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## G = D12⋊C4order 96 = 25·3

### 4th semidirect product of D12 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — D12⋊C4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4○D12 — D12⋊C4
 Lower central C3 — C6 — C12 — D12⋊C4
 Upper central C1 — C4 — C2×C4 — M4(2)

Generators and relations for D12⋊C4
G = < a,b,c | a12=b2=c4=1, bab=a-1, cac-1=a5, cbc-1=a7b >

Character table of D12⋊C4

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 8A 8B 12A 12B 12C 24A 24B 24C 24D size 1 1 2 12 2 1 1 2 6 6 6 6 12 2 4 4 4 2 2 4 4 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 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 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 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 linear of order 2 ρ5 1 1 -1 1 1 -1 -1 1 i -i i -i -1 1 -1 -i i -1 -1 1 i -i -i i linear of order 4 ρ6 1 1 -1 -1 1 -1 -1 1 -i i -i i 1 1 -1 -i i -1 -1 1 i -i -i i linear of order 4 ρ7 1 1 -1 1 1 -1 -1 1 -i i -i i -1 1 -1 i -i -1 -1 1 -i i i -i linear of order 4 ρ8 1 1 -1 -1 1 -1 -1 1 i -i i -i 1 1 -1 i -i -1 -1 1 -i i i -i linear of order 4 ρ9 2 2 2 0 -1 2 2 2 0 0 0 0 0 -1 -1 2 2 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 2 0 -1 2 2 2 0 0 0 0 0 -1 -1 -2 -2 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ11 2 2 2 0 2 -2 -2 -2 0 0 0 0 0 2 2 0 0 -2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 0 2 2 2 -2 0 0 0 0 0 2 -2 0 0 2 2 -2 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 0 -1 -2 -2 -2 0 0 0 0 0 -1 -1 0 0 1 1 1 -√3 √3 -√3 √3 orthogonal lifted from D12 ρ14 2 2 2 0 -1 -2 -2 -2 0 0 0 0 0 -1 -1 0 0 1 1 1 √3 -√3 √3 -√3 orthogonal lifted from D12 ρ15 2 -2 0 0 2 -2i 2i 0 -1+i 1+i 1-i -1-i 0 -2 0 0 0 -2i 2i 0 0 0 0 0 complex lifted from C4≀C2 ρ16 2 -2 0 0 2 2i -2i 0 -1-i 1-i 1+i -1+i 0 -2 0 0 0 2i -2i 0 0 0 0 0 complex lifted from C4≀C2 ρ17 2 2 -2 0 -1 2 2 -2 0 0 0 0 0 -1 1 0 0 -1 -1 1 -√-3 -√-3 √-3 √-3 complex lifted from C3⋊D4 ρ18 2 2 -2 0 -1 2 2 -2 0 0 0 0 0 -1 1 0 0 -1 -1 1 √-3 √-3 -√-3 -√-3 complex lifted from C3⋊D4 ρ19 2 -2 0 0 2 2i -2i 0 1+i -1+i -1-i 1-i 0 -2 0 0 0 2i -2i 0 0 0 0 0 complex lifted from C4≀C2 ρ20 2 2 -2 0 -1 -2 -2 2 0 0 0 0 0 -1 1 -2i 2i 1 1 -1 -i i i -i complex lifted from C4×S3 ρ21 2 2 -2 0 -1 -2 -2 2 0 0 0 0 0 -1 1 2i -2i 1 1 -1 i -i -i i complex lifted from C4×S3 ρ22 2 -2 0 0 2 -2i 2i 0 1-i -1-i -1+i 1+i 0 -2 0 0 0 -2i 2i 0 0 0 0 0 complex lifted from C4≀C2 ρ23 4 -4 0 0 -2 4i -4i 0 0 0 0 0 0 2 0 0 0 -2i 2i 0 0 0 0 0 complex faithful ρ24 4 -4 0 0 -2 -4i 4i 0 0 0 0 0 0 2 0 0 0 2i -2i 0 0 0 0 0 complex faithful

Permutation representations of D12⋊C4
On 24 points - transitive group 24T106
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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)
(2 6)(3 11)(5 9)(8 12)(13 16 19 22)(14 21 20 15)(17 24 23 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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13), (2,6)(3,11)(5,9)(8,12)(13,16,19,22)(14,21,20,15)(17,24,23,18)>;`

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

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

`G:=TransitiveGroup(24,106);`

Matrix representation of D12⋊C4 in GL4(𝔽5) generated by

 3 0 0 1 0 2 4 0 0 4 0 0 1 0 0 0
,
 0 3 0 0 2 0 0 0 4 0 0 3 0 1 2 0
,
 2 0 0 4 0 1 0 0 0 2 4 0 0 0 0 3
`G:=sub<GL(4,GF(5))| [3,0,0,1,0,2,4,0,0,4,0,0,1,0,0,0],[0,2,4,0,3,0,0,1,0,0,0,2,0,0,3,0],[2,0,0,0,0,1,2,0,0,0,4,0,4,0,0,3] >;`

D12⋊C4 in GAP, Magma, Sage, TeX

`D_{12}\rtimes C_4`
`% in TeX`

`G:=Group("D12:C4");`
`// GroupNames label`

`G:=SmallGroup(96,32);`
`// by ID`

`G=gap.SmallGroup(96,32);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,121,31,86,579,297,69,2309]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^7*b>;`
`// generators/relations`

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