Copied to
clipboard

## G = Q8⋊SD16order 128 = 27

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

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C42 — Q8⋊SD16
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×Q8 — Q82 — Q8⋊SD16
 Lower central C1 — C22 — C42 — Q8⋊SD16
 Upper central C1 — C22 — C42 — Q8⋊SD16
 Jennings C1 — C22 — C22 — C42 — Q8⋊SD16

Generators and relations for Q8⋊SD16
G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=cac-1=dad=a-1, cbc-1=dbd=a-1b, dcd=c3 >

Subgroups: 280 in 114 conjugacy classes, 38 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, SD16, C2×D4, C2×Q8, C2×Q8, C4×C8, D4⋊C4, C4⋊C8, C4×Q8, C4×Q8, C41D4, C4⋊Q8, C4⋊Q8, C2×SD16, Q8⋊C8, C4.6Q16, C4⋊SD16, C85D4, Q82, Q8⋊SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C22≀C2, C2×SD16, C8.C22, Q8⋊D4, D44D4, Q8⋊SD16

Character table of Q8⋊SD16

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 16 2 2 2 2 4 4 4 4 4 8 8 8 8 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 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 2 2 0 2 -2 -2 2 -2 2 0 0 2 0 0 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 -2 -2 -2 -2 2 0 0 0 0 0 2 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 -2 -2 -2 -2 2 0 0 0 0 0 -2 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 0 -2 2 2 -2 -2 0 -2 -2 0 2 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 -2 0 2 2 0 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 0 2 -2 -2 2 -2 -2 0 0 -2 0 0 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 -2 -2 2 0 2 0 0 -2 0 2 0 0 -2 0 0 0 0 -√-2 √-2 √-2 -√-2 -√-2 0 0 √-2 complex lifted from SD16 ρ16 2 -2 -2 2 0 2 0 0 -2 0 -2 0 0 2 0 0 0 0 -√-2 √-2 √-2 -√-2 √-2 0 0 -√-2 complex lifted from SD16 ρ17 2 2 -2 -2 0 0 2 -2 0 0 0 2 -2 0 0 0 0 0 -√-2 -√-2 √-2 √-2 0 √-2 -√-2 0 complex lifted from SD16 ρ18 2 -2 -2 2 0 2 0 0 -2 0 2 0 0 -2 0 0 0 0 √-2 -√-2 -√-2 √-2 √-2 0 0 -√-2 complex lifted from SD16 ρ19 2 -2 -2 2 0 2 0 0 -2 0 -2 0 0 2 0 0 0 0 √-2 -√-2 -√-2 √-2 -√-2 0 0 √-2 complex lifted from SD16 ρ20 2 2 -2 -2 0 0 2 -2 0 0 0 -2 2 0 0 0 0 0 -√-2 -√-2 √-2 √-2 0 -√-2 √-2 0 complex lifted from SD16 ρ21 2 2 -2 -2 0 0 2 -2 0 0 0 2 -2 0 0 0 0 0 √-2 √-2 -√-2 -√-2 0 -√-2 √-2 0 complex lifted from SD16 ρ22 2 2 -2 -2 0 0 2 -2 0 0 0 -2 2 0 0 0 0 0 √-2 √-2 -√-2 -√-2 0 √-2 -√-2 0 complex lifted from SD16 ρ23 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 2 -2 2 0 0 0 0 orthogonal lifted from D4⋊4D4 ρ24 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 -2 2 -2 0 0 0 0 orthogonal lifted from D4⋊4D4 ρ25 4 -4 -4 4 0 -4 0 0 4 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 ρ26 4 4 -4 -4 0 0 -4 4 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⋊SD16
On 64 points
Generators in S64
(1 9 27 44)(2 45 28 10)(3 11 29 46)(4 47 30 12)(5 13 31 48)(6 41 32 14)(7 15 25 42)(8 43 26 16)(17 61 54 37)(18 38 55 62)(19 63 56 39)(20 40 49 64)(21 57 50 33)(22 34 51 58)(23 59 52 35)(24 36 53 60)
(1 58 27 34)(2 23 28 52)(3 60 29 36)(4 17 30 54)(5 62 31 38)(6 19 32 56)(7 64 25 40)(8 21 26 50)(9 51 44 22)(10 59 45 35)(11 53 46 24)(12 61 47 37)(13 55 48 18)(14 63 41 39)(15 49 42 20)(16 57 43 33)
(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 4)(3 7)(6 8)(9 44)(10 47)(11 42)(12 45)(13 48)(14 43)(15 46)(16 41)(17 59)(18 62)(19 57)(20 60)(21 63)(22 58)(23 61)(24 64)(25 29)(26 32)(28 30)(33 56)(34 51)(35 54)(36 49)(37 52)(38 55)(39 50)(40 53)

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

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

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

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

 0 1 0 0 16 0 0 0 0 0 1 0 0 0 0 1
,
 5 5 0 0 5 12 0 0 0 0 16 0 0 0 0 16
,
 16 0 0 0 0 1 0 0 0 0 7 10 0 0 12 0
,
 1 0 0 0 0 16 0 0 0 0 1 0 0 0 1 16
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[5,5,0,0,5,12,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,1,0,0,0,0,7,12,0,0,10,0],[1,0,0,0,0,16,0,0,0,0,1,1,0,0,0,16] >;

Q8⋊SD16 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(128,355);
// by ID

G=gap.SmallGroup(128,355);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,232,422,352,1123,570,521,136,2804,1411,718,172]);
// Polycyclic

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

Export

׿
×
𝔽