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## G = SD16⋊Q8order 128 = 27

### 1st semidirect product of SD16 and Q8 acting via Q8/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — SD16⋊Q8
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C4×D4 — D4⋊3Q8 — SD16⋊Q8
 Lower central C1 — C2 — C2×C4 — SD16⋊Q8
 Upper central C1 — C22 — C4×Q8 — SD16⋊Q8
 Jennings C1 — C2 — C2 — C2×C4 — SD16⋊Q8

Generators and relations for SD16⋊Q8
G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=a3, ac=ca, dad-1=a5, bc=cb, dbd-1=a4b, dcd-1=c-1 >

Subgroups: 304 in 170 conjugacy classes, 96 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C2×Q8, C4×C8, C8⋊C4, D4⋊C4, D4⋊C4, Q8⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C4.Q8, C2.D8, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C4×Q8, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C4⋊Q8, C4⋊Q8, C2×SD16, C4×SD16, SD16⋊C4, C84Q8, D4⋊Q8, D42Q8, C4.Q16, Q8.Q8, C82Q8, C8⋊Q8, D43Q8, Q82, SD16⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C24, C8.C22, C22×D4, C22×Q8, 2- 1+4, D4×Q8, C2×C8.C22, D4○D8, SD16⋊Q8

Character table of SD16⋊Q8

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 4Q 8A 8B 8C 8D 8E 8F size 1 1 1 1 4 4 2 2 2 2 4 4 4 4 4 4 4 8 8 8 8 8 8 4 4 4 4 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 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 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 -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 -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 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 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 -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 -1 1 1 linear of order 2 ρ9 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 1 -1 1 linear of order 2 ρ10 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 1 1 -1 linear of order 2 ρ11 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 -1 1 -1 linear of order 2 ρ12 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 -1 -1 1 linear of order 2 ρ13 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 1 -1 1 linear of order 2 ρ14 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 1 1 -1 linear of order 2 ρ15 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 -1 1 -1 linear of order 2 ρ16 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 -1 -1 1 linear of order 2 ρ17 2 2 2 2 0 0 -2 2 -2 2 -2 0 2 2 0 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 0 0 -2 -2 -2 -2 -2 0 2 -2 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 0 0 -2 -2 -2 -2 2 0 -2 2 0 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 0 0 -2 2 -2 2 2 0 -2 -2 0 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 -2 2 -2 -2 2 -2 0 2 0 0 2 0 0 -2 0 0 0 0 0 0 0 0 -2 2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ22 2 -2 2 -2 2 -2 -2 0 2 0 0 2 0 0 -2 0 0 0 0 0 0 0 0 2 -2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ23 2 -2 2 -2 -2 2 -2 0 2 0 0 -2 0 0 2 0 0 0 0 0 0 0 0 2 -2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ24 2 -2 2 -2 2 -2 -2 0 2 0 0 -2 0 0 2 0 0 0 0 0 0 0 0 -2 2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ25 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2√2 2√2 0 0 orthogonal lifted from D4○D8 ρ26 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2√2 -2√2 0 0 orthogonal lifted from D4○D8 ρ27 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 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ28 4 -4 4 -4 0 0 4 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from 2- 1+4, Schur index 2 ρ29 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 0 0 0 symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of SD16⋊Q8
On 64 points
Generators in S64
(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 19)(2 22)(3 17)(4 20)(5 23)(6 18)(7 21)(8 24)(9 57)(10 60)(11 63)(12 58)(13 61)(14 64)(15 59)(16 62)(25 53)(26 56)(27 51)(28 54)(29 49)(30 52)(31 55)(32 50)(33 43)(34 46)(35 41)(36 44)(37 47)(38 42)(39 45)(40 48)
(1 31 23 51)(2 32 24 52)(3 25 17 53)(4 26 18 54)(5 27 19 55)(6 28 20 56)(7 29 21 49)(8 30 22 50)(9 36 61 48)(10 37 62 41)(11 38 63 42)(12 39 64 43)(13 40 57 44)(14 33 58 45)(15 34 59 46)(16 35 60 47)
(1 38 23 42)(2 35 24 47)(3 40 17 44)(4 37 18 41)(5 34 19 46)(6 39 20 43)(7 36 21 48)(8 33 22 45)(9 49 61 29)(10 54 62 26)(11 51 63 31)(12 56 64 28)(13 53 57 25)(14 50 58 30)(15 55 59 27)(16 52 60 32)

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

G:=Group( (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), (1,19)(2,22)(3,17)(4,20)(5,23)(6,18)(7,21)(8,24)(9,57)(10,60)(11,63)(12,58)(13,61)(14,64)(15,59)(16,62)(25,53)(26,56)(27,51)(28,54)(29,49)(30,52)(31,55)(32,50)(33,43)(34,46)(35,41)(36,44)(37,47)(38,42)(39,45)(40,48), (1,31,23,51)(2,32,24,52)(3,25,17,53)(4,26,18,54)(5,27,19,55)(6,28,20,56)(7,29,21,49)(8,30,22,50)(9,36,61,48)(10,37,62,41)(11,38,63,42)(12,39,64,43)(13,40,57,44)(14,33,58,45)(15,34,59,46)(16,35,60,47), (1,38,23,42)(2,35,24,47)(3,40,17,44)(4,37,18,41)(5,34,19,46)(6,39,20,43)(7,36,21,48)(8,33,22,45)(9,49,61,29)(10,54,62,26)(11,51,63,31)(12,56,64,28)(13,53,57,25)(14,50,58,30)(15,55,59,27)(16,52,60,32) );

G=PermutationGroup([[(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)], [(1,19),(2,22),(3,17),(4,20),(5,23),(6,18),(7,21),(8,24),(9,57),(10,60),(11,63),(12,58),(13,61),(14,64),(15,59),(16,62),(25,53),(26,56),(27,51),(28,54),(29,49),(30,52),(31,55),(32,50),(33,43),(34,46),(35,41),(36,44),(37,47),(38,42),(39,45),(40,48)], [(1,31,23,51),(2,32,24,52),(3,25,17,53),(4,26,18,54),(5,27,19,55),(6,28,20,56),(7,29,21,49),(8,30,22,50),(9,36,61,48),(10,37,62,41),(11,38,63,42),(12,39,64,43),(13,40,57,44),(14,33,58,45),(15,34,59,46),(16,35,60,47)], [(1,38,23,42),(2,35,24,47),(3,40,17,44),(4,37,18,41),(5,34,19,46),(6,39,20,43),(7,36,21,48),(8,33,22,45),(9,49,61,29),(10,54,62,26),(11,51,63,31),(12,56,64,28),(13,53,57,25),(14,50,58,30),(15,55,59,27),(16,52,60,32)]])

Matrix representation of SD16⋊Q8 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 5 12 0 0 0 0 5 5 0 0 0 0 0 0 12 5 0 0 0 0 12 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1
,
 0 16 0 0 0 0 1 0 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 0 13 0 0 0 0 13 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 16 0 0 0 0 0 0 16 0 0

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,0,0,0,0,12,12,0,0,0,0,5,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;

SD16⋊Q8 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes Q_8
% in TeX

G:=Group("SD16:Q8");
// GroupNames label

G:=SmallGroup(128,2117);
// by ID

G=gap.SmallGroup(128,2117);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,120,758,723,346,80,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=c^2,b*a*b=a^3,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations

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