋊S3, C4○D12, D4⋊2S3, C2×S32, C22×C3⋊S3, S3×C3⋊S3, D6.3D6, C12.59D6, C2×S3×C3⋊S3, C62.93D6
►On 72 points
(1 31 45)(2 32 46)(3 33 47)(4 34 48)(5 35 37)(6 36 38)(7 25 39)(8 26 40)(9 27 41)(10 28 42)(11 29 43)(12 30 44)(13 59 64 19 53 70)(14 60 65 20 54 71)(15 49 66 21 55 72)(16 50 67 22 56 61)(17 51 68 23 57 62)(18 52 69 24 58 63)
(1 3 5 7 9 11)(2 12 10 8 6 4)(13 15 17 19 21 23)(14 24 22 20 18 16)(25 27 29 31 33 35)(26 36 34 32 30 28)(37 39 41 43 45 47)(38 48 46 44 42 40)(49 51 53 55 57 59)(50 60 58 56 54 52)(61 71 69 67 65 63)(62 64 66 68 70 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 53 7 59)(2 58 8 52)(3 51 9 57)(4 56 10 50)(5 49 11 55)(6 54 12 60)(13 25 19 31)(14 30 20 36)(15 35 21 29)(16 28 22 34)(17 33 23 27)(18 26 24 32)(37 72 43 66)(38 65 44 71)(39 70 45 64)(40 63 46 69)(41 68 47 62)(42 61 48 67)
G:=sub<Sym(72)| (1,31,45)(2,32,46)(3,33,47)(4,34,48)(5,35,37)(6,36,38)(7,25,39)(8,26,40)(9,27,41)(10,28,42)(11,29,43)(12,30,44)(13,59,64,19,53,70)(14,60,65,20,54,71)(15,49,66,21,55,72)(16,50,67,22,56,61)(17,51,68,23,57,62)(18,52,69,24,58,63), (1,3,5,7,9,11)(2,12,10,8,6,4)(13,15,17,19,21,23)(14,24,22,20,18,16)(25,27,29,31,33,35)(26,36,34,32,30,28)(37,39,41,43,45,47)(38,48,46,44,42,40)(49,51,53,55,57,59)(50,60,58,56,54,52)(61,71,69,67,65,63)(62,64,66,68,70,72), (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), (1,53,7,59)(2,58,8,52)(3,51,9,57)(4,56,10,50)(5,49,11,55)(6,54,12,60)(13,25,19,31)(14,30,20,36)(15,35,21,29)(16,28,22,34)(17,33,23,27)(18,26,24,32)(37,72,43,66)(38,65,44,71)(39,70,45,64)(40,63,46,69)(41,68,47,62)(42,61,48,67)>;
G:=Group( (1,31,45)(2,32,46)(3,33,47)(4,34,48)(5,35,37)(6,36,38)(7,25,39)(8,26,40)(9,27,41)(10,28,42)(11,29,43)(12,30,44)(13,59,64,19,53,70)(14,60,65,20,54,71)(15,49,66,21,55,72)(16,50,67,22,56,61)(17,51,68,23,57,62)(18,52,69,24,58,63), (1,3,5,7,9,11)(2,12,10,8,6,4)(13,15,17,19,21,23)(14,24,22,20,18,16)(25,27,29,31,33,35)(26,36,34,32,30,28)(37,39,41,43,45,47)(38,48,46,44,42,40)(49,51,53,55,57,59)(50,60,58,56,54,52)(61,71,69,67,65,63)(62,64,66,68,70,72), (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), (1,53,7,59)(2,58,8,52)(3,51,9,57)(4,56,10,50)(5,49,11,55)(6,54,12,60)(13,25,19,31)(14,30,20,36)(15,35,21,29)(16,28,22,34)(17,33,23,27)(18,26,24,32)(37,72,43,66)(38,65,44,71)(39,70,45,64)(40,63,46,69)(41,68,47,62)(42,61,48,67) );
G=PermutationGroup([[(1,31,45),(2,32,46),(3,33,47),(4,34,48),(5,35,37),(6,36,38),(7,25,39),(8,26,40),(9,27,41),(10,28,42),(11,29,43),(12,30,44),(13,59,64,19,53,70),(14,60,65,20,54,71),(15,49,66,21,55,72),(16,50,67,22,56,61),(17,51,68,23,57,62),(18,52,69,24,58,63)], [(1,3,5,7,9,11),(2,12,10,8,6,4),(13,15,17,19,21,23),(14,24,22,20,18,16),(25,27,29,31,33,35),(26,36,34,32,30,28),(37,39,41,43,45,47),(38,48,46,44,42,40),(49,51,53,55,57,59),(50,60,58,56,54,52),(61,71,69,67,65,63),(62,64,66,68,70,72)], [(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)], [(1,53,7,59),(2,58,8,52),(3,51,9,57),(4,56,10,50),(5,49,11,55),(6,54,12,60),(13,25,19,31),(14,30,20,36),(15,35,21,29),(16,28,22,34),(17,33,23,27),(18,26,24,32),(37,72,43,66),(38,65,44,71),(39,70,45,64),(40,63,46,69),(41,68,47,62),(42,61,48,67)]])
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | ··· | 3E | 3F | 3G | 3H | 3I | 4A | 4B | 4C | 4D | 4E | 6A | ··· | 6M | 6N | ··· | 6Z | 6AA | 12A | ··· | 12P | 12Q |
| order | 1 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 12 | ··· | 12 | 12 |
| size | 1 | 1 | 2 | 18 | 54 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 3 | 3 | 6 | 18 | 54 | 2 | ··· | 2 | 4 | ··· | 4 | 36 | 6 | ··· | 6 | 36 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4○D12 | S32 | D4⋊2S3 | C2×S32 | D6.3D6 |
| kernel | C62.93D6 | Dic3×C3⋊S3 | C33⋊8(C2×C4) | C33⋊8D4 | C33⋊4Q8 | Dic3×C3×C6 | C3×C32⋊7D4 | C33⋊15D4 | C6×Dic3 | C32⋊7D4 | C3×Dic3 | C3⋊Dic3 | C2×C3⋊S3 | C62 | C33 | C32 | C2×C6 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 8 | 1 | 1 | 5 | 2 | 16 | 4 | 1 | 4 | 8 |
Matrix representation of C62.93D6 ►in GL6(𝔽13)
| 3 | 0 | 0 | 0 | 0 | 0 |
| 5 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 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 | 12 |
| 5 | 1 | 0 | 0 | 0 | 0 |
| 0 | 8 | 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 | 1 | 0 |
G:=sub<GL(6,GF(13))| [3,5,0,0,0,0,0,4,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[5,0,0,0,0,0,1,8,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C62.93D6 in GAP, Magma, Sage, TeX
C_6^2._{93}D_6 % in TeX
G:=Group("C6^2.93D6"); // GroupNames label
G:=SmallGroup(432,678);
// by ID
G=gap.SmallGroup(432,678);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,135,571,2028,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=1,c^6=d^2=b^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^3,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations