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G = D4oD12order 96 = 25·3

Central product of D4 and D12

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4oD12, D4:8D6, Q8:8D6, Q8oDic6, C6.12C24, D6.7C23, C3:22+ 1+4, D12:11C22, C12.26C23, Dic6:12C22, Dic3.7C23, (C2xC4):4D6, C4oD4:5S3, (S3xD4):5C2, C4oD12:8C2, (C2xD12):13C2, (C4xS3):2C22, (C2xC12):5C22, Q8:3S3:5C2, (C3xD4):9C22, C3:D4:5C22, (C2xC6).4C23, (C3xQ8):8C22, C2.13(S3xC23), C4.33(C22xS3), (C22xS3):4C22, C22.3(C22xS3), (C3xC4oD4):4C2, SmallGroup(96,216)

Series: Derived Chief Lower central Upper central

C1C6 — D4oD12
C1C3C6D6C22xS3S3xD4 — D4oD12
C3C6 — D4oD12
C1C2C4oD4

Generators and relations for D4oD12
 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, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, D6, C2xC6, C2xD4, C4oD4, C4oD4, Dic6, C4xS3, D12, C3:D4, C2xC12, C3xD4, C3xQ8, C22xS3, 2+ 1+4, C2xD12, C4oD12, S3xD4, Q8:3S3, C3xC4oD4, D4oD12
Quotients: C1, C2, C22, S3, C23, D6, C24, C22xS3, 2+ 1+4, S3xC23, D4oD12

Character table of D4oD12

 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
ρ264-4000000000-2000000200023-23000    orthogonal faithful
ρ274-4000000000-20000002000-2323000    orthogonal faithful

Permutation representations of D4oD12
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);

D4oD12 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  S3x2+ 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
D4oD12 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  D4xD12  D12:23D4  Dic6:24D4  C42:19D6  C42.116D6  C42.117D6  C42.119D6  Q8xDic6  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 D4oD12 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] >;

D4oD12 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 D4oD12 in TeX

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