p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8⋊5D8, C42.525C23, C4.1422+ (1+4), (C8×Q8)⋊9C2, (C4×D8)⋊18C2, C4.48(C2×D8), C8⋊17(C4○D4), C8⋊4D4⋊13C2, C4⋊D8⋊16C2, C4⋊C4.283D4, Q8○2(C2.D8), Q8⋊6D4⋊12C2, C2.70(D4○D8), (C4×C8).97C22, (C2×Q8).273D4, C2.23(C22×D8), C4⋊C8.306C22, C4⋊C4.442C23, (C2×C4).583C24, (C2×C8).218C23, (C2×D8).40C22, C2.37(Q8⋊6D4), (C2×D4).278C23, (C4×D4).217C22, (C4×Q8).314C22, C2.D8.236C22, C4⋊1D4.104C22, C22.843(C22×D4), D4⋊C4.174C22, C4.161(C2×C4○D4), (C2×C4).1105(C2×D4), SmallGroup(128,2123)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 552 in 230 conjugacy classes, 96 normal (14 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×6], C4 [×5], C22, C22 [×18], C8 [×2], C8 [×3], C2×C4, C2×C4 [×6], C2×C4 [×14], D4 [×30], Q8 [×4], C23 [×6], C42 [×3], C22⋊C4 [×6], C4⋊C4 [×5], C2×C8, C2×C8 [×3], D8 [×12], C22×C4 [×6], C2×D4 [×6], C2×D4 [×12], C2×Q8, C4○D4 [×8], C4×C8 [×3], D4⋊C4 [×6], C4⋊C8 [×3], C2.D8, C4×D4 [×6], C4×Q8, C4⋊D4 [×6], C4⋊1D4 [×6], C2×D8 [×9], C2×C4○D4 [×2], C4×D8 [×3], C8×Q8, C4⋊D8 [×6], C8⋊4D4 [×3], Q8⋊6D4 [×2], Q8⋊5D8
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C4○D4 [×2], C24, C2×D8 [×6], C22×D4, C2×C4○D4, 2+ (1+4), Q8⋊6D4, C22×D8, D4○D8, Q8⋊5D8
Generators and relations
G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=cac-1=a-1, ad=da, cbc-1=dbd=a2b, dcd=c-1 >
►On 64 points
(1 26 62 55)(2 56 63 27)(3 28 64 49)(4 50 57 29)(5 30 58 51)(6 52 59 31)(7 32 60 53)(8 54 61 25)(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 38 62 15)(2 16 63 39)(3 40 64 9)(4 10 57 33)(5 34 58 11)(6 12 59 35)(7 36 60 13)(8 14 61 37)(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 46)(2 45)(3 44)(4 43)(5 42)(6 41)(7 48)(8 47)(9 56)(10 55)(11 54)(12 53)(13 52)(14 51)(15 50)(16 49)(17 58)(18 57)(19 64)(20 63)(21 62)(22 61)(23 60)(24 59)(25 34)(26 33)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)
G:=sub<Sym(64)| (1,26,62,55)(2,56,63,27)(3,28,64,49)(4,50,57,29)(5,30,58,51)(6,52,59,31)(7,32,60,53)(8,54,61,25)(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), (1,38,62,15)(2,16,63,39)(3,40,64,9)(4,10,57,33)(5,34,58,11)(6,12,59,35)(7,36,60,13)(8,14,61,37)(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), (1,46)(2,45)(3,44)(4,43)(5,42)(6,41)(7,48)(8,47)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,58)(18,57)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,34)(26,33)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)>;
G:=Group( (1,26,62,55)(2,56,63,27)(3,28,64,49)(4,50,57,29)(5,30,58,51)(6,52,59,31)(7,32,60,53)(8,54,61,25)(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), (1,38,62,15)(2,16,63,39)(3,40,64,9)(4,10,57,33)(5,34,58,11)(6,12,59,35)(7,36,60,13)(8,14,61,37)(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), (1,46)(2,45)(3,44)(4,43)(5,42)(6,41)(7,48)(8,47)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,58)(18,57)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,34)(26,33)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35) );
G=PermutationGroup([(1,26,62,55),(2,56,63,27),(3,28,64,49),(4,50,57,29),(5,30,58,51),(6,52,59,31),(7,32,60,53),(8,54,61,25),(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)], [(1,38,62,15),(2,16,63,39),(3,40,64,9),(4,10,57,33),(5,34,58,11),(6,12,59,35),(7,36,60,13),(8,14,61,37),(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)], [(1,46),(2,45),(3,44),(4,43),(5,42),(6,41),(7,48),(8,47),(9,56),(10,55),(11,54),(12,53),(13,52),(14,51),(15,50),(16,49),(17,58),(18,57),(19,64),(20,63),(21,62),(22,61),(23,60),(24,59),(25,34),(26,33),(27,40),(28,39),(29,38),(30,37),(31,36),(32,35)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 4 | 13 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 15 |
| 0 | 0 | 1 | 16 |
| 3 | 14 | 0 | 0 |
| 3 | 3 | 0 | 0 |
| 0 | 0 | 4 | 9 |
| 0 | 0 | 0 | 13 |
| 3 | 14 | 0 | 0 |
| 14 | 14 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 16 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,4,4,0,0,0,13],[16,0,0,0,0,16,0,0,0,0,1,1,0,0,15,16],[3,3,0,0,14,3,0,0,0,0,4,0,0,0,9,13],[3,14,0,0,14,14,0,0,0,0,1,1,0,0,0,16] >;
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | ··· | 4H | 4I | ··· | 4O | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | D8 | 2+ (1+4) | D4○D8 |
| kernel | Q8⋊5D8 | C4×D8 | C8×Q8 | C4⋊D8 | C8⋊4D4 | Q8⋊6D4 | C4⋊C4 | C2×Q8 | C8 | Q8 | C4 | C2 |
| # reps | 1 | 3 | 1 | 6 | 3 | 2 | 3 | 1 | 4 | 8 | 1 | 2 |
In GAP, Magma, Sage, TeX
Q_8\rtimes_5D_8
% in TeX
G:=Group("Q8:5D8"); // GroupNames label
G:=SmallGroup(128,2123);
// by ID
G=gap.SmallGroup(128,2123);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,100,346,80,4037,1027,124]);
// 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=a^-1,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations