p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8⋊9SD16, C42.526C23, C4.1432+ (1+4), (C8×Q8)⋊24C2, C8⋊18(C4○D4), C8⋊5D4⋊28C2, C4⋊C4.284D4, Q8○2(C4.Q8), Q8⋊3Q8⋊12C2, C4⋊SD16⋊45C2, (C4×SD16)⋊49C2, (C2×Q8).274D4, C4.50(C2×SD16), D4.D4⋊46C2, C4⋊C4.443C23, C4⋊C8.352C22, (C2×C4).584C24, (C2×C8).378C23, (C4×C8).283C22, Q8⋊6D4.10C2, C4⋊Q8.212C22, C2.38(Q8⋊6D4), (C4×D4).218C22, (C2×D4).279C23, (C4×Q8).315C22, (C2×Q8).263C23, C2.34(C22×SD16), C4.Q8.186C22, C2.111(D4○SD16), C4⋊1D4.105C22, C22.844(C22×D4), D4⋊C4.191C22, Q8⋊C4.189C22, (C2×SD16).102C22, C4.162(C2×C4○D4), (C2×C4).1106(C2×D4), SmallGroup(128,2124)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 416 in 202 conjugacy classes, 96 normal (18 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×6], C4 [×8], C22, C22 [×9], C8 [×2], C8 [×3], C2×C4, C2×C4 [×6], C2×C4 [×11], D4 [×15], Q8 [×4], Q8 [×6], C23 [×3], C42 [×3], C42 [×3], C22⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×3], C4⋊C4 [×9], C2×C8, C2×C8 [×3], SD16 [×12], C22×C4 [×3], C2×D4 [×3], C2×D4 [×6], C2×Q8, C2×Q8 [×3], C4○D4 [×4], C4×C8 [×3], D4⋊C4 [×3], Q8⋊C4 [×3], C4⋊C8 [×3], C4.Q8, C4×D4 [×3], C4×Q8, C4×Q8 [×3], C4×Q8, C4⋊D4 [×3], C42.C2 [×3], C4⋊1D4 [×3], C4⋊Q8 [×3], C2×SD16 [×9], C2×C4○D4, C4×SD16 [×3], C8×Q8, C4⋊SD16 [×3], D4.D4 [×3], C8⋊5D4 [×3], Q8⋊6D4, Q8⋊3Q8, Q8⋊9SD16
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], SD16 [×4], C2×D4 [×6], C4○D4 [×2], C24, C2×SD16 [×6], C22×D4, C2×C4○D4, 2+ (1+4), Q8⋊6D4, C22×SD16, D4○SD16, Q8⋊9SD16
Generators and relations
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 >
►On 64 points
(1 26 59 51)(2 52 60 27)(3 28 61 53)(4 54 62 29)(5 30 63 55)(6 56 64 31)(7 32 57 49)(8 50 58 25)(9 20 38 45)(10 46 39 21)(11 22 40 47)(12 48 33 23)(13 24 34 41)(14 42 35 17)(15 18 36 43)(16 44 37 19)
(1 40 59 11)(2 12 60 33)(3 34 61 13)(4 14 62 35)(5 36 63 15)(6 16 64 37)(7 38 57 9)(8 10 58 39)(17 29 42 54)(18 55 43 30)(19 31 44 56)(20 49 45 32)(21 25 46 50)(22 51 47 26)(23 27 48 52)(24 53 41 28)
(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 33)(26 36)(27 39)(28 34)(29 37)(30 40)(31 35)(32 38)
G:=sub<Sym(64)| (1,26,59,51)(2,52,60,27)(3,28,61,53)(4,54,62,29)(5,30,63,55)(6,56,64,31)(7,32,57,49)(8,50,58,25)(9,20,38,45)(10,46,39,21)(11,22,40,47)(12,48,33,23)(13,24,34,41)(14,42,35,17)(15,18,36,43)(16,44,37,19), (1,40,59,11)(2,12,60,33)(3,34,61,13)(4,14,62,35)(5,36,63,15)(6,16,64,37)(7,38,57,9)(8,10,58,39)(17,29,42,54)(18,55,43,30)(19,31,44,56)(20,49,45,32)(21,25,46,50)(22,51,47,26)(23,27,48,52)(24,53,41,28), (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)(25,33)(26,36)(27,39)(28,34)(29,37)(30,40)(31,35)(32,38)>;
G:=Group( (1,26,59,51)(2,52,60,27)(3,28,61,53)(4,54,62,29)(5,30,63,55)(6,56,64,31)(7,32,57,49)(8,50,58,25)(9,20,38,45)(10,46,39,21)(11,22,40,47)(12,48,33,23)(13,24,34,41)(14,42,35,17)(15,18,36,43)(16,44,37,19), (1,40,59,11)(2,12,60,33)(3,34,61,13)(4,14,62,35)(5,36,63,15)(6,16,64,37)(7,38,57,9)(8,10,58,39)(17,29,42,54)(18,55,43,30)(19,31,44,56)(20,49,45,32)(21,25,46,50)(22,51,47,26)(23,27,48,52)(24,53,41,28), (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)(25,33)(26,36)(27,39)(28,34)(29,37)(30,40)(31,35)(32,38) );
G=PermutationGroup([(1,26,59,51),(2,52,60,27),(3,28,61,53),(4,54,62,29),(5,30,63,55),(6,56,64,31),(7,32,57,49),(8,50,58,25),(9,20,38,45),(10,46,39,21),(11,22,40,47),(12,48,33,23),(13,24,34,41),(14,42,35,17),(15,18,36,43),(16,44,37,19)], [(1,40,59,11),(2,12,60,33),(3,34,61,13),(4,14,62,35),(5,36,63,15),(6,16,64,37),(7,38,57,9),(8,10,58,39),(17,29,42,54),(18,55,43,30),(19,31,44,56),(20,49,45,32),(21,25,46,50),(22,51,47,26),(23,27,48,52),(24,53,41,28)], [(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),(25,33),(26,36),(27,39),(28,34),(29,37),(30,40),(31,35),(32,38)])
Matrix representation ►G ⊆ 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] >;
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 | C4○D4 | SD16 | 2+ (1+4) | D4○SD16 |
| kernel | Q8⋊9SD16 | C4×SD16 | C8×Q8 | C4⋊SD16 | D4.D4 | C8⋊5D4 | Q8⋊6D4 | Q8⋊3Q8 | C4⋊C4 | C2×Q8 | C8 | Q8 | C4 | C2 |
| # reps | 1 | 3 | 1 | 3 | 3 | 3 | 1 | 1 | 3 | 1 | 4 | 8 | 1 | 2 |
In GAP, Magma, Sage, TeX
Q_8\rtimes_9SD_{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