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 [×3], C2 [×3], C4 [×4], C4 [×7], C22, C22 [×7], C8 [×3], C2×C4 [×3], C2×C4 [×11], D4 [×2], D4 [×7], Q8 [×2], Q8 [×7], C23 [×2], C42, C42, C22⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×3], D8 [×2], SD16 [×2], C22×C4 [×3], C2×D4, C2×D4 [×3], C2×Q8, C2×Q8 [×7], C4×C8, D4⋊C4 [×4], C4⋊C8 [×2], C4×D4, C4×D4, C4×Q8, C22⋊Q8 [×3], 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 [×7], C22 [×7], D4 [×6], C23, D8 [×2], SD16 [×2], C2×D4 [×3], 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 |
(1 10 54 32)(2 25 55 11)(3 12 56 26)(4 27 49 13)(5 14 50 28)(6 29 51 15)(7 16 52 30)(8 31 53 9)(17 33 42 58)(18 59 43 34)(19 35 44 60)(20 61 45 36)(21 37 46 62)(22 63 47 38)(23 39 48 64)(24 57 41 40)
(1 43 54 18)(2 35 55 60)(3 45 56 20)(4 37 49 62)(5 47 50 22)(6 39 51 64)(7 41 52 24)(8 33 53 58)(9 42 31 17)(10 59 32 34)(11 44 25 19)(12 61 26 36)(13 46 27 21)(14 63 28 38)(15 48 29 23)(16 57 30 40)
(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 25)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)(17 60)(18 59)(19 58)(20 57)(21 64)(22 63)(23 62)(24 61)(33 44)(34 43)(35 42)(36 41)(37 48)(38 47)(39 46)(40 45)(49 51)(52 56)(53 55)
G:=sub<Sym(64)| (1,10,54,32)(2,25,55,11)(3,12,56,26)(4,27,49,13)(5,14,50,28)(6,29,51,15)(7,16,52,30)(8,31,53,9)(17,33,42,58)(18,59,43,34)(19,35,44,60)(20,61,45,36)(21,37,46,62)(22,63,47,38)(23,39,48,64)(24,57,41,40), (1,43,54,18)(2,35,55,60)(3,45,56,20)(4,37,49,62)(5,47,50,22)(6,39,51,64)(7,41,52,24)(8,33,53,58)(9,42,31,17)(10,59,32,34)(11,44,25,19)(12,61,26,36)(13,46,27,21)(14,63,28,38)(15,48,29,23)(16,57,30,40), (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)(9,25)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,60)(18,59)(19,58)(20,57)(21,64)(22,63)(23,62)(24,61)(33,44)(34,43)(35,42)(36,41)(37,48)(38,47)(39,46)(40,45)(49,51)(52,56)(53,55)>;
G:=Group( (1,10,54,32)(2,25,55,11)(3,12,56,26)(4,27,49,13)(5,14,50,28)(6,29,51,15)(7,16,52,30)(8,31,53,9)(17,33,42,58)(18,59,43,34)(19,35,44,60)(20,61,45,36)(21,37,46,62)(22,63,47,38)(23,39,48,64)(24,57,41,40), (1,43,54,18)(2,35,55,60)(3,45,56,20)(4,37,49,62)(5,47,50,22)(6,39,51,64)(7,41,52,24)(8,33,53,58)(9,42,31,17)(10,59,32,34)(11,44,25,19)(12,61,26,36)(13,46,27,21)(14,63,28,38)(15,48,29,23)(16,57,30,40), (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)(9,25)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,60)(18,59)(19,58)(20,57)(21,64)(22,63)(23,62)(24,61)(33,44)(34,43)(35,42)(36,41)(37,48)(38,47)(39,46)(40,45)(49,51)(52,56)(53,55) );
G=PermutationGroup([(1,10,54,32),(2,25,55,11),(3,12,56,26),(4,27,49,13),(5,14,50,28),(6,29,51,15),(7,16,52,30),(8,31,53,9),(17,33,42,58),(18,59,43,34),(19,35,44,60),(20,61,45,36),(21,37,46,62),(22,63,47,38),(23,39,48,64),(24,57,41,40)], [(1,43,54,18),(2,35,55,60),(3,45,56,20),(4,37,49,62),(5,47,50,22),(6,39,51,64),(7,41,52,24),(8,33,53,58),(9,42,31,17),(10,59,32,34),(11,44,25,19),(12,61,26,36),(13,46,27,21),(14,63,28,38),(15,48,29,23),(16,57,30,40)], [(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),(9,25),(10,32),(11,31),(12,30),(13,29),(14,28),(15,27),(16,26),(17,60),(18,59),(19,58),(20,57),(21,64),(22,63),(23,62),(24,61),(33,44),(34,43),(35,42),(36,41),(37,48),(38,47),(39,46),(40,45),(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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