metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6:5SD16, D12.8D4, D4.6D12, C4:C4:2D6, D6:C8:9C2, (C2xC8):16D6, (C3xD4).1D4, C12.1(C2xD4), C4.85(S3xD4), C4.2(C2xD12), C4.D12:1C2, C6.D8:7C2, D4:C4:10S3, (C2xC24):15C22, C6.20C22wrC2, (C2xD4).135D6, C3:2(C22:SD16), C6.23(C2xSD16), C2.11(S3xSD16), C2.13(D8:S3), C6.31(C8:C22), (C2xDic3).20D4, (C22xS3).72D4, (C6xD4).37C22, C22.174(S3xD4), C2.23(D6:D4), (C2xC12).216C23, (C2xDic6):13C22, (C2xD12).50C22, (C2xS3xD4).5C2, (C2xC3:C8):3C22, (C3xC4:C4):4C22, (C2xD4.S3):3C2, (C2xC24:C2):14C2, (S3xC2xC4).9C22, (C2xC6).229(C2xD4), (C3xD4:C4):10C2, (C2xC4).323(C22xS3), SmallGroup(192,335)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6:5SD16
G = < a,b,c,d | a6=b2=c8=d2=1, bab=cac-1=a-1, ad=da, cbc-1=ab, bd=db, dcd=c3 >
Subgroups: 712 in 188 conjugacy classes, 45 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, D4, Q8, C23, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, SD16, C22xC4, C2xD4, C2xD4, C2xQ8, C24, C3:C8, C24, Dic6, C4xS3, D12, D12, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C3xD4, C22xS3, C22xS3, C22xC6, C22:C8, D4:C4, D4:C4, C22:Q8, C2xSD16, C22xD4, C24:C2, C2xC3:C8, C4:Dic3, D6:C4, D4.S3, C3xC4:C4, C2xC24, C2xDic6, S3xC2xC4, C2xD12, S3xD4, C2xC3:D4, C6xD4, S3xC23, C22:SD16, C6.D8, D6:C8, C3xD4:C4, C4.D12, C2xC24:C2, C2xD4.S3, C2xS3xD4, D6:5SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2xD4, D12, C22xS3, C22wrC2, C2xSD16, C8:C22, C2xD12, S3xD4, C22:SD16, D6:D4, D8:S3, S3xSD16, D6:5SD16
►On 48 points
(1 40 26 22 42 12)(2 13 43 23 27 33)(3 34 28 24 44 14)(4 15 45 17 29 35)(5 36 30 18 46 16)(6 9 47 19 31 37)(7 38 32 20 48 10)(8 11 41 21 25 39)
(1 16)(2 47)(3 10)(4 41)(5 12)(6 43)(7 14)(8 45)(9 13)(11 15)(17 39)(18 26)(19 33)(20 28)(21 35)(22 30)(23 37)(24 32)(25 29)(27 31)(34 48)(36 42)(38 44)(40 46)
(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)
(1 22)(2 17)(3 20)(4 23)(5 18)(6 21)(7 24)(8 19)(9 25)(10 28)(11 31)(12 26)(13 29)(14 32)(15 27)(16 30)(33 45)(34 48)(35 43)(36 46)(37 41)(38 44)(39 47)(40 42)
G:=sub<Sym(48)| (1,40,26,22,42,12)(2,13,43,23,27,33)(3,34,28,24,44,14)(4,15,45,17,29,35)(5,36,30,18,46,16)(6,9,47,19,31,37)(7,38,32,20,48,10)(8,11,41,21,25,39), (1,16)(2,47)(3,10)(4,41)(5,12)(6,43)(7,14)(8,45)(9,13)(11,15)(17,39)(18,26)(19,33)(20,28)(21,35)(22,30)(23,37)(24,32)(25,29)(27,31)(34,48)(36,42)(38,44)(40,46), (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), (1,22)(2,17)(3,20)(4,23)(5,18)(6,21)(7,24)(8,19)(9,25)(10,28)(11,31)(12,26)(13,29)(14,32)(15,27)(16,30)(33,45)(34,48)(35,43)(36,46)(37,41)(38,44)(39,47)(40,42)>;
G:=Group( (1,40,26,22,42,12)(2,13,43,23,27,33)(3,34,28,24,44,14)(4,15,45,17,29,35)(5,36,30,18,46,16)(6,9,47,19,31,37)(7,38,32,20,48,10)(8,11,41,21,25,39), (1,16)(2,47)(3,10)(4,41)(5,12)(6,43)(7,14)(8,45)(9,13)(11,15)(17,39)(18,26)(19,33)(20,28)(21,35)(22,30)(23,37)(24,32)(25,29)(27,31)(34,48)(36,42)(38,44)(40,46), (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), (1,22)(2,17)(3,20)(4,23)(5,18)(6,21)(7,24)(8,19)(9,25)(10,28)(11,31)(12,26)(13,29)(14,32)(15,27)(16,30)(33,45)(34,48)(35,43)(36,46)(37,41)(38,44)(39,47)(40,42) );
G=PermutationGroup([[(1,40,26,22,42,12),(2,13,43,23,27,33),(3,34,28,24,44,14),(4,15,45,17,29,35),(5,36,30,18,46,16),(6,9,47,19,31,37),(7,38,32,20,48,10),(8,11,41,21,25,39)], [(1,16),(2,47),(3,10),(4,41),(5,12),(6,43),(7,14),(8,45),(9,13),(11,15),(17,39),(18,26),(19,33),(20,28),(21,35),(22,30),(23,37),(24,32),(25,29),(27,31),(34,48),(36,42),(38,44),(40,46)], [(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)], [(1,22),(2,17),(3,20),(4,23),(5,18),(6,21),(7,24),(8,19),(9,25),(10,28),(11,31),(12,26),(13,29),(14,32),(15,27),(16,30),(33,45),(34,48),(35,43),(36,46),(37,41),(38,44),(39,47),(40,42)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 8 | 12 | 24 | 2 | 2 | 2 | 8 | 8 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D4 | D6 | D6 | D6 | SD16 | D12 | C8:C22 | S3xD4 | S3xD4 | D8:S3 | S3xSD16 |
| kernel | D6:5SD16 | C6.D8 | D6:C8 | C3xD4:C4 | C4.D12 | C2xC24:C2 | C2xD4.S3 | C2xS3xD4 | D4:C4 | D12 | C2xDic3 | C3xD4 | C22xS3 | C4:C4 | C2xC8 | C2xD4 | D6 | D4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of D6:5SD16 ►in GL4(F73) generated by
| 0 | 1 | 0 | 0 |
| 72 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 72 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 59 | 7 | 0 | 0 |
| 66 | 14 | 0 | 0 |
| 0 | 0 | 67 | 67 |
| 0 | 0 | 6 | 67 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [0,72,0,0,1,1,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,1,1,0,0,0,0,72,0,0,0,0,72],[59,66,0,0,7,14,0,0,0,0,67,6,0,0,67,67],[72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,72] >;
D6:5SD16 in GAP, Magma, Sage, TeX
D_6\rtimes_5{\rm SD}_{16} % in TeX
G:=Group("D6:5SD16"); // GroupNames label
G:=SmallGroup(192,335);
// by ID
G=gap.SmallGroup(192,335);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,422,135,268,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^8=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a*b,b*d=d*b,d*c*d=c^3>;
// generators/relations