Copied to
clipboard

## G = Q16⋊5D4order 128 = 27

### 4th semidirect product of Q16 and D4 acting via D4/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — Q16⋊5D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×Q8 — C22×Q8 — C2×C8.C22 — Q16⋊5D4
 Lower central C1 — C2 — C2×C4 — Q16⋊5D4
 Upper central C1 — C22 — C4×D4 — Q16⋊5D4
 Jennings C1 — C2 — C2 — C2×C4 — Q16⋊5D4

Generators and relations for Q165D4
G = < a,b,c,d | a8=c4=d2=1, b2=a4, bab-1=a-1, ac=ca, dad=a5, bc=cb, bd=db, dcd=c-1 >

Subgroups: 488 in 241 conjugacy classes, 96 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, Q16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, Q8⋊C4, C4⋊C8, C2.D8, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C41D4, C41D4, C4⋊Q8, C4⋊Q8, C2×M4(2), C2×SD16, C2×Q16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, C86D4, C4×Q16, Q8⋊D4, D4.7D4, C4⋊SD16, C42Q16, C8⋊D4, C85D4, D4×Q8, Q86D4, C2×C8.C22, Q165D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C8.C22, C22×D4, 2+ 1+4, D42, C2×C8.C22, D4○SD16, Q165D4

Character table of Q165D4

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 8A 8B 8C 8D 8E 8F size 1 1 1 1 4 4 8 8 2 2 2 2 4 4 4 4 4 4 4 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 2 -2 0 0 -2 -2 -2 -2 0 -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 0 0 0 2 -2 0 -2 0 2 0 0 -2 2 0 0 0 0 2 -2 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 -2 -2 0 0 2 -2 -2 2 0 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 0 0 0 2 -2 0 2 0 -2 0 0 -2 2 0 0 0 0 -2 2 0 0 0 0 orthogonal lifted from D4 ρ21 2 -2 2 -2 0 0 0 0 0 2 -2 0 2 0 -2 0 0 2 -2 0 0 0 0 2 -2 0 0 0 0 orthogonal lifted from D4 ρ22 2 2 2 2 -2 2 0 0 -2 -2 -2 -2 0 2 0 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ23 2 -2 2 -2 0 0 0 0 0 2 -2 0 -2 0 2 0 0 2 -2 0 0 0 0 -2 2 0 0 0 0 orthogonal lifted from D4 ρ24 2 2 2 2 2 2 0 0 2 -2 -2 2 0 -2 0 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ25 4 -4 4 -4 0 0 0 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 2+ 1+4 ρ26 4 -4 -4 4 0 0 0 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 ρ27 4 -4 -4 4 0 0 0 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 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 Q165D4
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 11 5 15)(2 10 6 14)(3 9 7 13)(4 16 8 12)(17 49 21 53)(18 56 22 52)(19 55 23 51)(20 54 24 50)(25 40 29 36)(26 39 30 35)(27 38 31 34)(28 37 32 33)(41 62 45 58)(42 61 46 57)(43 60 47 64)(44 59 48 63)
(1 36 19 60)(2 37 20 61)(3 38 21 62)(4 39 22 63)(5 40 23 64)(6 33 24 57)(7 34 17 58)(8 35 18 59)(9 31 53 45)(10 32 54 46)(11 25 55 47)(12 26 56 48)(13 27 49 41)(14 28 50 42)(15 29 51 43)(16 30 52 44)
(1 12)(2 9)(3 14)(4 11)(5 16)(6 13)(7 10)(8 15)(17 54)(18 51)(19 56)(20 53)(21 50)(22 55)(23 52)(24 49)(25 63)(26 60)(27 57)(28 62)(29 59)(30 64)(31 61)(32 58)(33 41)(34 46)(35 43)(36 48)(37 45)(38 42)(39 47)(40 44)

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

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

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

Matrix representation of Q165D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 12 5 0 0 0 0 5 5 0 0 12 12 0 0 0 0 12 5 0 0
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 12 5 0 0 0 0 5 5 0 0 0 0 0 0 5 5 0 0 0 0 5 12
,
 4 4 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 16 0 0 0 0 16 0 0 0
,
 1 0 0 0 0 0 15 16 0 0 0 0 0 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16 0 0 0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,12,12,0,0,0,0,12,5,0,0,12,5,0,0,0,0,5,5,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,12,5,0,0,0,0,5,5,0,0,0,0,0,0,5,5,0,0,0,0,5,12],[4,0,0,0,0,0,4,13,0,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1,0,0,0],[1,15,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16,0,0,0] >;

Q165D4 in GAP, Magma, Sage, TeX

Q_{16}\rtimes_5D_4
% in TeX

G:=Group("Q16:5D4");
// GroupNames label

G:=SmallGroup(128,2010);
// by ID

G=gap.SmallGroup(128,2010);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,723,346,248,2804,1411,375,172]);
// Polycyclic

G:=Group<a,b,c,d|a^8=c^4=d^2=1,b^2=a^4,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^5,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

Export

׿
×
𝔽