Copied to
clipboard

## G = Q8⋊6SD16order 128 = 27

### 2nd 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⋊6SD16
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×Q8 — Q8⋊3Q8 — Q8⋊6SD16
 Lower central C1 — C22 — C42 — Q8⋊6SD16
 Upper central C1 — C22 — C42 — Q8⋊6SD16
 Jennings C1 — C22 — C22 — C42 — Q8⋊6SD16

Generators and relations for Q86SD16
G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=cac-1=dad=a-1, cbc-1=ab, dbd=a-1b, dcd=c3 >

Subgroups: 248 in 105 conjugacy classes, 36 normal (32 characteristic)
C1, C2 [×3], C2, C4 [×4], C4 [×8], C22, C22 [×3], C8 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×6], Q8 [×2], Q8 [×5], C23, C42, C42 [×4], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×9], C2×C8 [×3], SD16 [×4], C2×D4 [×3], C2×Q8 [×2], C2×Q8, C4×C8, D4⋊C4 [×4], C4⋊C8 [×2], C4×Q8 [×2], C4×Q8 [×2], C42.C2 [×3], C41D4, C4⋊Q8, C4⋊Q8, C2×SD16 [×2], Q8⋊C8 [×2], C4.10D8, C4⋊SD16 [×2], C4.4D8, Q83Q8, Q86SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, SD16 [×2], C2×D4 [×3], C22≀C2, C2×SD16, C4○D8, C8⋊C22, C8.C22, Q8⋊D4, D4⋊D4, D4.8D4, Q86SD16

Character table of Q86SD16

 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 -2i 2i 0 0 0 0 0 √2 √2 -√2 -√2 0 √-2 -√-2 0 complex lifted from C4○D8 ρ18 2 2 -2 -2 0 0 2 -2 0 0 0 -2i 2i 0 0 0 0 0 -√2 -√2 √2 √2 0 -√-2 √-2 0 complex lifted from C4○D8 ρ19 2 2 -2 -2 0 0 2 -2 0 0 0 2i -2i 0 0 0 0 0 -√2 -√2 √2 √2 0 √-2 -√-2 0 complex lifted from C4○D8 ρ20 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 ρ21 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 ρ22 2 2 -2 -2 0 0 2 -2 0 0 0 2i -2i 0 0 0 0 0 √2 √2 -√2 -√2 0 -√-2 √-2 0 complex lifted from C4○D8 ρ23 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 orthogonal lifted from C8⋊C22 ρ24 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 ρ25 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2i 2i -2i 2i 0 0 0 0 complex lifted from D4.8D4 ρ26 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2i -2i 2i -2i 0 0 0 0 complex lifted from D4.8D4

Smallest permutation representation of Q86SD16
On 64 points
Generators in S64
(1 15 33 19)(2 20 34 16)(3 9 35 21)(4 22 36 10)(5 11 37 23)(6 24 38 12)(7 13 39 17)(8 18 40 14)(25 64 55 48)(26 41 56 57)(27 58 49 42)(28 43 50 59)(29 60 51 44)(30 45 52 61)(31 62 53 46)(32 47 54 63)
(1 42 33 58)(2 50 34 28)(3 44 35 60)(4 52 36 30)(5 46 37 62)(6 54 38 32)(7 48 39 64)(8 56 40 26)(9 51 21 29)(10 61 22 45)(11 53 23 31)(12 63 24 47)(13 55 17 25)(14 57 18 41)(15 49 19 27)(16 59 20 43)
(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 17)(10 20)(11 23)(12 18)(13 21)(14 24)(15 19)(16 22)(25 44)(26 47)(27 42)(28 45)(29 48)(30 43)(31 46)(32 41)(34 36)(35 39)(38 40)(49 58)(50 61)(51 64)(52 59)(53 62)(54 57)(55 60)(56 63)

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

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

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

Matrix representation of Q86SD16 in GL4(𝔽17) generated by

 1 0 0 0 0 1 0 0 0 0 16 15 0 0 1 1
,
 1 0 0 0 0 1 0 0 0 0 0 10 0 0 5 0
,
 5 5 0 0 12 5 0 0 0 0 13 9 0 0 0 4
,
 1 0 0 0 0 16 0 0 0 0 1 0 0 0 16 16
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,16,1,0,0,15,1],[1,0,0,0,0,1,0,0,0,0,0,5,0,0,10,0],[5,12,0,0,5,5,0,0,0,0,13,0,0,0,9,4],[1,0,0,0,0,16,0,0,0,0,1,16,0,0,0,16] >;

Q86SD16 in GAP, Magma, Sage, TeX

Q_8\rtimes_6{\rm SD}_{16}
% in TeX

G:=Group("Q8:6SD16");
// GroupNames label

G:=SmallGroup(128,358);
// by ID

G=gap.SmallGroup(128,358);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,456,422,184,1123,570,521,136,2804,1411,718,172]);
// 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=a*b,d*b*d=a^-1*b,d*c*d=c^3>;
// generators/relations

Export

׿
×
𝔽