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## G = SD16⋊2Q8order 128 = 27

### 2nd 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⋊2Q8
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C4×D4 — D4×Q8 — SD16⋊2Q8
 Lower central C1 — C2 — C2×C4 — SD16⋊2Q8
 Upper central C1 — C22 — C4×Q8 — SD16⋊2Q8
 Jennings C1 — C2 — C2 — C2×C4 — SD16⋊2Q8

Generators and relations for SD162Q8
G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=a3, ac=ca, dad-1=a5, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 320 in 180 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, C4×D4, C4×D4, C4×Q8, C4×Q8, C4×Q8, C22⋊Q8, C42.C2, C42.C2, C4⋊Q8, C4⋊Q8, C4⋊Q8, C2×SD16, C22×Q8, C4×SD16, SD16⋊C4, C84Q8, D4⋊Q8, Q8⋊Q8, C4.Q16, D4.Q8, C82Q8, C8⋊Q8, D4×Q8, Q83Q8, SD162Q8
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, Q8○D8, SD162Q8

Character table of SD162Q8

 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 4 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ26 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 orthogonal lifted from C8⋊C22 ρ27 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 ρ28 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 symplectic lifted from Q8○D8, Schur index 2 ρ29 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 symplectic lifted from Q8○D8, Schur index 2

Smallest permutation representation of SD162Q8
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 55)(10 50)(11 53)(12 56)(13 51)(14 54)(15 49)(16 52)(25 62)(26 57)(27 60)(28 63)(29 58)(30 61)(31 64)(32 59)(33 43)(34 46)(35 41)(36 44)(37 47)(38 42)(39 45)(40 48)
(1 9 23 51)(2 10 24 52)(3 11 17 53)(4 12 18 54)(5 13 19 55)(6 14 20 56)(7 15 21 49)(8 16 22 50)(25 35 60 47)(26 36 61 48)(27 37 62 41)(28 38 63 42)(29 39 64 43)(30 40 57 44)(31 33 58 45)(32 34 59 46)
(1 36 23 48)(2 33 24 45)(3 38 17 42)(4 35 18 47)(5 40 19 44)(6 37 20 41)(7 34 21 46)(8 39 22 43)(9 26 51 61)(10 31 52 58)(11 28 53 63)(12 25 54 60)(13 30 55 57)(14 27 56 62)(15 32 49 59)(16 29 50 64)

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

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

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

Matrix representation of SD162Q8 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 8 7 7 0 0 9 16 10 7 0 0 13 4 1 9 0 0 13 13 8 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 16 0 16
,
 1 15 0 0 0 0 1 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 10 7 0 0 0 0 5 7 0 0 0 0 0 0 0 16 0 15 0 0 1 0 2 0 0 0 0 1 0 1 0 0 16 0 16 0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,9,13,13,0,0,8,16,4,13,0,0,7,10,1,8,0,0,7,7,9,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,1,0,0,0,0,1,0,16,0,0,0,0,1,0,0,0,0,0,0,16],[1,1,0,0,0,0,15,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[10,5,0,0,0,0,7,7,0,0,0,0,0,0,0,1,0,16,0,0,16,0,1,0,0,0,0,2,0,16,0,0,15,0,1,0] >;

SD162Q8 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes_2Q_8
% in TeX

G:=Group("SD16:2Q8");
// GroupNames label

G:=SmallGroup(128,2118);
// by ID

G=gap.SmallGroup(128,2118);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,568,758,723,346,80,4037,1027,124]);
// 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,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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