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G = D6⋊D4  order 96 = 25·3

1st semidirect product of D6 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6⋊4D4, C22⋊3D12, C23.20D6, (C2×C6)⋊1D4, (C2×C4)⋊1D6, D6⋊C4⋊4C2, C3⋊1C22≀C2, C6.5(C2×D4), C2.7(S3×D4), (C2×D12)⋊2C2, C22⋊C4⋊2S3, C2.7(C2×D12), (S3×C23)⋊1C2, (C2×C12)⋊1C22, (C2×C6).23C23, (C22×S3)⋊1C22, (C2×Dic3)⋊1C22, (C22×C6).12C22, C22.41(C22×S3), (C2×C3⋊D4)⋊1C2, (C3×C22⋊C4)⋊3C2, SmallGroup(96,89)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — D6⋊D4
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — D6⋊D4
Lower central
C3 — C2×C6 — D6⋊D4
Upper central
C1 — C22 — C22⋊C4

Generators and relations for D6⋊D4
 G = < a,b,c,d | a6=b2=c4=d2=1, bab=dad=a-1, ac=ca, cbc-1=a3b, dbd=ab, dcd=c-1 >

Subgroups: 378 in 130 conjugacy classes, 37 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C2×D4, C24, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×S3, C22×S3, C22×C6, C22≀C2, D6⋊C4, C3×C22⋊C4, C2×D12, C2×C3⋊D4, S3×C23, D6⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C22≀C2, C2×D12, S3×D4, D6⋊D4

Character table of D6⋊D4

 class 12A2B2C2D2E2F2G2H2I2J34A4B4C6A6B6C6D6E12A12B12C12D
 size 11112266661224412222444444
ρ1111111111111111111111111    trivial
ρ2111111-1-1-1-1-1111-1111111111    linear of order 2
ρ31111-1-1-111-1111-1-1111-1-1-1-111    linear of order 2
ρ41111-1-11-1-11-111-11111-1-1-1-111    linear of order 2
ρ51111111111-11-1-1-111111-1-1-1-1    linear of order 2
ρ6111111-1-1-1-111-1-1111111-1-1-1-1    linear of order 2
ρ71111-1-1-111-1-11-111111-1-111-1-1    linear of order 2
ρ81111-1-11-1-1111-11-1111-1-111-1-1    linear of order 2
ρ92-2-2200-200202000-2-22000000    orthogonal lifted from D4
ρ1022222200000-1-2-20-1-1-1-1-11111    orthogonal lifted from D6
ρ112-22-2000-220020002-2-2000000    orthogonal lifted from D4
ρ1222222200000-1220-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ132-22-20002-20020002-2-2000000    orthogonal lifted from D4
ρ142222-2-200000-12-20-1-1-11111-1-1    orthogonal lifted from D6
ρ1522-2-22-2000002000-22-22-20000    orthogonal lifted from D4
ρ162-2-2200200-202000-2-22000000    orthogonal lifted from D4
ρ172222-2-200000-1-220-1-1-111-1-111    orthogonal lifted from D6
ρ1822-2-2-22000002000-22-2-220000    orthogonal lifted from D4
ρ1922-2-22-200000-10001-11-11-√3√3√3-√3    orthogonal lifted from D12
ρ2022-2-2-2200000-10001-111-1-√3√3-√3√3    orthogonal lifted from D12
ρ2122-2-2-2200000-10001-111-1√3-√3√3-√3    orthogonal lifted from D12
ρ2222-2-22-200000-10001-11-11√3-√3-√3√3    orthogonal lifted from D12
ρ234-44-40000000-2000-222000000    orthogonal lifted from S3×D4
ρ244-4-440000000-200022-2000000    orthogonal lifted from S3×D4

Permutation representations of D6⋊D4
►On 24 points - transitive group 24T144
Generators in S24
(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 14)(2 13)(3 18)(4 17)(5 16)(6 15)(7 20)(8 19)(9 24)(10 23)(11 22)(12 21)

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

(1 22)(2 21)(3 20)(4 19)(5 24)(6 23)(7 17)(8 16)(9 15)(10 14)(11 13)(12 18)
 
G:=sub<Sym(24)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,14)(2,13)(3,18)(4,17)(5,16)(6,15)(7,20)(8,19)(9,24)(10,23)(11,22)(12,21), (1,19,15,12)(2,20,16,7)(3,21,17,8)(4,22,18,9)(5,23,13,10)(6,24,14,11), (1,22)(2,21)(3,20)(4,19)(5,24)(6,23)(7,17)(8,16)(9,15)(10,14)(11,13)(12,18)>;
 
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), (1,14)(2,13)(3,18)(4,17)(5,16)(6,15)(7,20)(8,19)(9,24)(10,23)(11,22)(12,21), (1,19,15,12)(2,20,16,7)(3,21,17,8)(4,22,18,9)(5,23,13,10)(6,24,14,11), (1,22)(2,21)(3,20)(4,19)(5,24)(6,23)(7,17)(8,16)(9,15)(10,14)(11,13)(12,18) );
 
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)], [(1,14),(2,13),(3,18),(4,17),(5,16),(6,15),(7,20),(8,19),(9,24),(10,23),(11,22),(12,21)], [(1,19,15,12),(2,20,16,7),(3,21,17,8),(4,22,18,9),(5,23,13,10),(6,24,14,11)], [(1,22),(2,21),(3,20),(4,19),(5,24),(6,23),(7,17),(8,16),(9,15),(10,14),(11,13),(12,18)]])
 
G:=TransitiveGroup(24,144);
 

D6⋊D4 is a maximal subgroup of
 C23⋊D12  C24.38D6  C23⋊4D12  C42⋊10D6  C42⋊11D6  C42⋊12D6  C42⋊14D6  D4×D12  D4⋊5D12  C42⋊19D6  S3×C22≀C2  C24⋊8D6  C24.45D6  C6.372+ 1+4  C6.382+ 1+4  D12⋊19D4  C6.482+ 1+4  C4⋊C4⋊26D6  D12⋊21D4  C6.532+ 1+4  C6.562+ 1+4  C6.1202+ 1+4  C6.1212+ 1+4  C4⋊C4⋊28D6  C6.612+ 1+4  C6.682+ 1+4  C42⋊20D6  D12⋊10D4  C42⋊22D6  C42⋊24D6  C42⋊25D6  C42⋊26D6  C42⋊27D6  C22⋊3D36  D6⋊4D12  D6⋊5D12  C62⋊5D4  C62⋊8D4  C62⋊12D4  A4⋊D12  D30⋊4D4  D30⋊5D4  (C2×C10)⋊11D12  D30⋊19D4  D30⋊16D4
D6⋊D4 is a maximal quotient of
 (C2×C4)⋊Dic6  (C2×C4)⋊9D12  D6⋊C4⋊C4  (C2×C12)⋊5D4  C6.C22≀C2  (C22×S3)⋊Q8  D12.31D4  D12⋊13D4  D12.32D4  D12⋊14D4  Dic6⋊14D4  Dic6.32D4  C23⋊D12  C23.5D12  M4(2)⋊D6  D12⋊1D4  D12.4D4  D12.5D4  D4⋊D12  D6⋊5SD16  D4⋊3D12  D4.D12  Q8⋊3D12  Q8.11D12  D6⋊Q16  Q8⋊4D12  Q8⋊5D12  C42⋊5D6  Q8.14D12  D4.10D12  C24.58D6  C24.59D6  C24.60D6  C23⋊3D12  C24.27D6  C22⋊3D36  D6⋊4D12  D6⋊5D12  C62⋊5D4  C62⋊8D4  C62⋊12D4  D30⋊4D4  D30⋊5D4  (C2×C10)⋊11D12  D30⋊19D4  D30⋊16D4

Matrix representation of D6⋊D4 ►in GL6(ℤ)

-100000
0-10000
000-100
001-100
000010
000001
,
100000
0-10000
001-100
000-100
0000-10
00000-1
,
010000
-100000
00-1000
000-100
000001
0000-10
,
010000
100000
000-100
00-1000
000001
000010

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,-1,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[0,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
 

D6⋊D4 in GAP, Magma, Sage, TeX 

D_6\rtimes D_4
 
% in TeX
 
G:=Group("D6:D4");
 
// GroupNames label
 
G:=SmallGroup(96,89);
 
// by ID
 
G=gap.SmallGroup(96,89);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,218,188,50,2309]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^2=c^4=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^3*b,d*b*d=a*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of D6⋊D4 in TeX

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