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## G = SD16⋊3Q8order 128 = 27

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

Generators and relations for SD163Q8
G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=a3, ac=ca, dad-1=a5, cbc-1=dbd-1=a4b, dcd-1=c-1 >

Subgroups: 288 in 167 conjugacy classes, 94 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C2×Q8, C4×C8, C8⋊C4, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C4⋊C4, C4×D4, C4×Q8, C4×Q8, C22⋊Q8, C42.C2, C42.C2, C4⋊Q8, C4⋊Q8, C2×SD16, C4×SD16, SD16⋊C4, C84Q8, Q8⋊Q8, D42Q8, D4.Q8, Q8.Q8, C8.5Q8, C8⋊Q8, D43Q8, Q83Q8, SD163Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C24, C22×D4, C22×Q8, 2- 1+4, D4×Q8, D8⋊C22, D4○SD16, SD163Q8

Character table of SD163Q8

 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 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 ρ26 4 -4 -4 4 0 0 0 -4i 0 4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from D8⋊C22 ρ27 4 -4 -4 4 0 0 0 4i 0 -4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from D8⋊C22 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 complex lifted from D4○SD16 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 complex lifted from D4○SD16

Smallest permutation representation of SD163Q8
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 23)(2 18)(3 21)(4 24)(5 19)(6 22)(7 17)(8 20)(9 52)(10 55)(11 50)(12 53)(13 56)(14 51)(15 54)(16 49)(25 43)(26 46)(27 41)(28 44)(29 47)(30 42)(31 45)(32 48)(33 61)(34 64)(35 59)(36 62)(37 57)(38 60)(39 63)(40 58)
(1 16 23 53)(2 9 24 54)(3 10 17 55)(4 11 18 56)(5 12 19 49)(6 13 20 50)(7 14 21 51)(8 15 22 52)(25 64 41 36)(26 57 42 37)(27 58 43 38)(28 59 44 39)(29 60 45 40)(30 61 46 33)(31 62 47 34)(32 63 48 35)
(1 26 23 42)(2 31 24 47)(3 28 17 44)(4 25 18 41)(5 30 19 46)(6 27 20 43)(7 32 21 48)(8 29 22 45)(9 34 54 62)(10 39 55 59)(11 36 56 64)(12 33 49 61)(13 38 50 58)(14 35 51 63)(15 40 52 60)(16 37 53 57)

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

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

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

Matrix representation of SD163Q8 in GL6(𝔽17)

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

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

SD163Q8 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes_3Q_8
% in TeX

G:=Group("SD16:3Q8");
// GroupNames label

G:=SmallGroup(128,2120);
// by ID

G=gap.SmallGroup(128,2120);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,723,346,304,4037,1027,124]);
// 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,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations

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