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G = D4○D12order 96 = 25·3

Central product of D4 and D12

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4D12, D48D6, Q88D6, Q8Dic6, C6.12C24, D6.7C23, C322+ 1+4, D1211C22, C12.26C23, Dic612C22, Dic3.7C23, (C2×C4)⋊4D6, C4○D45S3, (S3×D4)⋊5C2, C4○D128C2, (C2×D12)⋊13C2, (C4×S3)⋊2C22, (C2×C12)⋊5C22, Q83S35C2, (C3×D4)⋊9C22, C3⋊D45C22, (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

C1C6 — D4○D12
C1C3C6D6C22×S3S3×D4 — D4○D12
C3C6 — D4○D12
C1C2C4○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 [×9], C3, C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], S3 [×6], C6, C6 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×3], D4 [×15], Q8, Q8, C23 [×6], Dic3 [×2], C12, C12 [×3], D6 [×6], D6 [×6], C2×C6 [×3], C2×D4 [×9], C4○D4, C4○D4 [×5], Dic6, C4×S3 [×6], D12 [×9], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C22×S3 [×6], 2+ 1+4, C2×D12 [×3], C4○D12 [×3], S3×D4 [×6], Q83S3 [×2], C3×C4○D4, D4○D12
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C24, C22×S3 [×7], 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
ρ264-4000000000-2000000200023-23000    orthogonal faithful
ρ274-4000000000-20000002000-2323000    orthogonal faithful

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

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

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

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

G:=TransitiveGroup(24,102);

D4○D12 is a maximal subgroup of
Q85D12  C425D6  D1218D4  D12.39D4  D4.11D12  D4.12D12  D815D6  D811D6  D85D6  C24.C23  D12.32C23  D12.34C23  C6.C25  S3×2+ 1+4  D12.39C23  D48D18  Dic6.A4  D1224D6  D1227D6  D1213D6  D1216D6  C62.154C23  D2026D6  D2029D6  D1214D10  D2017D6  D48D30
D4○D12 is a maximal quotient of
C42.90D6  C429D6  C42.91D6  C4211D6  C4212D6  C42.95D6  C42.97D6  C42.99D6  C42.100D6  D46Dic6  C4213D6  D4×D12  D1223D4  Dic624D4  C4219D6  C42.116D6  C42.117D6  C42.119D6  Q8×Dic6  C42.126D6  Q87D12  D1210Q8  C42.133D6  C42.136D6  C6.372+ 1+4  C6.382+ 1+4  D1219D4  C6.462+ 1+4  C6.1152+ 1+4  C6.472+ 1+4  C6.482+ 1+4  C6.172- 1+4  D1221D4  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  C4220D6  D1210D4  C4222D6  C42.143D6  C42.144D6  C4224D6  C42.145D6  C42.148D6  D127Q8  C42.150D6  C42.153D6  C42.155D6  C42.157D6  C42.158D6  C4225D6  C4226D6  C42.161D6  C42.163D6  C42.164D6  C4227D6  C42.165D6  C6.1442+ 1+4  C6.1452+ 1+4  C6.1462+ 1+4  C6.1082- 1+4  C6.1482+ 1+4  D48D18  D1224D6  D1227D6  D1213D6  D1216D6  C62.154C23  D2026D6  D2029D6  D1214D10  D2017D6  D48D30

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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