p-group, metabelian, nilpotent (class 3), monomial
Aliases: D8⋊5Q8, C42.70C23, C4.1042- 1+4, C8⋊Q8⋊35C2, C8.9(C2×Q8), C2.47(D4×Q8), C4⋊C4.392D4, C8⋊4Q8⋊18C2, (C4×D8).18C2, D4.Q8⋊52C2, D4.14(C2×Q8), D4⋊3Q8⋊17C2, D4⋊Q8⋊47C2, D8⋊C4.3C2, C2.69(D4○D8), (C2×Q8).141D4, C8.5Q8⋊25C2, C4.47(C22×Q8), C4⋊C8.148C22, C4⋊C4.278C23, (C2×C8).216C23, (C4×C8).205C22, (C2×C4).581C24, C4⋊Q8.210C22, C8⋊C4.74C22, C4.Q8.79C22, C2.D8.76C22, (C2×D4).441C23, (C4×D4).216C22, (C2×D8).169C22, (C4×Q8).208C22, D4⋊C4.97C22, C22.841(C22×D4), C42.C2.79C22, C2.106(D8⋊C22), (C2×C4).651(C2×D4), SmallGroup(128,2121)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D8⋊5Q8
G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a5, cbc-1=a4b, bd=db, dcd-1=c-1 >
Subgroups: 328 in 174 conjugacy classes, 94 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C2×Q8, C2×Q8, C4×C8, C8⋊C4, D4⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C2.D8, C2.D8, C2×C4⋊C4, C4×D4, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C2×D8, C4×D8, D8⋊C4, C8⋊4Q8, D4⋊Q8, D4.Q8, C8.5Q8, C8⋊Q8, D4⋊3Q8, D8⋊5Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C24, C22×D4, C22×Q8, 2- 1+4, D4×Q8, D8⋊C22, D4○D8, D8⋊5Q8
Character table of D8⋊5Q8
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ10 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | linear of order 2 |
| ρ11 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | linear of order 2 |
| ρ12 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ13 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ14 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | linear of order 2 |
| ρ15 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | linear of order 2 |
| ρ16 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ17 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ22 | 2 | -2 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ23 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ24 | 2 | -2 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | orthogonal lifted from D4○D8 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | orthogonal lifted from D4○D8 |
| ρ27 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
| ρ28 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| ρ29 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
►On 64 points
Generators in S64
(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 25)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(16 40)(17 53)(18 52)(19 51)(20 50)(21 49)(22 56)(23 55)(24 54)(41 60)(42 59)(43 58)(44 57)(45 64)(46 63)(47 62)(48 61)
(1 36 29 12)(2 37 30 13)(3 38 31 14)(4 39 32 15)(5 40 25 16)(6 33 26 9)(7 34 27 10)(8 35 28 11)(17 45 55 62)(18 46 56 63)(19 47 49 64)(20 48 50 57)(21 41 51 58)(22 42 52 59)(23 43 53 60)(24 44 54 61)
(1 57 29 48)(2 62 30 45)(3 59 31 42)(4 64 32 47)(5 61 25 44)(6 58 26 41)(7 63 27 46)(8 60 28 43)(9 21 33 51)(10 18 34 56)(11 23 35 53)(12 20 36 50)(13 17 37 55)(14 22 38 52)(15 19 39 49)(16 24 40 54)
G:=sub<Sym(64)| (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), (1,25)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,40)(17,53)(18,52)(19,51)(20,50)(21,49)(22,56)(23,55)(24,54)(41,60)(42,59)(43,58)(44,57)(45,64)(46,63)(47,62)(48,61), (1,36,29,12)(2,37,30,13)(3,38,31,14)(4,39,32,15)(5,40,25,16)(6,33,26,9)(7,34,27,10)(8,35,28,11)(17,45,55,62)(18,46,56,63)(19,47,49,64)(20,48,50,57)(21,41,51,58)(22,42,52,59)(23,43,53,60)(24,44,54,61), (1,57,29,48)(2,62,30,45)(3,59,31,42)(4,64,32,47)(5,61,25,44)(6,58,26,41)(7,63,27,46)(8,60,28,43)(9,21,33,51)(10,18,34,56)(11,23,35,53)(12,20,36,50)(13,17,37,55)(14,22,38,52)(15,19,39,49)(16,24,40,54)>;
G:=Group( (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), (1,25)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,40)(17,53)(18,52)(19,51)(20,50)(21,49)(22,56)(23,55)(24,54)(41,60)(42,59)(43,58)(44,57)(45,64)(46,63)(47,62)(48,61), (1,36,29,12)(2,37,30,13)(3,38,31,14)(4,39,32,15)(5,40,25,16)(6,33,26,9)(7,34,27,10)(8,35,28,11)(17,45,55,62)(18,46,56,63)(19,47,49,64)(20,48,50,57)(21,41,51,58)(22,42,52,59)(23,43,53,60)(24,44,54,61), (1,57,29,48)(2,62,30,45)(3,59,31,42)(4,64,32,47)(5,61,25,44)(6,58,26,41)(7,63,27,46)(8,60,28,43)(9,21,33,51)(10,18,34,56)(11,23,35,53)(12,20,36,50)(13,17,37,55)(14,22,38,52)(15,19,39,49)(16,24,40,54) );
G=PermutationGroup([[(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)], [(1,25),(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,39),(10,38),(11,37),(12,36),(13,35),(14,34),(15,33),(16,40),(17,53),(18,52),(19,51),(20,50),(21,49),(22,56),(23,55),(24,54),(41,60),(42,59),(43,58),(44,57),(45,64),(46,63),(47,62),(48,61)], [(1,36,29,12),(2,37,30,13),(3,38,31,14),(4,39,32,15),(5,40,25,16),(6,33,26,9),(7,34,27,10),(8,35,28,11),(17,45,55,62),(18,46,56,63),(19,47,49,64),(20,48,50,57),(21,41,51,58),(22,42,52,59),(23,43,53,60),(24,44,54,61)], [(1,57,29,48),(2,62,30,45),(3,59,31,42),(4,64,32,47),(5,61,25,44),(6,58,26,41),(7,63,27,46),(8,60,28,43),(9,21,33,51),(10,18,34,56),(11,23,35,53),(12,20,36,50),(13,17,37,55),(14,22,38,52),(15,19,39,49),(16,24,40,54)]])
Matrix representation of D8⋊5Q8 ►in GL8(𝔽17)
| 4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 16 | 2 | 0 | 0 | 0 | 0 |
| 16 | 5 | 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 16 | 16 | 15 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 1 | 2 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 16 |
| 13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 14 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 | 4 | 8 |
| 0 | 0 | 0 | 0 | 0 | 13 | 13 | 13 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 6 | 0 | 6 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 6 | 16 | 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 2 | 0 | 4 |
| 0 | 0 | 0 | 0 | 15 | 1 | 13 | 13 |
| 0 | 0 | 0 | 0 | 2 | 2 | 14 | 0 |
| 0 | 0 | 0 | 0 | 12 | 14 | 2 | 14 |
G:=sub<GL(8,GF(17))| [4,0,12,16,0,0,0,0,2,13,0,5,0,0,0,0,0,0,16,16,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,16,0,1,0,0,0,0,0,16,0,0,1,0,0,0,0,16,1,0,0,0,0,0,0,15,0,0,1],[16,4,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,1,0,0,0,0,0,0,1,1,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,16],[13,0,0,14,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,13,4,0,0,0,0,0,4,0,4,13,0,0,0,0,0,0,4,13,0,0,0,0,0,0,8,13],[0,6,16,6,0,0,0,0,0,0,0,16,0,0,0,0,1,6,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,5,15,2,12,0,0,0,0,2,1,2,14,0,0,0,0,0,13,14,2,0,0,0,0,4,13,0,14] >;
D8⋊5Q8 in GAP, Magma, Sage, TeX
D_8\rtimes_5Q_8
% in TeX
G:=Group("D8:5Q8"); // GroupNames label
G:=SmallGroup(128,2121);
// by ID
G=gap.SmallGroup(128,2121);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,120,758,723,346,304,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^5,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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