metabelian, supersoluble, monomial
Aliases: C62.73D4, (C3xD4):15D6, (C3xQ8):17D6, C3:5(D4:D6), (C2xC12).161D6, (C3xC12).154D4, C32:7D8:10C2, C32:25(C8:C22), C32:11SD16:10C2, C12.58D6:15C2, C12.119(C3:D4), C12.103(C22xS3), (C6xC12).153C22, (C3xC12).107C23, C32:4C8:13C22, (D4xC32):17C22, C4.24(C32:7D4), (Q8xC32):16C22, C12:S3.31C22, C22.5(C32:7D4), D4:4(C2xC3:S3), Q8:5(C2xC3:S3), (C3xC4oD4):5S3, C4oD4:3(C3:S3), (C3xC6).293(C2xD4), (C32xC4oD4):3C2, C6.134(C2xC3:D4), (C2xC12:S3):16C2, C4.17(C22xC3:S3), (C2xC6).26(C3:D4), C2.23(C2xC32:7D4), (C2xC4).21(C2xC3:S3), SmallGroup(288,806)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.73D4
G = < a,b,c,d | a6=b6=d2=1, c4=b3, ab=ba, cac-1=a-1b3, dad=a-1, cbc-1=dbd=b-1, dcd=b3c3 >
Subgroups: 908 in 204 conjugacy classes, 65 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, D4, Q8, C23, C32, C12, C12, D6, C2xC6, C2xC6, M4(2), D8, SD16, C2xD4, C4oD4, C3:S3, C3xC6, C3xC6, C3:C8, D12, C2xC12, C2xC12, C3xD4, C3xD4, C3xQ8, C22xS3, C8:C22, C3xC12, C3xC12, C2xC3:S3, C62, C62, C4.Dic3, D4:S3, Q8:2S3, C2xD12, C3xC4oD4, C32:4C8, C12:S3, C12:S3, C6xC12, C6xC12, D4xC32, D4xC32, Q8xC32, C22xC3:S3, D4:D6, C12.58D6, C32:7D8, C32:11SD16, C2xC12:S3, C32xC4oD4, C62.73D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C3:S3, C3:D4, C22xS3, C8:C22, C2xC3:S3, C2xC3:D4, C32:7D4, C22xC3:S3, D4:D6, C2xC32:7D4, C62.73D4
►On 72 points
(1 25 15 5 29 11)(2 16 30)(3 27 9 7 31 13)(4 10 32)(6 12 26)(8 14 28)(17 64 50)(18 55 57 22 51 61)(19 58 52)(20 49 59 24 53 63)(21 60 54)(23 62 56)(33 48 69)(34 66 41 38 70 45)(35 42 71)(36 68 43 40 72 47)(37 44 65)(39 46 67)
(1 45 20 5 41 24)(2 17 42 6 21 46)(3 47 22 7 43 18)(4 19 44 8 23 48)(9 68 61 13 72 57)(10 58 65 14 62 69)(11 70 63 15 66 59)(12 60 67 16 64 71)(25 34 49 29 38 53)(26 54 39 30 50 35)(27 36 51 31 40 55)(28 56 33 32 52 37)
(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)
(2 8)(3 7)(4 6)(9 27)(10 26)(11 25)(12 32)(13 31)(14 30)(15 29)(16 28)(17 44)(18 43)(19 42)(20 41)(21 48)(22 47)(23 46)(24 45)(33 60)(34 59)(35 58)(36 57)(37 64)(38 63)(39 62)(40 61)(49 66)(50 65)(51 72)(52 71)(53 70)(54 69)(55 68)(56 67)
G:=sub<Sym(72)| (1,25,15,5,29,11)(2,16,30)(3,27,9,7,31,13)(4,10,32)(6,12,26)(8,14,28)(17,64,50)(18,55,57,22,51,61)(19,58,52)(20,49,59,24,53,63)(21,60,54)(23,62,56)(33,48,69)(34,66,41,38,70,45)(35,42,71)(36,68,43,40,72,47)(37,44,65)(39,46,67), (1,45,20,5,41,24)(2,17,42,6,21,46)(3,47,22,7,43,18)(4,19,44,8,23,48)(9,68,61,13,72,57)(10,58,65,14,62,69)(11,70,63,15,66,59)(12,60,67,16,64,71)(25,34,49,29,38,53)(26,54,39,30,50,35)(27,36,51,31,40,55)(28,56,33,32,52,37), (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), (2,8)(3,7)(4,6)(9,27)(10,26)(11,25)(12,32)(13,31)(14,30)(15,29)(16,28)(17,44)(18,43)(19,42)(20,41)(21,48)(22,47)(23,46)(24,45)(33,60)(34,59)(35,58)(36,57)(37,64)(38,63)(39,62)(40,61)(49,66)(50,65)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)>;
G:=Group( (1,25,15,5,29,11)(2,16,30)(3,27,9,7,31,13)(4,10,32)(6,12,26)(8,14,28)(17,64,50)(18,55,57,22,51,61)(19,58,52)(20,49,59,24,53,63)(21,60,54)(23,62,56)(33,48,69)(34,66,41,38,70,45)(35,42,71)(36,68,43,40,72,47)(37,44,65)(39,46,67), (1,45,20,5,41,24)(2,17,42,6,21,46)(3,47,22,7,43,18)(4,19,44,8,23,48)(9,68,61,13,72,57)(10,58,65,14,62,69)(11,70,63,15,66,59)(12,60,67,16,64,71)(25,34,49,29,38,53)(26,54,39,30,50,35)(27,36,51,31,40,55)(28,56,33,32,52,37), (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), (2,8)(3,7)(4,6)(9,27)(10,26)(11,25)(12,32)(13,31)(14,30)(15,29)(16,28)(17,44)(18,43)(19,42)(20,41)(21,48)(22,47)(23,46)(24,45)(33,60)(34,59)(35,58)(36,57)(37,64)(38,63)(39,62)(40,61)(49,66)(50,65)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67) );
G=PermutationGroup([[(1,25,15,5,29,11),(2,16,30),(3,27,9,7,31,13),(4,10,32),(6,12,26),(8,14,28),(17,64,50),(18,55,57,22,51,61),(19,58,52),(20,49,59,24,53,63),(21,60,54),(23,62,56),(33,48,69),(34,66,41,38,70,45),(35,42,71),(36,68,43,40,72,47),(37,44,65),(39,46,67)], [(1,45,20,5,41,24),(2,17,42,6,21,46),(3,47,22,7,43,18),(4,19,44,8,23,48),(9,68,61,13,72,57),(10,58,65,14,62,69),(11,70,63,15,66,59),(12,60,67,16,64,71),(25,34,49,29,38,53),(26,54,39,30,50,35),(27,36,51,31,40,55),(28,56,33,32,52,37)], [(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)], [(2,8),(3,7),(4,6),(9,27),(10,26),(11,25),(12,32),(13,31),(14,30),(15,29),(16,28),(17,44),(18,43),(19,42),(20,41),(21,48),(22,47),(23,46),(24,45),(33,60),(34,59),(35,58),(36,57),(37,64),(38,63),(39,62),(40,61),(49,66),(50,65),(51,72),(52,71),(53,70),(54,69),(55,68),(56,67)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | ··· | 6P | 8A | 8B | 12A | ··· | 12H | 12I | ··· | 12T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 4 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 36 | 36 | 2 | ··· | 2 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | C3:D4 | C3:D4 | C8:C22 | D4:D6 |
| kernel | C62.73D4 | C12.58D6 | C32:7D8 | C32:11SD16 | C2xC12:S3 | C32xC4oD4 | C3xC4oD4 | C3xC12 | C62 | C2xC12 | C3xD4 | C3xQ8 | C12 | C2xC6 | C32 | C3 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 1 | 1 | 4 | 4 | 4 | 8 | 8 | 1 | 8 |
Matrix representation of C62.73D4 ►in GL6(F73)
| 1 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 12 | 57 | 1 | 0 |
| 0 | 0 | 28 | 12 | 0 | 1 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 62 | 39 | 59 | 0 |
| 0 | 0 | 50 | 11 | 0 | 14 |
| 0 | 0 | 64 | 72 | 11 | 39 |
| 0 | 0 | 72 | 8 | 50 | 62 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 28 | 66 | 66 |
| 0 | 0 | 28 | 16 | 59 | 7 |
G:=sub<GL(6,GF(73))| [1,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,12,28,0,0,0,72,57,12,0,0,0,0,1,0,0,0,0,0,0,1],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,72,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,1,1],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,62,50,64,72,0,0,39,11,72,8,0,0,59,0,11,50,0,0,0,14,39,62],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,72,0,12,28,0,0,72,1,28,16,0,0,0,0,66,59,0,0,0,0,66,7] >;
C62.73D4 in GAP, Magma, Sage, TeX
C_6^2._{73}D_4 % in TeX
G:=Group("C6^2.73D4"); // GroupNames label
G:=SmallGroup(288,806);
// by ID
G=gap.SmallGroup(288,806);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,219,675,185,80,2693,9414]);
// 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=a^-1*b^3,d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=b^3*c^3>;
// generators/relations