p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8⋊4D8, C42.499C23, C4.202- 1+4, (C8×Q8)⋊7C2, (C4×D8)⋊16C2, (D4×Q8)⋊10C2, C4.46(C2×D8), D4⋊8(C4○D4), C4⋊D8⋊15C2, C4⋊C4.272D4, Q8○2(D4⋊C4), D4⋊Q8⋊16C2, Q8⋊6D4⋊10C2, C4.4D8⋊17C2, (C4×C8).91C22, (C2×Q8).266D4, C2.21(C22×D8), C4⋊C4.426C23, C4⋊C8.302C22, (C2×C4).550C24, (C2×C8).204C23, C4⋊Q8.179C22, C2.58(Q8⋊5D4), C2.96(D4○SD16), (C4×D4).190C22, (C2×D8).145C22, (C2×D4).265C23, C4⋊1D4.94C22, (C4×Q8).304C22, C2.D8.198C22, D4⋊C4.17C22, C22.810(C22×D4), C4.251(C2×C4○D4), (C2×C4).1098(C2×D4), SmallGroup(128,2090)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8⋊4D8
G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd=a2b, dcd=c-1 >
Subgroups: 464 in 213 conjugacy classes, 96 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, D4⋊C4, D4⋊C4, C4⋊C8, C2.D8, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C4⋊1D4, C4⋊Q8, C2×D8, C22×Q8, C2×C4○D4, C4×D8, C8×Q8, C4⋊D8, D4⋊Q8, C4.4D8, D4×Q8, Q8⋊6D4, Q8⋊4D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4○D4, C24, C2×D8, C22×D4, C2×C4○D4, 2- 1+4, Q8⋊5D4, C22×D8, D4○SD16, Q8⋊4D8
►On 64 points
(1 49 62 29)(2 50 63 30)(3 51 64 31)(4 52 57 32)(5 53 58 25)(6 54 59 26)(7 55 60 27)(8 56 61 28)(9 23 40 47)(10 24 33 48)(11 17 34 41)(12 18 35 42)(13 19 36 43)(14 20 37 44)(15 21 38 45)(16 22 39 46)
(1 46 62 22)(2 23 63 47)(3 48 64 24)(4 17 57 41)(5 42 58 18)(6 19 59 43)(7 44 60 20)(8 21 61 45)(9 30 40 50)(10 51 33 31)(11 32 34 52)(12 53 35 25)(13 26 36 54)(14 55 37 27)(15 28 38 56)(16 49 39 29)
(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 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 40)(9 61)(10 60)(11 59)(12 58)(13 57)(14 64)(15 63)(16 62)(17 26)(18 25)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 56)(48 55)
G:=sub<Sym(64)| (1,49,62,29)(2,50,63,30)(3,51,64,31)(4,52,57,32)(5,53,58,25)(6,54,59,26)(7,55,60,27)(8,56,61,28)(9,23,40,47)(10,24,33,48)(11,17,34,41)(12,18,35,42)(13,19,36,43)(14,20,37,44)(15,21,38,45)(16,22,39,46), (1,46,62,22)(2,23,63,47)(3,48,64,24)(4,17,57,41)(5,42,58,18)(6,19,59,43)(7,44,60,20)(8,21,61,45)(9,30,40,50)(10,51,33,31)(11,32,34,52)(12,53,35,25)(13,26,36,54)(14,55,37,27)(15,28,38,56)(16,49,39,29), (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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,40)(9,61)(10,60)(11,59)(12,58)(13,57)(14,64)(15,63)(16,62)(17,26)(18,25)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,56)(48,55)>;
G:=Group( (1,49,62,29)(2,50,63,30)(3,51,64,31)(4,52,57,32)(5,53,58,25)(6,54,59,26)(7,55,60,27)(8,56,61,28)(9,23,40,47)(10,24,33,48)(11,17,34,41)(12,18,35,42)(13,19,36,43)(14,20,37,44)(15,21,38,45)(16,22,39,46), (1,46,62,22)(2,23,63,47)(3,48,64,24)(4,17,57,41)(5,42,58,18)(6,19,59,43)(7,44,60,20)(8,21,61,45)(9,30,40,50)(10,51,33,31)(11,32,34,52)(12,53,35,25)(13,26,36,54)(14,55,37,27)(15,28,38,56)(16,49,39,29), (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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,40)(9,61)(10,60)(11,59)(12,58)(13,57)(14,64)(15,63)(16,62)(17,26)(18,25)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,56)(48,55) );
G=PermutationGroup([[(1,49,62,29),(2,50,63,30),(3,51,64,31),(4,52,57,32),(5,53,58,25),(6,54,59,26),(7,55,60,27),(8,56,61,28),(9,23,40,47),(10,24,33,48),(11,17,34,41),(12,18,35,42),(13,19,36,43),(14,20,37,44),(15,21,38,45),(16,22,39,46)], [(1,46,62,22),(2,23,63,47),(3,48,64,24),(4,17,57,41),(5,42,58,18),(6,19,59,43),(7,44,60,20),(8,21,61,45),(9,30,40,50),(10,51,33,31),(11,32,34,52),(12,53,35,25),(13,26,36,54),(14,55,37,27),(15,28,38,56),(16,49,39,29)], [(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)], [(1,39),(2,38),(3,37),(4,36),(5,35),(6,34),(7,33),(8,40),(9,61),(10,60),(11,59),(12,58),(13,57),(14,64),(15,63),(16,62),(17,26),(18,25),(19,32),(20,31),(21,30),(22,29),(23,28),(24,27),(41,54),(42,53),(43,52),(44,51),(45,50),(46,49),(47,56),(48,55)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4H | 4I | ··· | 4M | 4N | 4O | 4P | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 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 | D8 | 2- 1+4 | D4○SD16 |
| kernel | Q8⋊4D8 | C4×D8 | C8×Q8 | C4⋊D8 | D4⋊Q8 | C4.4D8 | D4×Q8 | Q8⋊6D4 | C4⋊C4 | C2×Q8 | D4 | Q8 | C4 | C2 |
| # reps | 1 | 3 | 1 | 3 | 3 | 3 | 1 | 1 | 3 | 1 | 4 | 8 | 1 | 2 |
Matrix representation of Q8⋊4D8 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 6 | 0 | 0 |
| 14 | 6 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 13 | 0 |
| 16 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 13 | 0 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,4],[0,14,0,0,6,6,0,0,0,0,0,13,0,0,4,0],[16,16,0,0,0,1,0,0,0,0,0,13,0,0,4,0] >;
Q8⋊4D8 in GAP, Magma, Sage, TeX
Q_8\rtimes_4D_8
% in TeX
G:=Group("Q8:4D8"); // GroupNames label
G:=SmallGroup(128,2090);
// by ID
G=gap.SmallGroup(128,2090);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,346,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^8=d^2=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations