p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8:9SD16, C42.526C23, C4.1432+ 1+4, (C8xQ8):24C2, C8:18(C4oD4), C8:5D4:28C2, C4:C4.284D4, Q8o2(C4.Q8), Q8:3Q8:12C2, C4:SD16:45C2, (C4xSD16):49C2, (C2xQ8).274D4, C4.50(C2xSD16), D4.D4:46C2, C4:C4.443C23, C4:C8.352C22, (C2xC4).584C24, (C2xC8).378C23, (C4xC8).283C22, Q8:6D4.10C2, C4:Q8.212C22, C2.38(Q8:6D4), (C2xD4).279C23, (C4xD4).218C22, (C4xQ8).315C22, (C2xQ8).263C23, C2.34(C22xSD16), C4.Q8.186C22, C2.111(D4oSD16), C4:1D4.105C22, C22.844(C22xD4), D4:C4.191C22, Q8:C4.189C22, (C2xSD16).102C22, C4.162(C2xC4oD4), (C2xC4).1106(C2xD4), SmallGroup(128,2124)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8:9SD16
G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=cac-1=dad=a-1, cbc-1=a2b, bd=db, dcd=c3 >
Subgroups: 416 in 202 conjugacy classes, 96 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, D4, Q8, Q8, C23, C42, C42, C22:C4, C4:C4, C4:C4, C4:C4, C2xC8, C2xC8, SD16, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C4oD4, C4xC8, D4:C4, Q8:C4, C4:C8, C4.Q8, C4xD4, C4xQ8, C4xQ8, C4xQ8, C4:D4, C42.C2, C4:1D4, C4:Q8, C2xSD16, C2xC4oD4, C4xSD16, C8xQ8, C4:SD16, D4.D4, C8:5D4, Q8:6D4, Q8:3Q8, Q8:9SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2xD4, C4oD4, C24, C2xSD16, C22xD4, C2xC4oD4, 2+ 1+4, Q8:6D4, C22xSD16, D4oSD16, Q8:9SD16
►On 64 points
(1 30 59 51)(2 52 60 31)(3 32 61 53)(4 54 62 25)(5 26 63 55)(6 56 64 27)(7 28 57 49)(8 50 58 29)(9 20 40 45)(10 46 33 21)(11 22 34 47)(12 48 35 23)(13 24 36 41)(14 42 37 17)(15 18 38 43)(16 44 39 19)
(1 34 59 11)(2 12 60 35)(3 36 61 13)(4 14 62 37)(5 38 63 15)(6 16 64 39)(7 40 57 9)(8 10 58 33)(17 25 42 54)(18 55 43 26)(19 27 44 56)(20 49 45 28)(21 29 46 50)(22 51 47 30)(23 31 48 52)(24 53 41 32)
(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)
(1 43)(2 46)(3 41)(4 44)(5 47)(6 42)(7 45)(8 48)(9 49)(10 52)(11 55)(12 50)(13 53)(14 56)(15 51)(16 54)(17 64)(18 59)(19 62)(20 57)(21 60)(22 63)(23 58)(24 61)(25 39)(26 34)(27 37)(28 40)(29 35)(30 38)(31 33)(32 36)
G:=sub<Sym(64)| (1,30,59,51)(2,52,60,31)(3,32,61,53)(4,54,62,25)(5,26,63,55)(6,56,64,27)(7,28,57,49)(8,50,58,29)(9,20,40,45)(10,46,33,21)(11,22,34,47)(12,48,35,23)(13,24,36,41)(14,42,37,17)(15,18,38,43)(16,44,39,19), (1,34,59,11)(2,12,60,35)(3,36,61,13)(4,14,62,37)(5,38,63,15)(6,16,64,39)(7,40,57,9)(8,10,58,33)(17,25,42,54)(18,55,43,26)(19,27,44,56)(20,49,45,28)(21,29,46,50)(22,51,47,30)(23,31,48,52)(24,53,41,32), (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), (1,43)(2,46)(3,41)(4,44)(5,47)(6,42)(7,45)(8,48)(9,49)(10,52)(11,55)(12,50)(13,53)(14,56)(15,51)(16,54)(17,64)(18,59)(19,62)(20,57)(21,60)(22,63)(23,58)(24,61)(25,39)(26,34)(27,37)(28,40)(29,35)(30,38)(31,33)(32,36)>;
G:=Group( (1,30,59,51)(2,52,60,31)(3,32,61,53)(4,54,62,25)(5,26,63,55)(6,56,64,27)(7,28,57,49)(8,50,58,29)(9,20,40,45)(10,46,33,21)(11,22,34,47)(12,48,35,23)(13,24,36,41)(14,42,37,17)(15,18,38,43)(16,44,39,19), (1,34,59,11)(2,12,60,35)(3,36,61,13)(4,14,62,37)(5,38,63,15)(6,16,64,39)(7,40,57,9)(8,10,58,33)(17,25,42,54)(18,55,43,26)(19,27,44,56)(20,49,45,28)(21,29,46,50)(22,51,47,30)(23,31,48,52)(24,53,41,32), (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), (1,43)(2,46)(3,41)(4,44)(5,47)(6,42)(7,45)(8,48)(9,49)(10,52)(11,55)(12,50)(13,53)(14,56)(15,51)(16,54)(17,64)(18,59)(19,62)(20,57)(21,60)(22,63)(23,58)(24,61)(25,39)(26,34)(27,37)(28,40)(29,35)(30,38)(31,33)(32,36) );
G=PermutationGroup([[(1,30,59,51),(2,52,60,31),(3,32,61,53),(4,54,62,25),(5,26,63,55),(6,56,64,27),(7,28,57,49),(8,50,58,29),(9,20,40,45),(10,46,33,21),(11,22,34,47),(12,48,35,23),(13,24,36,41),(14,42,37,17),(15,18,38,43),(16,44,39,19)], [(1,34,59,11),(2,12,60,35),(3,36,61,13),(4,14,62,37),(5,38,63,15),(6,16,64,39),(7,40,57,9),(8,10,58,33),(17,25,42,54),(18,55,43,26),(19,27,44,56),(20,49,45,28),(21,29,46,50),(22,51,47,30),(23,31,48,52),(24,53,41,32)], [(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)], [(1,43),(2,46),(3,41),(4,44),(5,47),(6,42),(7,45),(8,48),(9,49),(10,52),(11,55),(12,50),(13,53),(14,56),(15,51),(16,54),(17,64),(18,59),(19,62),(20,57),(21,60),(22,63),(23,58),(24,61),(25,39),(26,34),(27,37),(28,40),(29,35),(30,38),(31,33),(32,36)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4H | 4I | ··· | 4O | 4P | 4Q | 4R | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4oD4 | SD16 | 2+ 1+4 | D4oSD16 |
| kernel | Q8:9SD16 | C4xSD16 | C8xQ8 | C4:SD16 | D4.D4 | C8:5D4 | Q8:6D4 | Q8:3Q8 | C4:C4 | C2xQ8 | C8 | Q8 | C4 | C2 |
| # reps | 1 | 3 | 1 | 3 | 3 | 3 | 1 | 1 | 3 | 1 | 4 | 8 | 1 | 2 |
Matrix representation of Q8:9SD16 ►in GL4(F17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 4 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 |
| 0 | 0 | 12 | 10 |
| 0 | 13 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 16 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,13,0,0,0,0,16,0,0,0,0,16],[0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,4,0,0,4,0,0,0,0,0,0,12,0,0,10,10],[0,4,0,0,13,0,0,0,0,0,1,1,0,0,0,16] >;
Q8:9SD16 in GAP, Magma, Sage, TeX
Q_8\rtimes_9{\rm SD}_{16} % in TeX
G:=Group("Q8:9SD16"); // GroupNames label
G:=SmallGroup(128,2124);
// by ID
G=gap.SmallGroup(128,2124);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,100,346,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^8=d^2=1,b^2=a^2,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=c^3>;
// generators/relations