p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8⋊2D8, C42.182C23, Q8⋊C8⋊7C2, D4⋊C8⋊10C2, C4⋊D8⋊2C2, C8⋊4D4⋊2C2, C4⋊C4.47D4, C4.25(C2×D8), Q8⋊6D4⋊1C2, (C2×D4).43D4, C4.D8⋊6C2, C4⋊SD16⋊30C2, C4.80(C4○D8), (C4×C8).41C22, (C2×Q8).193D4, C4⋊C8.158C22, C4.56(C8⋊C22), (C4×D4).18C22, C4⋊1D4.6C22, (C4×Q8).18C22, C2.11(C22⋊D8), C2.14(D4⋊4D4), C2.13(D4⋊D4), C22.148C22≀C2, (C2×C4).939(C2×D4), SmallGroup(128,353)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8⋊D8
G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, cbc-1=dbd=a-1b, dcd=c-1 >
Subgroups: 400 in 139 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C4×C8, D4⋊C4, C4⋊C8, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊1D4, C4⋊1D4, C2×D8, C2×SD16, C2×C4○D4, D4⋊C8, Q8⋊C8, C4.D8, C4⋊D8, C4⋊SD16, C8⋊4D4, Q8⋊6D4, Q8⋊D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C22≀C2, C2×D8, C4○D8, C8⋊C22, C22⋊D8, D4⋊D4, D4⋊4D4, Q8⋊D8
Character table of Q8⋊D8
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 16 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 2 | 0 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | -2 | 0 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | -2 | 2 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | -2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | √2 | -√2 | √2 | -√2 | 0 | -√2 | 0 | √2 | orthogonal lifted from D8 |
| ρ16 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | -√2 | √2 | -√2 | √2 | 0 | √2 | 0 | -√2 | orthogonal lifted from D8 |
| ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | √2 | -√2 | √2 | -√2 | 0 | √2 | 0 | -√2 | orthogonal lifted from D8 |
| ρ18 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | -√2 | √2 | -√2 | √2 | 0 | -√2 | 0 | √2 | orthogonal lifted from D8 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | -√2 | -√2 | √2 | √2 | √-2 | 0 | -√-2 | 0 | complex lifted from C4○D8 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | -√2 | -√2 | √2 | √2 | -√-2 | 0 | √-2 | 0 | complex lifted from C4○D8 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | √2 | √2 | -√2 | -√2 | √-2 | 0 | -√-2 | 0 | complex lifted from C4○D8 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | √2 | √2 | -√2 | -√2 | -√-2 | 0 | √-2 | 0 | complex lifted from C4○D8 |
| ρ23 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊4D4 |
| ρ24 | 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 | orthogonal lifted from C8⋊C22 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊4D4 |
| ρ26 | 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 | orthogonal lifted from C8⋊C22 |
►On 64 points
Generators in S64
(1 63 30 48)(2 64 31 41)(3 57 32 42)(4 58 25 43)(5 59 26 44)(6 60 27 45)(7 61 28 46)(8 62 29 47)(9 56 35 18)(10 49 36 19)(11 50 37 20)(12 51 38 21)(13 52 39 22)(14 53 40 23)(15 54 33 24)(16 55 34 17)
(1 16 30 34)(2 56 31 18)(3 36 32 10)(4 20 25 50)(5 12 26 38)(6 52 27 22)(7 40 28 14)(8 24 29 54)(9 41 35 64)(11 58 37 43)(13 45 39 60)(15 62 33 47)(17 48 55 63)(19 57 49 42)(21 44 51 59)(23 61 53 46)
(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)
(2 8)(3 7)(4 6)(9 54)(10 53)(11 52)(12 51)(13 50)(14 49)(15 56)(16 55)(17 34)(18 33)(19 40)(20 39)(21 38)(22 37)(23 36)(24 35)(25 27)(28 32)(29 31)(41 62)(42 61)(43 60)(44 59)(45 58)(46 57)(47 64)(48 63)
G:=sub<Sym(64)| (1,63,30,48)(2,64,31,41)(3,57,32,42)(4,58,25,43)(5,59,26,44)(6,60,27,45)(7,61,28,46)(8,62,29,47)(9,56,35,18)(10,49,36,19)(11,50,37,20)(12,51,38,21)(13,52,39,22)(14,53,40,23)(15,54,33,24)(16,55,34,17), (1,16,30,34)(2,56,31,18)(3,36,32,10)(4,20,25,50)(5,12,26,38)(6,52,27,22)(7,40,28,14)(8,24,29,54)(9,41,35,64)(11,58,37,43)(13,45,39,60)(15,62,33,47)(17,48,55,63)(19,57,49,42)(21,44,51,59)(23,61,53,46), (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), (2,8)(3,7)(4,6)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,56)(16,55)(17,34)(18,33)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,27)(28,32)(29,31)(41,62)(42,61)(43,60)(44,59)(45,58)(46,57)(47,64)(48,63)>;
G:=Group( (1,63,30,48)(2,64,31,41)(3,57,32,42)(4,58,25,43)(5,59,26,44)(6,60,27,45)(7,61,28,46)(8,62,29,47)(9,56,35,18)(10,49,36,19)(11,50,37,20)(12,51,38,21)(13,52,39,22)(14,53,40,23)(15,54,33,24)(16,55,34,17), (1,16,30,34)(2,56,31,18)(3,36,32,10)(4,20,25,50)(5,12,26,38)(6,52,27,22)(7,40,28,14)(8,24,29,54)(9,41,35,64)(11,58,37,43)(13,45,39,60)(15,62,33,47)(17,48,55,63)(19,57,49,42)(21,44,51,59)(23,61,53,46), (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), (2,8)(3,7)(4,6)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,56)(16,55)(17,34)(18,33)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,27)(28,32)(29,31)(41,62)(42,61)(43,60)(44,59)(45,58)(46,57)(47,64)(48,63) );
G=PermutationGroup([[(1,63,30,48),(2,64,31,41),(3,57,32,42),(4,58,25,43),(5,59,26,44),(6,60,27,45),(7,61,28,46),(8,62,29,47),(9,56,35,18),(10,49,36,19),(11,50,37,20),(12,51,38,21),(13,52,39,22),(14,53,40,23),(15,54,33,24),(16,55,34,17)], [(1,16,30,34),(2,56,31,18),(3,36,32,10),(4,20,25,50),(5,12,26,38),(6,52,27,22),(7,40,28,14),(8,24,29,54),(9,41,35,64),(11,58,37,43),(13,45,39,60),(15,62,33,47),(17,48,55,63),(19,57,49,42),(21,44,51,59),(23,61,53,46)], [(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)], [(2,8),(3,7),(4,6),(9,54),(10,53),(11,52),(12,51),(13,50),(14,49),(15,56),(16,55),(17,34),(18,33),(19,40),(20,39),(21,38),(22,37),(23,36),(24,35),(25,27),(28,32),(29,31),(41,62),(42,61),(43,60),(44,59),(45,58),(46,57),(47,64),(48,63)]])
Matrix representation of Q8⋊D8 ►in GL4(𝔽17) generated by
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 12 | 0 | 0 |
| 12 | 5 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 3 | 14 | 0 | 0 |
| 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 11 |
| 0 | 0 | 3 | 6 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 16 | 16 |
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[12,12,0,0,12,5,0,0,0,0,1,0,0,0,0,1],[3,3,0,0,14,3,0,0,0,0,0,3,0,0,11,6],[1,0,0,0,0,16,0,0,0,0,1,16,0,0,0,16] >;
Q8⋊D8 in GAP, Magma, Sage, TeX
Q_8\rtimes D_8
% in TeX
G:=Group("Q8:D8"); // GroupNames label
G:=SmallGroup(128,353);
// by ID
G=gap.SmallGroup(128,353);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,422,352,1123,570,521,136,2804,1411,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^8=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,c*b*c^-1=d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations
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