p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8.6D8, C42.202C23, D4⋊C8⋊9C2, Q8⋊C8⋊11C2, C4⋊C4.55D4, C4.29(C2×D8), D4⋊Q8⋊2C2, C4⋊2Q16⋊2C2, (C2×D4).51D4, C4.D8⋊3C2, C4.4D8⋊5C2, C4.59(C4○D8), C4⋊C8.11C22, (C4×C8).23C22, Q8⋊6D4.2C2, (C2×Q8).199D4, C4⋊Q8.22C22, C4.63(C8⋊C22), (C4×D4).32C22, (C4×Q8).32C22, C2.15(C22⋊D8), C4⋊1D4.19C22, C4.64(C8.C22), C22.168C22≀C2, C2.20(D4.7D4), C2.14(D4.9D4), (C2×C4).959(C2×D4), SmallGroup(128,373)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8.D8
G = < a,b,c,d | a4=c8=1, b2=d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=dbd-1=a-1b, dcd-1=a2c-1 >
Subgroups: 328 in 125 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, Q16, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, D4⋊C4, Q8⋊C4, C4⋊C8, C2.D8, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊1D4, C4⋊1D4, C4⋊Q8, C2×Q16, C2×C4○D4, D4⋊C8, Q8⋊C8, C4.D8, C4⋊2Q16, D4⋊Q8, C4.4D8, Q8⋊6D4, Q8.D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C22≀C2, C2×D8, C4○D8, C8⋊C22, C8.C22, C22⋊D8, D4.7D4, D4.9D4, Q8.D8
Character table of Q8.D8
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 16 | 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 | 2 | 0 | 0 | 2 | -2 | 2 | -2 | -2 | 0 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | -2 | 2 | -2 | -2 | 0 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | -2 | 2 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | -2 | -2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | 0 | √2 | 0 | -√2 | orthogonal lifted from D8 |
| ρ16 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | 0 | -√2 | 0 | √2 | orthogonal lifted from D8 |
| ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | 0 | √2 | 0 | -√2 | orthogonal lifted from D8 |
| ρ18 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | 0 | -√2 | 0 | √2 | orthogonal lifted from D8 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | √-2 | √-2 | -√-2 | -√-2 | √2 | 0 | -√2 | 0 | complex lifted from C4○D8 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | -√-2 | -√-2 | √-2 | √-2 | √2 | 0 | -√2 | 0 | complex lifted from C4○D8 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | √-2 | √-2 | -√-2 | -√-2 | -√2 | 0 | √2 | 0 | complex lifted from C4○D8 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | -√-2 | -√-2 | √-2 | √-2 | -√2 | 0 | √2 | 0 | complex lifted from C4○D8 |
| ρ23 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | complex lifted from D4.9D4 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | complex lifted from D4.9D4 |
►On 64 points
Generators in S64
(1 9 32 61)(2 10 25 62)(3 11 26 63)(4 12 27 64)(5 13 28 57)(6 14 29 58)(7 15 30 59)(8 16 31 60)(17 35 51 42)(18 36 52 43)(19 37 53 44)(20 38 54 45)(21 39 55 46)(22 40 56 47)(23 33 49 48)(24 34 50 41)
(1 22 32 56)(2 33 25 48)(3 50 26 24)(4 42 27 35)(5 18 28 52)(6 37 29 44)(7 54 30 20)(8 46 31 39)(9 47 61 40)(10 23 62 49)(11 34 63 41)(12 51 64 17)(13 43 57 36)(14 19 58 53)(15 38 59 45)(16 55 60 21)
(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 31 32 8)(2 7 25 30)(3 29 26 6)(4 5 27 28)(9 16 61 60)(10 59 62 15)(11 14 63 58)(12 57 64 13)(17 18 51 52)(19 24 53 50)(20 49 54 23)(21 22 55 56)(33 45 48 38)(34 37 41 44)(35 43 42 36)(39 47 46 40)
G:=sub<Sym(64)| (1,9,32,61)(2,10,25,62)(3,11,26,63)(4,12,27,64)(5,13,28,57)(6,14,29,58)(7,15,30,59)(8,16,31,60)(17,35,51,42)(18,36,52,43)(19,37,53,44)(20,38,54,45)(21,39,55,46)(22,40,56,47)(23,33,49,48)(24,34,50,41), (1,22,32,56)(2,33,25,48)(3,50,26,24)(4,42,27,35)(5,18,28,52)(6,37,29,44)(7,54,30,20)(8,46,31,39)(9,47,61,40)(10,23,62,49)(11,34,63,41)(12,51,64,17)(13,43,57,36)(14,19,58,53)(15,38,59,45)(16,55,60,21), (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), (1,31,32,8)(2,7,25,30)(3,29,26,6)(4,5,27,28)(9,16,61,60)(10,59,62,15)(11,14,63,58)(12,57,64,13)(17,18,51,52)(19,24,53,50)(20,49,54,23)(21,22,55,56)(33,45,48,38)(34,37,41,44)(35,43,42,36)(39,47,46,40)>;
G:=Group( (1,9,32,61)(2,10,25,62)(3,11,26,63)(4,12,27,64)(5,13,28,57)(6,14,29,58)(7,15,30,59)(8,16,31,60)(17,35,51,42)(18,36,52,43)(19,37,53,44)(20,38,54,45)(21,39,55,46)(22,40,56,47)(23,33,49,48)(24,34,50,41), (1,22,32,56)(2,33,25,48)(3,50,26,24)(4,42,27,35)(5,18,28,52)(6,37,29,44)(7,54,30,20)(8,46,31,39)(9,47,61,40)(10,23,62,49)(11,34,63,41)(12,51,64,17)(13,43,57,36)(14,19,58,53)(15,38,59,45)(16,55,60,21), (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), (1,31,32,8)(2,7,25,30)(3,29,26,6)(4,5,27,28)(9,16,61,60)(10,59,62,15)(11,14,63,58)(12,57,64,13)(17,18,51,52)(19,24,53,50)(20,49,54,23)(21,22,55,56)(33,45,48,38)(34,37,41,44)(35,43,42,36)(39,47,46,40) );
G=PermutationGroup([[(1,9,32,61),(2,10,25,62),(3,11,26,63),(4,12,27,64),(5,13,28,57),(6,14,29,58),(7,15,30,59),(8,16,31,60),(17,35,51,42),(18,36,52,43),(19,37,53,44),(20,38,54,45),(21,39,55,46),(22,40,56,47),(23,33,49,48),(24,34,50,41)], [(1,22,32,56),(2,33,25,48),(3,50,26,24),(4,42,27,35),(5,18,28,52),(6,37,29,44),(7,54,30,20),(8,46,31,39),(9,47,61,40),(10,23,62,49),(11,34,63,41),(12,51,64,17),(13,43,57,36),(14,19,58,53),(15,38,59,45),(16,55,60,21)], [(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)], [(1,31,32,8),(2,7,25,30),(3,29,26,6),(4,5,27,28),(9,16,61,60),(10,59,62,15),(11,14,63,58),(12,57,64,13),(17,18,51,52),(19,24,53,50),(20,49,54,23),(21,22,55,56),(33,45,48,38),(34,37,41,44),(35,43,42,36),(39,47,46,40)]])
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 |
| 13 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 5 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 6 | 6 |
| 0 | 0 | 14 | 0 |
| 12 | 5 | 0 | 0 |
| 5 | 5 | 0 | 0 |
| 0 | 0 | 6 | 6 |
| 0 | 0 | 14 | 11 |
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[13,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[12,12,0,0,5,12,0,0,0,0,6,14,0,0,6,0],[12,5,0,0,5,5,0,0,0,0,6,14,0,0,6,11] >;
Q8.D8 in GAP, Magma, Sage, TeX
Q_8.D_8
% in TeX
G:=Group("Q8.D8"); // GroupNames label
G:=SmallGroup(128,373);
// by ID
G=gap.SmallGroup(128,373);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,422,352,1123,570,521,136,2804,1411,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^8=1,b^2=d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=d*b*d^-1=a^-1*b,d*c*d^-1=a^2*c^-1>;
// generators/relations
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