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G = D4○D12  order 96 = 25·3

Central product of D4 and D12

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4○D12, D4⋊8D6, Q8⋊8D6, Q8○Dic6, C6.12C24, D6.7C23, C3⋊22+ 1+4, D12⋊11C22, C12.26C23, Dic6⋊12C22, Dic3.7C23, (C2×C4)⋊4D6, C4○D4⋊5S3, (S3×D4)⋊5C2, C4○D12⋊8C2, (C2×D12)⋊13C2, (C4×S3)⋊2C22, (C2×C12)⋊5C22, Q8⋊3S3⋊5C2, (C3×D4)⋊9C22, C3⋊D4⋊5C22, (C2×C6).4C23, (C3×Q8)⋊8C22, C2.13(S3×C23), C4.33(C22×S3), (C22×S3)⋊4C22, C22.3(C22×S3), (C3×C4○D4)⋊4C2, SmallGroup(96,216)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — D4○D12
Chief series
C1 — C3 — C6 — D6 — C22×S3 — S3×D4 — D4○D12
Lower central
C3 — C6 — D4○D12
Upper central
C1 — C2 — C4○D4

Generators and relations for D4○D12
 G = < a,b,c,d | a4=b2=d2=1, c6=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c5 >

Subgroups: 394 in 166 conjugacy classes, 85 normal (12 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×D4, C4○D4, C4○D4, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3, 2+ 1+4, C2×D12, C4○D12, S3×D4, Q8⋊3S3, C3×C4○D4, D4○D12
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, S3×C23, D4○D12

Character table of D4○D12

 class 12A2B2C2D2E2F2G2H2I2J34A4B4C4D4E4F6A6B6C6D12A12B12C12D12E
 size 112226666662222266244422444
ρ1111111111111111111111111111    trivial
ρ2111-1111-11-1-111-1-1-11-11-111-1-1-11-1    linear of order 2
ρ311-1-1-111-1-11-11111-1-111-1-1-1-1-1111    linear of order 2
ρ411-11-1111-1-1111-1-11-1-111-1-111-11-1    linear of order 2
ρ511-1111-111-1-11-11-1-1-11111-1-1-11-1-1    linear of order 2
ρ611-1-111-1-11111-1-111-1-11-11-111-1-11    linear of order 2
ρ7111-1-11-1-1-1-111-11-11111-1-11111-1-1    linear of order 2
ρ81111-11-11-11-11-1-11-11-111-11-1-1-1-11    linear of order 2
ρ911-1-1-1-1-111-111111-11-11-1-1-1-1-1111    linear of order 2
ρ1011-11-1-1-1-111-111-1-111111-1-111-11-1    linear of order 2
ρ1111111-1-1-1-1-1-111111-1-1111111111    linear of order 2
ρ12111-11-1-11-11111-1-1-1-111-111-1-1-11-1    linear of order 2
ρ13111-1-1-11111-11-11-11-1-11-1-11111-1-1    linear of order 2
ρ141111-1-11-11-111-1-11-1-1111-11-1-1-1-11    linear of order 2
ρ1511-111-11-1-1111-11-1-11-1111-1-1-11-1-1    linear of order 2
ρ1611-1-11-111-1-1-11-1-111111-11-111-1-11    linear of order 2
ρ17222-2-2000000-1-22-2200-111-1-1-1-111    orthogonal lifted from D6
ρ1822-2-22000000-1-2-22200-11-11-1-111-1    orthogonal lifted from D6
ρ1922-222000000-1-22-2-200-1-1-1111-111    orthogonal lifted from D6
ρ202222-2000000-1-2-22-200-1-11-11111-1    orthogonal lifted from D6
ρ2122-22-2000000-12-2-2200-1-111-1-11-11    orthogonal lifted from D6
ρ2222222000000-1222200-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ23222-22000000-12-2-2-200-11-1-1111-11    orthogonal lifted from D6
ρ2422-2-2-2000000-1222-200-111111-1-1-1    orthogonal lifted from D6
ρ254-40000000004000000-400000000    orthogonal lifted from 2+ 1+4
ρ264-4000000000-200000020002√3-2√3000    orthogonal faithful
ρ274-4000000000-20000002000-2√32√3000    orthogonal faithful

Permutation representations of D4○D12
►On 24 points - transitive group 24T102
Generators in S24
(1 13 7 19)(2 14 8 20)(3 15 9 21)(4 16 10 22)(5 17 11 23)(6 18 12 24)

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

(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)
 
G:=sub<Sym(24)| (1,13,7,19)(2,14,8,20)(3,15,9,21)(4,16,10,22)(5,17,11,23)(6,18,12,24), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)>;
 
G:=Group( (1,13,7,19)(2,14,8,20)(3,15,9,21)(4,16,10,22)(5,17,11,23)(6,18,12,24), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19) );
 
G=PermutationGroup([[(1,13,7,19),(2,14,8,20),(3,15,9,21),(4,16,10,22),(5,17,11,23),(6,18,12,24)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19)]])
 
G:=TransitiveGroup(24,102);
 

D4○D12 is a maximal subgroup of
 Q8⋊5D12  C42⋊5D6  D12⋊18D4  D12.39D4  D4.11D12  D4.12D12  D8⋊15D6  D8⋊11D6  D8⋊5D6  C24.C23  D12.32C23  D12.34C23  C6.C25  S3×2+ 1+4  D12.39C23  D4⋊8D18  Dic6.A4  D12⋊24D6  D12⋊27D6  D12⋊13D6  D12⋊16D6  C62.154C23  D20⋊26D6  D20⋊29D6  D12⋊14D10  D20⋊17D6  D4⋊8D30
D4○D12 is a maximal quotient of
 C42.90D6  C42⋊9D6  C42.91D6  C42⋊11D6  C42⋊12D6  C42.95D6  C42.97D6  C42.99D6  C42.100D6  D4⋊6Dic6  C42⋊13D6  D4×D12  D12⋊23D4  Dic6⋊24D4  C42⋊19D6  C42.116D6  C42.117D6  C42.119D6  Q8×Dic6  C42.126D6  Q8⋊7D12  D12⋊10Q8  C42.133D6  C42.136D6  C6.372+ 1+4  C6.382+ 1+4  D12⋊19D4  C6.462+ 1+4  C6.1152+ 1+4  C6.472+ 1+4  C6.482+ 1+4  C6.172- 1+4  D12⋊21D4  C6.512+ 1+4  C6.1182+ 1+4  C6.242- 1+4  C6.562+ 1+4  C6.592+ 1+4  C6.1202+ 1+4  C6.1212+ 1+4  C6.612+ 1+4  C6.1222+ 1+4  C6.662+ 1+4  C6.852- 1+4  C6.682+ 1+4  C6.692+ 1+4  C42⋊20D6  D12⋊10D4  C42⋊22D6  C42.143D6  C42.144D6  C42⋊24D6  C42.145D6  C42.148D6  D12⋊7Q8  C42.150D6  C42.153D6  C42.155D6  C42.157D6  C42.158D6  C42⋊25D6  C42⋊26D6  C42.161D6  C42.163D6  C42.164D6  C42⋊27D6  C42.165D6  C6.1442+ 1+4  C6.1452+ 1+4  C6.1462+ 1+4  C6.1082- 1+4  C6.1482+ 1+4  D4⋊8D18  D12⋊24D6  D12⋊27D6  D12⋊13D6  D12⋊16D6  C62.154C23  D20⋊26D6  D20⋊29D6  D12⋊14D10  D20⋊17D6  D4⋊8D30

Matrix representation of D4○D12 ►in GL4(𝔽13) generated by

0010
0001
12000
01200
,
0010
0001
1000
0100
,
10300
10700
00103
00107
,
1100
01200
0011
00012
G:=sub<GL(4,GF(13))| [0,0,12,0,0,0,0,12,1,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[10,10,0,0,3,7,0,0,0,0,10,10,0,0,3,7],[1,0,0,0,1,12,0,0,0,0,1,0,0,0,1,12] >;
 

D4○D12 in GAP, Magma, Sage, TeX 

D_4\circ D_{12}
 
% in TeX
 
G:=Group("D4oD12");
 
// GroupNames label
 
G:=SmallGroup(96,216);
 
// by ID
 
G=gap.SmallGroup(96,216);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,188,579,69,2309]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^6=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^2*c^5>;
 
// generators/relations
 

Export

Character table of D4○D12 in TeX

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