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## G = D4○D12order 96 = 25·3

### Central product of D4 and D12

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, Q83S3, 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 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 12A 12B 12C 12D 12E size 1 1 2 2 2 6 6 6 6 6 6 2 2 2 2 2 6 6 2 4 4 4 2 2 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 -1 1 1 1 -1 1 -1 -1 1 1 -1 -1 -1 1 -1 1 -1 1 1 -1 -1 -1 1 -1 linear of order 2 ρ3 1 1 -1 -1 -1 1 1 -1 -1 1 -1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 -1 1 1 1 linear of order 2 ρ4 1 1 -1 1 -1 1 1 1 -1 -1 1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 1 -1 linear of order 2 ρ5 1 1 -1 1 1 1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 1 -1 -1 linear of order 2 ρ6 1 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 1 linear of order 2 ρ7 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 -1 1 -1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 linear of order 2 ρ8 1 1 1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 -1 -1 -1 1 linear of order 2 ρ9 1 1 -1 -1 -1 -1 -1 1 1 -1 1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 linear of order 2 ρ10 1 1 -1 1 -1 -1 -1 -1 1 1 -1 1 1 -1 -1 1 1 1 1 1 -1 -1 1 1 -1 1 -1 linear of order 2 ρ11 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ12 1 1 1 -1 1 -1 -1 1 -1 1 1 1 1 -1 -1 -1 -1 1 1 -1 1 1 -1 -1 -1 1 -1 linear of order 2 ρ13 1 1 1 -1 -1 -1 1 1 1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 -1 1 1 1 1 -1 -1 linear of order 2 ρ14 1 1 1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 1 -1 -1 1 1 1 -1 1 -1 -1 -1 -1 1 linear of order 2 ρ15 1 1 -1 1 1 -1 1 -1 -1 1 1 1 -1 1 -1 -1 1 -1 1 1 1 -1 -1 -1 1 -1 -1 linear of order 2 ρ16 1 1 -1 -1 1 -1 1 1 -1 -1 -1 1 -1 -1 1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 linear of order 2 ρ17 2 2 2 -2 -2 0 0 0 0 0 0 -1 -2 2 -2 2 0 0 -1 1 1 -1 -1 -1 -1 1 1 orthogonal lifted from D6 ρ18 2 2 -2 -2 2 0 0 0 0 0 0 -1 -2 -2 2 2 0 0 -1 1 -1 1 -1 -1 1 1 -1 orthogonal lifted from D6 ρ19 2 2 -2 2 2 0 0 0 0 0 0 -1 -2 2 -2 -2 0 0 -1 -1 -1 1 1 1 -1 1 1 orthogonal lifted from D6 ρ20 2 2 2 2 -2 0 0 0 0 0 0 -1 -2 -2 2 -2 0 0 -1 -1 1 -1 1 1 1 1 -1 orthogonal lifted from D6 ρ21 2 2 -2 2 -2 0 0 0 0 0 0 -1 2 -2 -2 2 0 0 -1 -1 1 1 -1 -1 1 -1 1 orthogonal lifted from D6 ρ22 2 2 2 2 2 0 0 0 0 0 0 -1 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ23 2 2 2 -2 2 0 0 0 0 0 0 -1 2 -2 -2 -2 0 0 -1 1 -1 -1 1 1 1 -1 1 orthogonal lifted from D6 ρ24 2 2 -2 -2 -2 0 0 0 0 0 0 -1 2 2 2 -2 0 0 -1 1 1 1 1 1 -1 -1 -1 orthogonal lifted from D6 ρ25 4 -4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 -4 0 0 0 0 0 0 0 0 orthogonal lifted from 2+ 1+4 ρ26 4 -4 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 2 0 0 0 2√3 -2√3 0 0 0 orthogonal faithful ρ27 4 -4 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 2 0 0 0 -2√3 2√3 0 0 0 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);`

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

 0 0 1 0 0 0 0 1 12 0 0 0 0 12 0 0
,
 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0
,
 10 3 0 0 10 7 0 0 0 0 10 3 0 0 10 7
,
 1 1 0 0 0 12 0 0 0 0 1 1 0 0 0 12
`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`

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