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## G = A4⋊D12order 288 = 25·32

### The semidirect product of A4 and D12 acting via D12/D6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C6×A4 — A4⋊D12
 Chief series C1 — C22 — C2×C6 — C3×A4 — C6×A4 — C2×S3×A4 — A4⋊D12
 Lower central C3×A4 — C6×A4 — A4⋊D12
 Upper central C1 — C2

Generators and relations for A4⋊D12
G = < a,b,c,d,e | a2=b2=c3=d12=e2=1, cac-1=dad-1=eae=ab=ba, cbc-1=a, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 942 in 134 conjugacy classes, 19 normal (all characteristic)
C1, C2, C2 [×5], C3, C3 [×2], C4 [×3], C22, C22 [×12], S3 [×6], C6, C6 [×5], C2×C4 [×3], D4 [×6], C23, C23 [×5], C32, Dic3 [×2], C12 [×2], A4, A4, D6, D6 [×11], C2×C6, C2×C6 [×3], C22⋊C4 [×3], C2×D4 [×3], C24, C3×S3, C3⋊S3, C3×C6, D12 [×4], C2×Dic3, C3⋊D4 [×3], C2×C12 [×2], S4 [×3], C2×A4, C2×A4 [×2], C22×S3 [×5], C22×C6, C22≀C2, C3×Dic3, C3×A4, S3×C6, C2×C3⋊S3, D6⋊C4 [×2], C3×C22⋊C4, A4⋊C4, C2×D12 [×2], C2×C3⋊D4, C2×S4 [×2], C22×A4, S3×C23, C3⋊D12, C3⋊S4, S3×A4, C6×A4, D6⋊D4, A4⋊D4, C3×A4⋊C4, C2×C3⋊S4, C2×S3×A4, A4⋊D12
Quotients: C1, C2 [×3], C22, S3 [×2], D4, D6 [×2], D12, C3⋊D4, S4, S32, C2×S4, C3⋊D12, A4⋊D4, S3×S4, A4⋊D12

Character table of A4⋊D12

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 3C 4A 4B 4C 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D size 1 1 3 3 6 18 36 2 8 16 12 12 36 2 6 6 8 16 24 24 12 12 12 12 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 2 2 2 2 0 0 0 -1 2 -1 2 2 0 -1 -1 -1 2 -1 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 2 2 2 0 0 0 -1 2 -1 -2 -2 0 -1 -1 -1 2 -1 0 0 1 1 1 1 orthogonal lifted from D6 ρ7 2 -2 -2 2 0 0 0 2 2 2 0 0 0 -2 -2 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ8 2 2 2 2 2 2 0 2 -1 -1 0 0 0 2 2 2 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ9 2 2 2 2 -2 -2 0 2 -1 -1 0 0 0 2 2 2 -1 -1 1 1 0 0 0 0 orthogonal lifted from D6 ρ10 2 -2 -2 2 0 0 0 -1 2 -1 0 0 0 1 1 -1 -2 1 0 0 -√3 -√3 √3 √3 orthogonal lifted from D12 ρ11 2 -2 -2 2 0 0 0 -1 2 -1 0 0 0 1 1 -1 -2 1 0 0 √3 √3 -√3 -√3 orthogonal lifted from D12 ρ12 2 -2 -2 2 0 0 0 2 -1 -1 0 0 0 -2 -2 2 1 1 -√-3 √-3 0 0 0 0 complex lifted from C3⋊D4 ρ13 2 -2 -2 2 0 0 0 2 -1 -1 0 0 0 -2 -2 2 1 1 √-3 -√-3 0 0 0 0 complex lifted from C3⋊D4 ρ14 3 3 -1 -1 -3 1 -1 3 0 0 1 -1 1 3 -1 -1 0 0 0 0 1 -1 1 -1 orthogonal lifted from C2×S4 ρ15 3 3 -1 -1 3 -1 1 3 0 0 1 -1 -1 3 -1 -1 0 0 0 0 1 -1 1 -1 orthogonal lifted from S4 ρ16 3 3 -1 -1 -3 1 1 3 0 0 -1 1 -1 3 -1 -1 0 0 0 0 -1 1 -1 1 orthogonal lifted from C2×S4 ρ17 3 3 -1 -1 3 -1 -1 3 0 0 -1 1 1 3 -1 -1 0 0 0 0 -1 1 -1 1 orthogonal lifted from S4 ρ18 4 4 4 4 0 0 0 -2 -2 1 0 0 0 -2 -2 -2 -2 1 0 0 0 0 0 0 orthogonal lifted from S32 ρ19 4 -4 -4 4 0 0 0 -2 -2 1 0 0 0 2 2 -2 2 -1 0 0 0 0 0 0 orthogonal lifted from C3⋊D12 ρ20 6 6 -2 -2 0 0 0 -3 0 0 2 -2 0 -3 1 1 0 0 0 0 -1 1 -1 1 orthogonal lifted from S3×S4 ρ21 6 6 -2 -2 0 0 0 -3 0 0 -2 2 0 -3 1 1 0 0 0 0 1 -1 1 -1 orthogonal lifted from S3×S4 ρ22 6 -6 2 -2 0 0 0 6 0 0 0 0 0 -6 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from A4⋊D4 ρ23 6 -6 2 -2 0 0 0 -3 0 0 0 0 0 3 -1 1 0 0 0 0 -√3 √3 √3 -√3 orthogonal faithful ρ24 6 -6 2 -2 0 0 0 -3 0 0 0 0 0 3 -1 1 0 0 0 0 √3 -√3 -√3 √3 orthogonal faithful

Smallest permutation representation of A4⋊D12
On 36 points
Generators in S36
```(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(14 20)(16 22)(18 24)(26 32)(28 34)(30 36)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(1 22 35)(2 36 23)(3 24 25)(4 26 13)(5 14 27)(6 28 15)(7 16 29)(8 30 17)(9 18 31)(10 32 19)(11 20 33)(12 34 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)(25 26 27 28 29 30 31 32 33 34 35 36)
(1 3)(4 12)(5 11)(6 10)(7 9)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 36)(24 35)```

`G:=sub<Sym(36)| (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(14,20)(16,22)(18,24)(26,32)(28,34)(30,36), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (1,22,35)(2,36,23)(3,24,25)(4,26,13)(5,14,27)(6,28,15)(7,16,29)(8,30,17)(9,18,31)(10,32,19)(11,20,33)(12,34,21), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36), (1,3)(4,12)(5,11)(6,10)(7,9)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,36)(24,35)>;`

`G:=Group( (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(14,20)(16,22)(18,24)(26,32)(28,34)(30,36), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (1,22,35)(2,36,23)(3,24,25)(4,26,13)(5,14,27)(6,28,15)(7,16,29)(8,30,17)(9,18,31)(10,32,19)(11,20,33)(12,34,21), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36), (1,3)(4,12)(5,11)(6,10)(7,9)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,36)(24,35) );`

`G=PermutationGroup([(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(14,20),(16,22),(18,24),(26,32),(28,34),(30,36)], [(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36)], [(1,22,35),(2,36,23),(3,24,25),(4,26,13),(5,14,27),(6,28,15),(7,16,29),(8,30,17),(9,18,31),(10,32,19),(11,20,33),(12,34,21)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,34),(14,33),(15,32),(16,31),(17,30),(18,29),(19,28),(20,27),(21,26),(22,25),(23,36),(24,35)])`

Matrix representation of A4⋊D12 in GL5(𝔽13)

 1 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 12 0 0 0 1 12 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 12 0 0 1 0 12 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 11 7 0 0 0 6 11 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1
,
 7 11 0 0 0 11 6 0 0 0 0 0 0 12 0 0 0 12 0 0 0 0 0 0 12

`G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,12,12,12,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,12,12,12],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[11,6,0,0,0,7,11,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1],[7,11,0,0,0,11,6,0,0,0,0,0,0,12,0,0,0,12,0,0,0,0,0,0,12] >;`

A4⋊D12 in GAP, Magma, Sage, TeX

`A_4\rtimes D_{12}`
`% in TeX`

`G:=Group("A4:D12");`
`// GroupNames label`

`G:=SmallGroup(288,858);`
`// by ID`

`G=gap.SmallGroup(288,858);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,85,36,234,1684,3036,782,1777,1350]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^12=e^2=1,c*a*c^-1=d*a*d^-1=e*a*e=a*b=b*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;`
`// generators/relations`

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