p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8⋊3D8, D4⋊6SD16, C42.188C23, Q8⋊C8⋊8C2, (D4×Q8)⋊1C2, D4⋊C8⋊16C2, C4⋊C4.24D4, C4.27(C2×D8), C4⋊D8.2C2, C4.4D8⋊2C2, C4⋊SD16⋊32C2, (C2×D4).250D4, C4.10D8⋊8C2, (C4×C8).16C22, (C2×Q8).195D4, C4.26(C2×SD16), C4⋊Q8.10C22, C4⋊C8.161C22, C4.59(C8⋊C22), (C4×D4).22C22, (C4×Q8).22C22, C2.13(C22⋊D8), C2.13(Q8⋊D4), C4⋊1D4.12C22, C4.58(C8.C22), C2.14(D4.8D4), C22.154C22≀C2, (C2×C4).945(C2×D4), SmallGroup(128,359)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8⋊3D8
G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=cac-1=dad=a-1, cbc-1=dbd=a-1b, dcd=c-1 >
Subgroups: 320 in 125 conjugacy classes, 38 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4×C8, D4⋊C4, C4⋊C8, C4×D4, C4×D4, C4×Q8, C22⋊Q8, C4⋊1D4, C4⋊Q8, C4⋊Q8, C2×D8, C2×SD16, C22×Q8, D4⋊C8, Q8⋊C8, C4.10D8, C4⋊D8, C4⋊SD16, C4.4D8, D4×Q8, Q8⋊3D8
Quotients: C1, C2, C22, D4, C23, D8, SD16, C2×D4, C22≀C2, C2×D8, C2×SD16, C8⋊C22, C8.C22, C22⋊D8, Q8⋊D4, D4.8D4, Q8⋊3D8
Character table of Q8⋊3D8
| 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 | 4 | 4 | 16 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 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 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 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 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | 0 | -√2 | √2 | 0 | orthogonal lifted from D8 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | 0 | √2 | -√2 | 0 | orthogonal lifted from D8 |
| ρ17 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | 0 | √2 | -√2 | 0 | orthogonal lifted from D8 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | 0 | -√2 | √2 | 0 | orthogonal lifted from D8 |
| ρ19 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | √-2 | -√-2 | -√-2 | 0 | 0 | √-2 | complex lifted from SD16 |
| ρ20 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√-2 | -√-2 | √-2 | -√-2 | 0 | 0 | √-2 | complex lifted from SD16 |
| ρ21 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√-2 | -√-2 | √-2 | √-2 | 0 | 0 | -√-2 | complex lifted from SD16 |
| ρ22 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | √-2 | -√-2 | √-2 | 0 | 0 | -√-2 | complex lifted from SD16 |
| ρ23 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | -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 | 4 | 0 | 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 D4.8D4 |
| 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 D4.8D4 |
►On 64 points
Generators in S64
(1 57 54 34)(2 35 55 58)(3 59 56 36)(4 37 49 60)(5 61 50 38)(6 39 51 62)(7 63 52 40)(8 33 53 64)(9 41 31 24)(10 17 32 42)(11 43 25 18)(12 19 26 44)(13 45 27 20)(14 21 28 46)(15 47 29 22)(16 23 30 48)
(1 43 54 18)(2 26 55 12)(3 45 56 20)(4 28 49 14)(5 47 50 22)(6 30 51 16)(7 41 52 24)(8 32 53 10)(9 40 31 63)(11 34 25 57)(13 36 27 59)(15 38 29 61)(17 64 42 33)(19 58 44 35)(21 60 46 37)(23 62 48 39)
(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 20)(10 19)(11 18)(12 17)(13 24)(14 23)(15 22)(16 21)(25 43)(26 42)(27 41)(28 48)(29 47)(30 46)(31 45)(32 44)(33 58)(34 57)(35 64)(36 63)(37 62)(38 61)(39 60)(40 59)(49 51)(52 56)(53 55)
G:=sub<Sym(64)| (1,57,54,34)(2,35,55,58)(3,59,56,36)(4,37,49,60)(5,61,50,38)(6,39,51,62)(7,63,52,40)(8,33,53,64)(9,41,31,24)(10,17,32,42)(11,43,25,18)(12,19,26,44)(13,45,27,20)(14,21,28,46)(15,47,29,22)(16,23,30,48), (1,43,54,18)(2,26,55,12)(3,45,56,20)(4,28,49,14)(5,47,50,22)(6,30,51,16)(7,41,52,24)(8,32,53,10)(9,40,31,63)(11,34,25,57)(13,36,27,59)(15,38,29,61)(17,64,42,33)(19,58,44,35)(21,60,46,37)(23,62,48,39), (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)(9,20)(10,19)(11,18)(12,17)(13,24)(14,23)(15,22)(16,21)(25,43)(26,42)(27,41)(28,48)(29,47)(30,46)(31,45)(32,44)(33,58)(34,57)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(49,51)(52,56)(53,55)>;
G:=Group( (1,57,54,34)(2,35,55,58)(3,59,56,36)(4,37,49,60)(5,61,50,38)(6,39,51,62)(7,63,52,40)(8,33,53,64)(9,41,31,24)(10,17,32,42)(11,43,25,18)(12,19,26,44)(13,45,27,20)(14,21,28,46)(15,47,29,22)(16,23,30,48), (1,43,54,18)(2,26,55,12)(3,45,56,20)(4,28,49,14)(5,47,50,22)(6,30,51,16)(7,41,52,24)(8,32,53,10)(9,40,31,63)(11,34,25,57)(13,36,27,59)(15,38,29,61)(17,64,42,33)(19,58,44,35)(21,60,46,37)(23,62,48,39), (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)(9,20)(10,19)(11,18)(12,17)(13,24)(14,23)(15,22)(16,21)(25,43)(26,42)(27,41)(28,48)(29,47)(30,46)(31,45)(32,44)(33,58)(34,57)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(49,51)(52,56)(53,55) );
G=PermutationGroup([[(1,57,54,34),(2,35,55,58),(3,59,56,36),(4,37,49,60),(5,61,50,38),(6,39,51,62),(7,63,52,40),(8,33,53,64),(9,41,31,24),(10,17,32,42),(11,43,25,18),(12,19,26,44),(13,45,27,20),(14,21,28,46),(15,47,29,22),(16,23,30,48)], [(1,43,54,18),(2,26,55,12),(3,45,56,20),(4,28,49,14),(5,47,50,22),(6,30,51,16),(7,41,52,24),(8,32,53,10),(9,40,31,63),(11,34,25,57),(13,36,27,59),(15,38,29,61),(17,64,42,33),(19,58,44,35),(21,60,46,37),(23,62,48,39)], [(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),(9,20),(10,19),(11,18),(12,17),(13,24),(14,23),(15,22),(16,21),(25,43),(26,42),(27,41),(28,48),(29,47),(30,46),(31,45),(32,44),(33,58),(34,57),(35,64),(36,63),(37,62),(38,61),(39,60),(40,59),(49,51),(52,56),(53,55)]])
Matrix representation of Q8⋊3D8 ►in GL4(𝔽17) generated by
| 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 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 14 | 3 |
| 0 | 0 | 14 | 14 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [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],[0,1,0,0,1,0,0,0,0,0,14,14,0,0,3,14],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,16] >;
Q8⋊3D8 in GAP, Magma, Sage, TeX
Q_8\rtimes_3D_8
% in TeX
G:=Group("Q8:3D8"); // GroupNames label
G:=SmallGroup(128,359);
// by ID
G=gap.SmallGroup(128,359);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,456,422,352,1123,570,521,136,2804,1411,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^8=d^2=1,b^2=a^2,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations
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