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## G = Q8×SD16order 128 = 27

### Direct product of Q8 and SD16

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for Q8×SD16
G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >

Subgroups: 336 in 185 conjugacy classes, 104 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C2×Q8, C4×C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C4×D4, C4×Q8, C4×Q8, C4×Q8, C22⋊Q8, C4⋊Q8, C4⋊Q8, C2×SD16, C22×Q8, C4×SD16, C8×Q8, Q8⋊Q8, D42Q8, C83Q8, D4×Q8, Q82, Q8×SD16
Quotients: C1, C2, C22, D4, Q8, C23, SD16, C2×D4, C2×Q8, C24, C2×SD16, C22×D4, C22×Q8, 2- 1+4, D4×Q8, C22×SD16, D4○SD16, Q8×SD16

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

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

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

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

35 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4H 4I ··· 4M 4N ··· 4S 8A 8B 8C 8D 8E ··· 8J order 1 2 2 2 2 2 4 ··· 4 4 ··· 4 4 ··· 4 8 8 8 8 8 ··· 8 size 1 1 1 1 4 4 2 ··· 2 4 ··· 4 8 ··· 8 2 2 2 2 4 ··· 4

35 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 Q8 D4 SD16 2- 1+4 D4○SD16 kernel Q8×SD16 C4×SD16 C8×Q8 Q8⋊Q8 D4⋊2Q8 C8⋊3Q8 D4×Q8 Q82 C4⋊C4 SD16 C2×Q8 Q8 C4 C2 # reps 1 3 1 3 3 3 1 1 3 4 1 8 1 2

Matrix representation of Q8×SD16 in GL4(𝔽17) generated by

 1 0 0 0 0 1 0 0 0 0 4 2 0 0 0 13
,
 1 0 0 0 0 1 0 0 0 0 3 15 0 0 5 14
,
 5 12 0 0 5 5 0 0 0 0 1 0 0 0 0 1
,
 16 0 0 0 0 1 0 0 0 0 16 0 0 0 0 16
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,4,0,0,0,2,13],[1,0,0,0,0,1,0,0,0,0,3,5,0,0,15,14],[5,5,0,0,12,5,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16] >;

Q8×SD16 in GAP, Magma, Sage, TeX

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

G:=Group("Q8xSD16");
// GroupNames label

G:=SmallGroup(128,2111);
// by ID

G=gap.SmallGroup(128,2111);
# by ID

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

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

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