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G = Q8⋊5D12  order 192 = 26·3

3rd semidirect product of Q8 and D12 acting via D12/D6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4⋊4D12, Q8⋊5D12, C42⋊4D6, D12⋊15D4, Dic6⋊15D4, M4(2)⋊3D6, C4≀C2⋊1S3, (C3×D4)⋊3D4, (C3×Q8)⋊3D4, D4○D12⋊1C2, C8⋊D6⋊8C2, C4.9(C2×D12), D4⋊D6⋊1C2, C4⋊D12⋊6C2, C3⋊2(D4⋊4D4), C4○D4.17D6, C4.125(S3×D4), C42⋊4S3⋊5C2, (C4×C12)⋊11C22, C6.27C22≀C2, C12.337(C2×D4), (C22×S3).2D4, C22.29(S3×D4), C12.46D4⋊1C2, (C2×D12)⋊13C22, C4.Dic3⋊4C22, C2.30(D6⋊D4), (C2×C12).262C23, C4○D12.11C22, (C3×M4(2))⋊10C22, (C3×C4≀C2)⋊1C2, (C2×C6).26(C2×D4), (C3×C4○D4).3C22, (C2×C4).109(C22×S3), SmallGroup(192,381)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C12 — Q8⋊5D12
Chief series
C1 — C3 — C6 — C2×C6 — C2×C12 — C2×D12 — D4○D12 — Q8⋊5D12
Lower central
C3 — C6 — C2×C12 — Q8⋊5D12
Upper central
C1 — C2 — C2×C4 — C4≀C2

Generators and relations for Q8⋊5D12
 G = < a,b,c,d | a4=c12=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, cbc-1=dbd=a-1b, dcd=c-1 >

Subgroups: 672 in 168 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, M4(2), M4(2), D8, SD16, C2×D4, C4○D4, C4○D4, C3⋊C8, C24, Dic6, C4×S3, D12, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×S3, C4.D4, C4≀C2, C4≀C2, C4⋊1D4, C8⋊C22, 2+ 1+4, C24⋊C2, D24, C4.Dic3, D4⋊S3, Q8⋊2S3, C4×C12, C3×M4(2), C2×D12, C2×D12, C4○D12, C4○D12, S3×D4, Q8⋊3S3, C3×C4○D4, D4⋊4D4, C42⋊4S3, C12.46D4, C3×C4≀C2, C4⋊D12, C8⋊D6, D4⋊D6, D4○D12, Q8⋊5D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C22≀C2, C2×D12, S3×D4, D4⋊4D4, D6⋊D4, Q8⋊5D12

Character table of Q8⋊5D12

 class 12A2B2C2D2E2F2G34A4B4C4D4E4F6A6B6C8A8B12A12B12C12D12E12F12G12H24A24B
 size 112412121224222444122488242244444888
ρ1111111111111111111111111111111    trivial
ρ21111-1-1-11111-1-11-1111-1111-1-1-1-111-1-1    linear of order 2
ρ3111-1-111111111-1-111-1-1-11111111-1-1-1    linear of order 2
ρ4111-11-1-11111-1-1-1111-11-111-1-1-1-11-111    linear of order 2
ρ5111-1-111-1111-1-1-1-111-11111-1-1-1-11-111    linear of order 2
ρ6111-11-1-1-111111-1111-1-111111111-1-1-1    linear of order 2
ρ71111111-1111-1-111111-1-111-1-1-1-111-1-1    linear of order 2
ρ81111-1-1-1-1111111-11111-11111111111    linear of order 2
ρ922220000-122-2-220-1-1-1-20-1-11111-1-111    orthogonal lifted from D6
ρ10222002-202-2-2000022000-2-20000-2000    orthogonal lifted from D4
ρ1122-2-2000022-200202-2-200-2-200002200    orthogonal lifted from D4
ρ1222-2020002-22000-22-2000220000-2000    orthogonal lifted from D4
ρ1322-22000022-200-202-2200-2-200002-200    orthogonal lifted from D4
ρ1422-20-20002-2200022-2000220000-2000    orthogonal lifted from D4
ρ15222-20000-12222-20-1-11-20-1-1-1-1-1-1-1111    orthogonal lifted from D6
ρ1622200-2202-2-2000022000-2-20000-2000    orthogonal lifted from D4
ρ17222-20000-122-2-2-20-1-1120-1-11111-11-1-1    orthogonal lifted from D6
ρ1822220000-1222220-1-1-120-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1922-220000-12-200-20-11-10011√3√3-√3-√3-11√3-√3    orthogonal lifted from D12
ρ2022-2-20000-12-20020-1110011√3√3-√3-√3-1-1-√3√3    orthogonal lifted from D12
ρ2122-220000-12-200-20-11-10011-√3-√3√3√3-11-√3√3    orthogonal lifted from D12
ρ2222-2-20000-12-20020-1110011-√3-√3√3√3-1-1√3-√3    orthogonal lifted from D12
ρ234-4000000400-2200-40000002-2-220000    orthogonal lifted from D4⋊4D4
ρ2444-400000-2-440000-22000-2-200002000    orthogonal lifted from S3×D4
ρ2544400000-2-4-40000-2-20002200002000    orthogonal lifted from S3×D4
ρ264-40000004002-200-4000000-222-20000    orthogonal lifted from D4⋊4D4
ρ274-4000000-2002-200200002√3-2√31+√3-1-√3-1+√31-√30000    orthogonal faithful
ρ284-4000000-200-220020000-2√32√3-1+√31-√31+√3-1-√30000    orthogonal faithful
ρ294-4000000-200-2200200002√3-2√3-1-√31+√31-√3-1+√30000    orthogonal faithful
ρ304-4000000-2002-20020000-2√32√31-√3-1+√3-1-√31+√30000    orthogonal faithful

Permutation representations of Q8⋊5D12
►On 24 points - transitive group 24T365
Generators in S24
(1 10 4 7)(2 11 5 8)(3 12 6 9)(13 22 19 16)(14 23 20 17)(15 24 21 18)

(1 22 4 16)(2 20 5 14)(3 18 6 24)(7 19 10 13)(8 17 11 23)(9 15 12 21)

(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)

(1 6)(2 5)(3 4)(7 9)(10 12)(13 18)(14 17)(15 16)(19 24)(20 23)(21 22)
 
G:=sub<Sym(24)| (1,10,4,7)(2,11,5,8)(3,12,6,9)(13,22,19,16)(14,23,20,17)(15,24,21,18), (1,22,4,16)(2,20,5,14)(3,18,6,24)(7,19,10,13)(8,17,11,23)(9,15,12,21), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,6)(2,5)(3,4)(7,9)(10,12)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)>;
 
G:=Group( (1,10,4,7)(2,11,5,8)(3,12,6,9)(13,22,19,16)(14,23,20,17)(15,24,21,18), (1,22,4,16)(2,20,5,14)(3,18,6,24)(7,19,10,13)(8,17,11,23)(9,15,12,21), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,6)(2,5)(3,4)(7,9)(10,12)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22) );
 
G=PermutationGroup([[(1,10,4,7),(2,11,5,8),(3,12,6,9),(13,22,19,16),(14,23,20,17),(15,24,21,18)], [(1,22,4,16),(2,20,5,14),(3,18,6,24),(7,19,10,13),(8,17,11,23),(9,15,12,21)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,6),(2,5),(3,4),(7,9),(10,12),(13,18),(14,17),(15,16),(19,24),(20,23),(21,22)]])
 
G:=TransitiveGroup(24,365);
 

Matrix representation of Q8⋊5D12 ►in GL4(𝔽73) generated by

71400
596600
006659
00147
,
0010
0001
72000
07200
,
727200
1000
00766
00714
,
727200
0100
00766
005966
G:=sub<GL(4,GF(73))| [7,59,0,0,14,66,0,0,0,0,66,14,0,0,59,7],[0,0,72,0,0,0,0,72,1,0,0,0,0,1,0,0],[72,1,0,0,72,0,0,0,0,0,7,7,0,0,66,14],[72,0,0,0,72,1,0,0,0,0,7,59,0,0,66,66] >;
 

Q8⋊5D12 in GAP, Magma, Sage, TeX 

Q_8\rtimes_5D_{12}
 
% in TeX
 
G:=Group("Q8:5D12");
 
// GroupNames label
 
G:=SmallGroup(192,381);
 
// by ID
 
G=gap.SmallGroup(192,381);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,58,570,1684,851,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=c^12=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,c*b*c^-1=d*b*d=a^-1*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of Q8⋊5D12 in TeX

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