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## G = Q8.2SD16order 128 = 27

### 2nd non-split extension by Q8 of SD16 acting via SD16/C8=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C42 — Q8.2SD16
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×Q8 — C8×Q8 — Q8.2SD16
 Lower central C1 — C22 — C42 — Q8.2SD16
 Upper central C1 — C22 — C42 — Q8.2SD16
 Jennings C1 — C22 — C22 — C42 — Q8.2SD16

Generators and relations for Q8.2SD16
G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a-1b, dcd=a2c3 >

Subgroups: 200 in 80 conjugacy classes, 34 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C2×D4, C2×Q8, C2×Q8, C4×C8, C4×C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C2.D8, C4×Q8, C41D4, C4⋊Q8, C2×SD16, C4.D8, C4.6Q16, C82C8, C8×Q8, C4⋊SD16, C4.Q16, C4.4D8, Q8.2SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C4⋊D4, C2×SD16, C4○D8, C8.C22, Q8.D4, C88D4, D4.4D4, Q8.2SD16

Character table of Q8.2SD16

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L 8M 8N size 1 1 1 1 16 2 2 2 2 4 4 4 4 4 16 2 2 2 2 4 4 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 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 2 2 2 2 0 -2 2 -2 2 0 0 -2 0 0 0 -2 -2 -2 -2 0 0 0 2 2 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 -2 2 -2 2 0 0 -2 0 0 0 2 2 2 2 0 0 0 -2 -2 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 2 -2 2 -2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 0 2 -2 2 -2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 0 -2 -2 -2 -2 0 0 2 0 0 0 0 0 0 0 -2i -2i 2i 0 0 2i 0 0 0 0 complex lifted from C4○D4 ρ14 2 2 2 2 0 -2 -2 -2 -2 0 0 2 0 0 0 0 0 0 0 2i 2i -2i 0 0 -2i 0 0 0 0 complex lifted from C4○D4 ρ15 2 -2 2 -2 0 0 2 0 -2 0 0 0 0 0 0 -2i 2i 2i -2i 0 0 0 0 0 0 -√2 -√-2 √2 √-2 complex lifted from C4○D8 ρ16 2 2 -2 -2 0 -2 0 2 0 2i 0 0 -2i 0 0 -√-2 √-2 -√-2 √-2 -√2 √2 -√2 -√-2 √-2 √2 0 0 0 0 complex lifted from C4○D8 ρ17 2 -2 2 -2 0 0 2 0 -2 0 0 0 0 0 0 2i -2i -2i 2i 0 0 0 0 0 0 √2 -√-2 -√2 √-2 complex lifted from C4○D8 ρ18 2 2 -2 -2 0 2 0 -2 0 0 2 0 0 -2 0 -√-2 √-2 -√-2 √-2 √-2 -√-2 -√-2 √-2 -√-2 √-2 0 0 0 0 complex lifted from SD16 ρ19 2 2 -2 -2 0 2 0 -2 0 0 -2 0 0 2 0 √-2 -√-2 √-2 -√-2 √-2 -√-2 -√-2 -√-2 √-2 √-2 0 0 0 0 complex lifted from SD16 ρ20 2 2 -2 -2 0 -2 0 2 0 -2i 0 0 2i 0 0 √-2 -√-2 √-2 -√-2 -√2 √2 -√2 √-2 -√-2 √2 0 0 0 0 complex lifted from C4○D8 ρ21 2 -2 2 -2 0 0 2 0 -2 0 0 0 0 0 0 -2i 2i 2i -2i 0 0 0 0 0 0 √2 √-2 -√2 -√-2 complex lifted from C4○D8 ρ22 2 2 -2 -2 0 -2 0 2 0 -2i 0 0 2i 0 0 -√-2 √-2 -√-2 √-2 √2 -√2 √2 -√-2 √-2 -√2 0 0 0 0 complex lifted from C4○D8 ρ23 2 2 -2 -2 0 -2 0 2 0 2i 0 0 -2i 0 0 √-2 -√-2 √-2 -√-2 √2 -√2 √2 √-2 -√-2 -√2 0 0 0 0 complex lifted from C4○D8 ρ24 2 2 -2 -2 0 2 0 -2 0 0 2 0 0 -2 0 √-2 -√-2 √-2 -√-2 -√-2 √-2 √-2 -√-2 √-2 -√-2 0 0 0 0 complex lifted from SD16 ρ25 2 -2 2 -2 0 0 2 0 -2 0 0 0 0 0 0 2i -2i -2i 2i 0 0 0 0 0 0 -√2 √-2 √2 -√-2 complex lifted from C4○D8 ρ26 2 2 -2 -2 0 2 0 -2 0 0 -2 0 0 2 0 -√-2 √-2 -√-2 √-2 -√-2 √-2 √-2 √-2 -√-2 -√-2 0 0 0 0 complex lifted from SD16 ρ27 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 2√2 2√2 -2√2 -2√2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4.4D4 ρ28 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 -2√2 -2√2 2√2 2√2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4.4D4 ρ29 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 C8.C22, Schur index 2

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

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

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

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

Matrix representation of Q8.2SD16 in GL4(𝔽17) generated by

 16 15 0 0 1 1 0 0 0 0 1 0 0 0 0 1
,
 0 10 0 0 5 0 0 0 0 0 16 0 0 0 0 16
,
 13 0 0 0 0 13 0 0 0 0 10 10 0 0 12 0
,
 1 0 0 0 16 16 0 0 0 0 1 0 0 0 16 16
G:=sub<GL(4,GF(17))| [16,1,0,0,15,1,0,0,0,0,1,0,0,0,0,1],[0,5,0,0,10,0,0,0,0,0,16,0,0,0,0,16],[13,0,0,0,0,13,0,0,0,0,10,12,0,0,10,0],[1,16,0,0,0,16,0,0,0,0,1,16,0,0,0,16] >;

Q8.2SD16 in GAP, Magma, Sage, TeX

Q_8._2{\rm SD}_{16}
% in TeX

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

G:=SmallGroup(128,410);
// by ID

G=gap.SmallGroup(128,410);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,512,422,352,1123,136,2804,718,172]);
// Polycyclic

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

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