Copied to
clipboard

G = Q8⋊4SD16  order 128 = 27

3rd semidirect product of Q8 and SD16 acting via SD16/D4=C2

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

Aliases: Q8⋊4SD16, D4.6SD16, C42.212C23, Q8⋊C8⋊27C2, C4⋊C4.34D4, (D4×Q8).3C2, D4⋊C8.11C2, C8⋊3Q8⋊12C2, Q8⋊Q8⋊34C2, (C2×D4).258D4, (C2×Q8).203D4, C4.35(C2×SD16), C4⋊Q8.32C22, C4.10D8⋊21C2, C4⋊C8.171C22, C4.39(C8⋊C22), (C4×C8).245C22, D4.D4.5C2, (C4×D4).40C22, (C4×Q8).40C22, C2.16(Q8⋊D4), C4.67(C8.C22), C22.178C22≀C2, C2.16(C22⋊SD16), C2.16(D4.10D4), (C2×C4).969(C2×D4), SmallGroup(128,383)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C42 — Q8⋊4SD16
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C4×D4 — D4×Q8 — Q8⋊4SD16
Lower central
C1 — C22 — C42 — Q8⋊4SD16
Upper central
C1 — C22 — C42 — Q8⋊4SD16
Jennings
C1 — C22 — C22 — C42 — Q8⋊4SD16

Generators and relations for Q8⋊4SD16
 G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=cac-1=a-1, ad=da, cbc-1=a-1b, bd=db, dcd=c3 >

Subgroups: 264 in 117 conjugacy classes, 38 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C2×Q8, C4×C8, Q8⋊C4, C4⋊C8, C4.Q8, C4×D4, C4×D4, C4×Q8, C22⋊Q8, C4⋊Q8, C4⋊Q8, C2×SD16, C22×Q8, D4⋊C8, Q8⋊C8, C4.10D8, D4.D4, Q8⋊Q8, C8⋊3Q8, D4×Q8, Q8⋊4SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C22≀C2, C2×SD16, C8⋊C22, C8.C22, Q8⋊D4, C22⋊SD16, D4.10D4, Q8⋊4SD16

Character table of Q8⋊4SD16

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F8G8H
 size 111144222244488881644448888
ρ111111111111111111111111111    trivial
ρ21111-1-111111111-1-1-11-1-1-1-1-11-11    linear of order 2
ρ311111111111111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ41111-1-111111111-1-1-1-111111-11-1    linear of order 2
ρ51111-1-11111-1-11-1-111-1-1-1-1-11111    linear of order 2
ρ61111111111-1-11-11-1-1-11111-11-11    linear of order 2
ρ71111-1-11111-1-11-1-11111111-1-1-1-1    linear of order 2
ρ81111111111-1-11-11-1-11-1-1-1-11-11-1    linear of order 2
ρ9222222-2-22200-20-200000000000    orthogonal lifted from D4
ρ10222200-2-2-2-2002002-2000000000    orthogonal lifted from D4
ρ11222200-2-2-2-200200-22000000000    orthogonal lifted from D4
ρ1222220022-2-2-2-2-22000000000000    orthogonal lifted from D4
ρ1322220022-2-222-2-2000000000000    orthogonal lifted from D4
ρ142222-2-2-2-22200-20200000000000    orthogonal lifted from D4
ρ152-22-2-2200-2200000000-√-2-√-2√-2√-20-√-20√-2    complex lifted from SD16
ρ162-2-22002-200-22000000√-2-√-2-√-2√-2√-20-√-20    complex lifted from SD16
ρ172-22-22-200-2200000000√-2√-2-√-2-√-20-√-20√-2    complex lifted from SD16
ρ182-22-2-2200-2200000000√-2√-2-√-2-√-20√-20-√-2    complex lifted from SD16
ρ192-2-22002-2002-2000000√-2-√-2-√-2√-2-√-20√-20    complex lifted from SD16
ρ202-22-22-200-2200000000-√-2-√-2√-2√-20√-20-√-2    complex lifted from SD16
ρ212-2-22002-200-22000000-√-2√-2√-2-√-2-√-20√-20    complex lifted from SD16
ρ222-2-22002-2002-2000000-√-2√-2√-2-√-2√-20-√-20    complex lifted from SD16
ρ234-4-4400-44000000000000000000    orthogonal lifted from C8⋊C22
ρ2444-4-400000000000000-22-220000    symplectic lifted from D4.10D4, Schur index 2
ρ254-44-400004-40000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2644-4-4000000000000002-22-20000    symplectic lifted from D4.10D4, Schur index 2

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

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

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

Matrix representation of Q8⋊4SD16 ►in GL4(𝔽17) generated by

13000
0400
0010
0001
,
0100
16000
0010
0001
,
01500
8000
00010
001210
,
1000
0100
00160
00161
G:=sub<GL(4,GF(17))| [13,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,8,0,0,15,0,0,0,0,0,0,12,0,0,10,10],[1,0,0,0,0,1,0,0,0,0,16,16,0,0,0,1] >;
 

Q8⋊4SD16 in GAP, Magma, Sage, TeX 

Q_8\rtimes_4{\rm SD}_{16}
 
% in TeX
 
G:=Group("Q8:4SD16");
 
// GroupNames label
 
G:=SmallGroup(128,383);
 
// by ID
 
G=gap.SmallGroup(128,383);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,680,422,1123,570,521,136,2804,1411,718,172]);
 
// 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=a^-1,a*d=d*a,c*b*c^-1=a^-1*b,b*d=d*b,d*c*d=c^3>;
 
// generators/relations
 

Export

Character table of Q8⋊4SD16 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁