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## G = Q8⋊3D8order 128 = 27

### 2nd semidirect product of Q8 and D8 acting via D8/D4=C2

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C42 — Q8⋊3D8
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×Q8 — D4×Q8 — Q8⋊3D8
 Lower central C1 — C22 — C42 — Q8⋊3D8
 Upper central C1 — C22 — C42 — Q8⋊3D8
 Jennings C1 — C22 — C22 — C42 — Q8⋊3D8

Generators and relations for Q83D8
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, C41D4, C4⋊Q8, C4⋊Q8, C2×D8, C2×SD16, C22×Q8, D4⋊C8, Q8⋊C8, C4.10D8, C4⋊D8, C4⋊SD16, C4.4D8, D4×Q8, Q83D8
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, Q83D8

Character table of Q83D8

 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

Smallest permutation representation of Q83D8
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 Q83D8 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] >;

Q83D8 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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