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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, (C2xC6):1D4, (C2xC4):1D6, D6:C4:4C2, C3:1C22wrC2, C6.5(C2xD4), C2.7(S3xD4), (C2xD12):2C2, C22:C4:2S3, C2.7(C2xD12), (S3xC23):1C2, (C2xC12):1C22, (C2xC6).23C23, (C22xS3):1C22, (C2xDic3):1C22, (C22xC6).12C22, C22.41(C22xS3), (C2xC3:D4):1C2, (C3xC22:C4):3C2, SmallGroup(96,89)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2xC6 — D6:D4
Chief series
C1 — C3 — C6 — C2xC6 — C22xS3 — S3xC23 — D6:D4
Lower central
C3 — C2xC6 — 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, C2xC4, C2xC4, D4, C23, C23, Dic3, C12, D6, D6, C2xC6, C2xC6, C2xC6, C22:C4, C22:C4, C2xD4, C24, D12, C2xDic3, C3:D4, C2xC12, C22xS3, C22xS3, C22xS3, C22xC6, C22wrC2, D6:C4, C3xC22:C4, C2xD12, C2xC3:D4, S3xC23, D6:D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, D12, C22xS3, C22wrC2, C2xD12, S3xD4, 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 S3xD4
ρ244-4-440000000-200022-2000000    orthogonal lifted from S3xD4

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  D4xD12  D4:5D12  C42:19D6  S3xC22wrC2  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  (C2xC10):11D12  D30:19D4  D30:16D4
D6:D4 is a maximal quotient of
 (C2xC4):Dic6  (C2xC4):9D12  D6:C4:C4  (C2xC12):5D4  C6.C22wrC2  (C22xS3):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  (C2xC10):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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