metabelian, supersoluble, monomial
Aliases: C62.131D4, (C6xD4):2S3, (C3xD4).41D6, (C3xC12).98D4, C32:7D8:9C2, (C2xC12).150D6, C32:9SD16:9C2, C3:5(D12:6C22), C12.59D6:5C2, C12.57(C3:D4), C32:23(C8:C22), C12.98(C22xS3), C12:S3:19C22, C12.58D6:12C2, (C6xC12).141C22, (C3xC12).102C23, C32:4C8:12C22, C4.16(C32:7D4), C32:4Q8:17C22, (D4xC32).26C22, C22.10(C32:7D4), (D4xC3xC6):6C2, D4.6(C2xC3:S3), (C2xD4):2(C3:S3), (C3xC6).279(C2xD4), C6.120(C2xC3:D4), C4.12(C22xC3:S3), C2.9(C2xC32:7D4), (C2xC6).99(C3:D4), (C2xC4).17(C2xC3:S3), SmallGroup(288,789)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.131D4
G = < a,b,c,d | a6=b6=d2=1, c4=b3, ab=ba, cac-1=dad=a-1b3, cbc-1=dbd=b-1, dcd=b3c3 >
Subgroups: 716 in 204 conjugacy classes, 65 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, D4, Q8, C23, C32, Dic3, C12, D6, C2xC6, C2xC6, M4(2), D8, SD16, C2xD4, C4oD4, C3:S3, C3xC6, C3xC6, C3:C8, Dic6, C4xS3, D12, C3:D4, C2xC12, C3xD4, C3xD4, C22xC6, C8:C22, C3:Dic3, C3xC12, C2xC3:S3, C62, C62, C4.Dic3, D4:S3, D4.S3, C4oD12, C6xD4, C32:4C8, C32:4Q8, C4xC3:S3, C12:S3, C32:7D4, C6xC12, D4xC32, D4xC32, C2xC62, D12:6C22, C12.58D6, C32:7D8, C32:9SD16, C12.59D6, D4xC3xC6, C62.131D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C3:S3, C3:D4, C22xS3, C8:C22, C2xC3:S3, C2xC3:D4, C32:7D4, C22xC3:S3, D12:6C22, C2xC32:7D4, C62.131D4
►On 72 points
(1 43 68 5 47 72)(2 69 48)(3 45 70 7 41 66)(4 71 42)(6 65 44)(8 67 46)(9 26 64)(10 61 27 14 57 31)(11 28 58)(12 63 29 16 59 25)(13 30 60)(15 32 62)(17 49 34)(18 39 50 22 35 54)(19 51 36)(20 33 52 24 37 56)(21 53 38)(23 55 40)
(1 18 25 5 22 29)(2 30 23 6 26 19)(3 20 27 7 24 31)(4 32 17 8 28 21)(9 36 48 13 40 44)(10 45 33 14 41 37)(11 38 42 15 34 46)(12 47 35 16 43 39)(49 67 58 53 71 62)(50 63 72 54 59 68)(51 69 60 55 65 64)(52 57 66 56 61 70)
(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 8)(2 7)(3 6)(4 5)(9 14)(10 13)(11 12)(15 16)(17 18)(19 24)(20 23)(21 22)(25 32)(26 31)(27 30)(28 29)(33 36)(34 35)(37 40)(38 39)(41 44)(42 43)(45 48)(46 47)(49 50)(51 56)(52 55)(53 54)(57 60)(58 59)(61 64)(62 63)(65 70)(66 69)(67 68)(71 72)
G:=sub<Sym(72)| (1,43,68,5,47,72)(2,69,48)(3,45,70,7,41,66)(4,71,42)(6,65,44)(8,67,46)(9,26,64)(10,61,27,14,57,31)(11,28,58)(12,63,29,16,59,25)(13,30,60)(15,32,62)(17,49,34)(18,39,50,22,35,54)(19,51,36)(20,33,52,24,37,56)(21,53,38)(23,55,40), (1,18,25,5,22,29)(2,30,23,6,26,19)(3,20,27,7,24,31)(4,32,17,8,28,21)(9,36,48,13,40,44)(10,45,33,14,41,37)(11,38,42,15,34,46)(12,47,35,16,43,39)(49,67,58,53,71,62)(50,63,72,54,59,68)(51,69,60,55,65,64)(52,57,66,56,61,70), (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), (1,8)(2,7)(3,6)(4,5)(9,14)(10,13)(11,12)(15,16)(17,18)(19,24)(20,23)(21,22)(25,32)(26,31)(27,30)(28,29)(33,36)(34,35)(37,40)(38,39)(41,44)(42,43)(45,48)(46,47)(49,50)(51,56)(52,55)(53,54)(57,60)(58,59)(61,64)(62,63)(65,70)(66,69)(67,68)(71,72)>;
G:=Group( (1,43,68,5,47,72)(2,69,48)(3,45,70,7,41,66)(4,71,42)(6,65,44)(8,67,46)(9,26,64)(10,61,27,14,57,31)(11,28,58)(12,63,29,16,59,25)(13,30,60)(15,32,62)(17,49,34)(18,39,50,22,35,54)(19,51,36)(20,33,52,24,37,56)(21,53,38)(23,55,40), (1,18,25,5,22,29)(2,30,23,6,26,19)(3,20,27,7,24,31)(4,32,17,8,28,21)(9,36,48,13,40,44)(10,45,33,14,41,37)(11,38,42,15,34,46)(12,47,35,16,43,39)(49,67,58,53,71,62)(50,63,72,54,59,68)(51,69,60,55,65,64)(52,57,66,56,61,70), (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), (1,8)(2,7)(3,6)(4,5)(9,14)(10,13)(11,12)(15,16)(17,18)(19,24)(20,23)(21,22)(25,32)(26,31)(27,30)(28,29)(33,36)(34,35)(37,40)(38,39)(41,44)(42,43)(45,48)(46,47)(49,50)(51,56)(52,55)(53,54)(57,60)(58,59)(61,64)(62,63)(65,70)(66,69)(67,68)(71,72) );
G=PermutationGroup([[(1,43,68,5,47,72),(2,69,48),(3,45,70,7,41,66),(4,71,42),(6,65,44),(8,67,46),(9,26,64),(10,61,27,14,57,31),(11,28,58),(12,63,29,16,59,25),(13,30,60),(15,32,62),(17,49,34),(18,39,50,22,35,54),(19,51,36),(20,33,52,24,37,56),(21,53,38),(23,55,40)], [(1,18,25,5,22,29),(2,30,23,6,26,19),(3,20,27,7,24,31),(4,32,17,8,28,21),(9,36,48,13,40,44),(10,45,33,14,41,37),(11,38,42,15,34,46),(12,47,35,16,43,39),(49,67,58,53,71,62),(50,63,72,54,59,68),(51,69,60,55,65,64),(52,57,66,56,61,70)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,14),(10,13),(11,12),(15,16),(17,18),(19,24),(20,23),(21,22),(25,32),(26,31),(27,30),(28,29),(33,36),(34,35),(37,40),(38,39),(41,44),(42,43),(45,48),(46,47),(49,50),(51,56),(52,55),(53,54),(57,60),(58,59),(61,64),(62,63),(65,70),(66,69),(67,68),(71,72)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 6A | ··· | 6L | 6M | ··· | 6AB | 8A | 8B | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 4 | 4 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 36 | 2 | ··· | 2 | 4 | ··· | 4 | 36 | 36 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C3:D4 | C3:D4 | C8:C22 | D12:6C22 |
| kernel | C62.131D4 | C12.58D6 | C32:7D8 | C32:9SD16 | C12.59D6 | D4xC3xC6 | C6xD4 | C3xC12 | C62 | C2xC12 | C3xD4 | C12 | C2xC6 | C32 | C3 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 1 | 1 | 4 | 8 | 8 | 8 | 1 | 8 |
Matrix representation of C62.131D4 ►in GL6(F73)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 65 | 0 | 0 | 0 | 0 |
| 0 | 0 | 65 | 0 | 0 | 0 |
| 0 | 0 | 0 | 65 | 0 | 0 |
| 0 | 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 0 | 64 |
| 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(73))| [9,0,0,0,0,0,0,65,0,0,0,0,0,0,65,0,0,0,0,0,0,65,0,0,0,0,0,0,64,0,0,0,0,0,0,64],[64,0,0,0,0,0,0,8,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,72,0,0,1,72,0,0,0,0,0,72,0,0],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
C62.131D4 in GAP, Magma, Sage, TeX
C_6^2._{131}D_4 % in TeX
G:=Group("C6^2.131D4"); // GroupNames label
G:=SmallGroup(288,789);
// by ID
G=gap.SmallGroup(288,789);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,675,185,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=d^2=1,c^4=b^3,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=b^3*c^3>;
// generators/relations