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## G = D4⋊6D6order 96 = 25·3

### 2nd semidirect product of D4 and D6 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — D4⋊6D6
 Chief series C1 — C3 — C6 — D6 — C22×S3 — S3×D4 — D4⋊6D6
 Lower central C3 — C6 — D4⋊6D6
 Upper central C1 — C2 — C2×D4

Generators and relations for D46D6
G = < a,b,c,d | a4=b2=c6=d2=1, bab=cac-1=a-1, ad=da, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 370 in 166 conjugacy classes, 85 normal (11 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C2×D4, C2×D4, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, 2+ 1+4, C4○D12, S3×D4, D42S3, C2×C3⋊D4, C6×D4, D46D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, S3×C23, D46D6

Character table of D46D6

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 6G 12A 12B size 1 1 2 2 2 2 2 6 6 6 6 2 2 2 6 6 6 6 2 2 2 4 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 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 -2 -2 0 0 0 0 -1 2 -2 0 0 0 0 1 1 -1 -1 1 1 -1 1 -1 orthogonal lifted from D6 ρ18 2 2 2 -2 -2 2 2 0 0 0 0 -1 -2 -2 0 0 0 0 -1 -1 -1 1 -1 1 -1 1 1 orthogonal lifted from D6 ρ19 2 2 -2 2 2 -2 2 0 0 0 0 -1 -2 -2 0 0 0 0 -1 -1 -1 -1 1 -1 1 1 1 orthogonal lifted from D6 ρ20 2 2 -2 -2 2 2 -2 0 0 0 0 -1 2 -2 0 0 0 0 1 1 -1 1 -1 -1 1 1 -1 orthogonal lifted from D6 ρ21 2 2 2 -2 2 -2 -2 0 0 0 0 -1 -2 2 0 0 0 0 1 1 -1 1 1 -1 -1 -1 1 orthogonal lifted from D6 ρ22 2 2 2 2 2 2 2 0 0 0 0 -1 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ23 2 2 -2 -2 -2 -2 2 0 0 0 0 -1 2 2 0 0 0 0 -1 -1 -1 1 1 1 1 -1 -1 orthogonal lifted from D6 ρ24 2 2 -2 2 -2 2 -2 0 0 0 0 -1 -2 2 0 0 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 0 0 -4 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√-3 -2√-3 2 0 0 0 0 0 0 complex faithful ρ27 4 -4 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 -2√-3 2√-3 2 0 0 0 0 0 0 complex faithful

Permutation representations of D46D6
On 24 points - transitive group 24T95
Generators in S24
```(1 23 7 20)(2 21 8 24)(3 19 9 22)(4 18 11 15)(5 16 12 13)(6 14 10 17)
(1 23)(2 21)(3 19)(4 15)(5 13)(6 17)(7 20)(8 24)(9 22)(10 14)(11 18)(12 16)
(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 8)(5 7)(6 9)(13 23)(14 22)(15 21)(16 20)(17 19)(18 24)```

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

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

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

`G:=TransitiveGroup(24,95);`

On 24 points - transitive group 24T100
Generators in S24
```(1 22 14 9)(2 10 15 23)(3 24 16 11)(4 12 17 19)(5 20 18 7)(6 8 13 21)
(1 9)(2 23)(3 11)(4 19)(5 7)(6 21)(8 13)(10 15)(12 17)(14 22)(16 24)(18 20)
(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 8)(2 7)(3 12)(4 11)(5 10)(6 9)(13 22)(14 21)(15 20)(16 19)(17 24)(18 23)```

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

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

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

`G:=TransitiveGroup(24,100);`

Matrix representation of D46D6 in GL4(𝔽7) generated by

 3 1 0 0 4 4 0 0 4 5 6 3 0 3 4 1
,
 6 6 5 6 4 0 1 3 1 1 6 5 1 6 3 2
,
 4 4 1 5 5 4 6 5 6 4 6 3 4 4 5 0
,
 6 6 1 1 0 6 0 1 0 5 1 1 0 0 0 1
`G:=sub<GL(4,GF(7))| [3,4,4,0,1,4,5,3,0,0,6,4,0,0,3,1],[6,4,1,1,6,0,1,6,5,1,6,3,6,3,5,2],[4,5,6,4,4,4,4,4,1,6,6,5,5,5,3,0],[6,0,0,0,6,6,5,0,1,0,1,0,1,1,1,1] >;`

D46D6 in GAP, Magma, Sage, TeX

`D_4\rtimes_6D_6`
`% in TeX`

`G:=Group("D4:6D6");`
`// GroupNames label`

`G:=SmallGroup(96,211);`
`// by ID`

`G=gap.SmallGroup(96,211);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,188,579,2309]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=c^6=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;`
`// generators/relations`

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