p-group, metabelian, nilpotent (class 3), monomial
Aliases: D8⋊4Q8, C42.65C23, C4.992- 1+4, C8⋊2(C2×Q8), C8⋊Q8⋊31C2, D4⋊5(C2×Q8), (D4×Q8)⋊14C2, C2.42(D4×Q8), C4⋊C4.387D4, C8⋊4Q8⋊14C2, C8⋊3Q8⋊11C2, (C4×D8).17C2, D4.Q8⋊51C2, D4⋊Q8⋊46C2, D4⋊3Q8⋊16C2, D4⋊2Q8⋊28C2, D8⋊C4.2C2, (C2×Q8).248D4, C4.42(C22×Q8), C4⋊C4.273C23, C4⋊C8.143C22, C4.52(C8⋊C22), (C2×C8).375C23, (C2×C4).576C24, (C4×C8).200C22, C4⋊Q8.205C22, C8⋊C4.69C22, (C2×D8).168C22, (C4×D4).212C22, (C2×D4).437C23, (C4×Q8).203C22, C4.Q8.118C22, C2.D8.138C22, C2.108(D4○SD16), D4⋊C4.93C22, C22.836(C22×D4), C42.C2.74C22, (C2×C4).646(C2×D4), C2.90(C2×C8⋊C22), SmallGroup(128,2116)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D8⋊4Q8
G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a5, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 360 in 187 conjugacy classes, 96 normal (38 characteristic)
C1, C2, C2, C4, 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, D4⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C2.D8, C2.D8, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C4⋊Q8, C2×D8, C22×Q8, C4×D8, D8⋊C4, C8⋊4Q8, D4⋊Q8, D4⋊2Q8, D4⋊2Q8, D4.Q8, C8⋊3Q8, C8⋊Q8, D4×Q8, D4⋊3Q8, D8⋊4Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C24, C8⋊C22, C22×D4, C22×Q8, 2- 1+4, D4×Q8, C2×C8⋊C22, D4○SD16, D8⋊4Q8
Character table of D8⋊4Q8
| 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 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| 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 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√-2 | -2√-2 | 0 | 0 | complex lifted from D4○SD16 |
| ρ29 | 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 | complex lifted from D4○SD16 |
►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 29)(2 28)(3 27)(4 26)(5 25)(6 32)(7 31)(8 30)(9 36)(10 35)(11 34)(12 33)(13 40)(14 39)(15 38)(16 37)(17 49)(18 56)(19 55)(20 54)(21 53)(22 52)(23 51)(24 50)(41 64)(42 63)(43 62)(44 61)(45 60)(46 59)(47 58)(48 57)
(1 34 29 11)(2 35 30 12)(3 36 31 13)(4 37 32 14)(5 38 25 15)(6 39 26 16)(7 40 27 9)(8 33 28 10)(17 47 55 64)(18 48 56 57)(19 41 49 58)(20 42 50 59)(21 43 51 60)(22 44 52 61)(23 45 53 62)(24 46 54 63)
(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 24 40 54)(10 21 33 51)(11 18 34 56)(12 23 35 53)(13 20 36 50)(14 17 37 55)(15 22 38 52)(16 19 39 49)
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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,32)(7,31)(8,30)(9,36)(10,35)(11,34)(12,33)(13,40)(14,39)(15,38)(16,37)(17,49)(18,56)(19,55)(20,54)(21,53)(22,52)(23,51)(24,50)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57), (1,34,29,11)(2,35,30,12)(3,36,31,13)(4,37,32,14)(5,38,25,15)(6,39,26,16)(7,40,27,9)(8,33,28,10)(17,47,55,64)(18,48,56,57)(19,41,49,58)(20,42,50,59)(21,43,51,60)(22,44,52,61)(23,45,53,62)(24,46,54,63), (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)(9,24,40,54)(10,21,33,51)(11,18,34,56)(12,23,35,53)(13,20,36,50)(14,17,37,55)(15,22,38,52)(16,19,39,49)>;
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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,32)(7,31)(8,30)(9,36)(10,35)(11,34)(12,33)(13,40)(14,39)(15,38)(16,37)(17,49)(18,56)(19,55)(20,54)(21,53)(22,52)(23,51)(24,50)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57), (1,34,29,11)(2,35,30,12)(3,36,31,13)(4,37,32,14)(5,38,25,15)(6,39,26,16)(7,40,27,9)(8,33,28,10)(17,47,55,64)(18,48,56,57)(19,41,49,58)(20,42,50,59)(21,43,51,60)(22,44,52,61)(23,45,53,62)(24,46,54,63), (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)(9,24,40,54)(10,21,33,51)(11,18,34,56)(12,23,35,53)(13,20,36,50)(14,17,37,55)(15,22,38,52)(16,19,39,49) );
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)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,32),(7,31),(8,30),(9,36),(10,35),(11,34),(12,33),(13,40),(14,39),(15,38),(16,37),(17,49),(18,56),(19,55),(20,54),(21,53),(22,52),(23,51),(24,50),(41,64),(42,63),(43,62),(44,61),(45,60),(46,59),(47,58),(48,57)], [(1,34,29,11),(2,35,30,12),(3,36,31,13),(4,37,32,14),(5,38,25,15),(6,39,26,16),(7,40,27,9),(8,33,28,10),(17,47,55,64),(18,48,56,57),(19,41,49,58),(20,42,50,59),(21,43,51,60),(22,44,52,61),(23,45,53,62),(24,46,54,63)], [(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),(9,24,40,54),(10,21,33,51),(11,18,34,56),(12,23,35,53),(13,20,36,50),(14,17,37,55),(15,22,38,52),(16,19,39,49)]])
Matrix representation of D8⋊4Q8 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 8 | 14 | 0 | 0 | 0 | 0 |
| 16 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 16 | 0 |
| 0 | 0 | 0 | 7 | 0 | 16 |
| 0 | 0 | 16 | 0 | 10 | 0 |
| 0 | 0 | 0 | 16 | 0 | 10 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,12,12,0,0,0,0,5,12,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[1,16,0,0,0,0,2,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[8,16,0,0,0,0,14,9,0,0,0,0,0,0,7,0,16,0,0,0,0,7,0,16,0,0,16,0,10,0,0,0,0,16,0,10] >;
D8⋊4Q8 in GAP, Magma, Sage, TeX
D_8\rtimes_4Q_8
% in TeX
G:=Group("D8:4Q8"); // GroupNames label
G:=SmallGroup(128,2116);
// by ID
G=gap.SmallGroup(128,2116);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,723,346,80,4037,1027,124]);
// 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,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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