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## G = A4⋊M4(2)  order 192 = 26·3

### The semidirect product of A4 and M4(2) acting via M4(2)/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×A4 — A4⋊M4(2)
 Chief series C1 — C22 — A4 — C2×A4 — C4×A4 — A4⋊C8 — A4⋊M4(2)
 Lower central A4 — C2×A4 — A4⋊M4(2)
 Upper central C1 — C4 — C2×C4

Generators and relations for A4⋊M4(2)
G = < a,b,c,d,e | a2=b2=c3=d8=e2=1, cac-1=dad-1=ab=ba, ae=ea, cbc-1=a, bd=db, be=eb, dcd-1=c-1, ce=ec, ede=d5 >

Subgroups: 254 in 81 conjugacy classes, 21 normal (19 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4 [×2], C22 [×2], C22 [×8], C6 [×2], C8 [×4], C2×C4, C2×C4 [×8], C23, C23 [×4], C12 [×2], A4, C2×C6, C2×C8 [×4], M4(2) [×4], C22×C4 [×2], C22×C4 [×4], C24, C3⋊C8 [×2], C2×C12, C2×A4, C2×A4, C22⋊C8 [×4], C2×M4(2) [×2], C23×C4, C4.Dic3, C4×A4 [×2], C22×A4, C24.4C4, A4⋊C8 [×2], C2×C4×A4, A4⋊M4(2)
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, Dic3 [×2], D6, M4(2), C2×Dic3, S4, C4.Dic3, A4⋊C4 [×2], C2×S4, C2×A4⋊C4, A4⋊M4(2)

Character table of A4⋊M4(2)

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D size 1 1 2 3 3 6 8 1 1 2 3 3 6 8 8 8 12 12 12 12 12 12 12 12 8 8 8 8 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 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 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 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 1 linear of order 2 ρ5 1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 -1 i i i -i -i -i -i i 1 1 -1 -1 linear of order 4 ρ6 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 -i i i -i -i i i -i -1 -1 -1 -1 linear of order 4 ρ7 1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 -1 -i -i -i i i i i -i 1 1 -1 -1 linear of order 4 ρ8 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 i -i -i i i -i -i i -1 -1 -1 -1 linear of order 4 ρ9 2 2 2 2 2 2 -1 2 2 2 2 2 2 -1 -1 -1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 -2 2 2 -2 -1 2 2 -2 2 2 -2 1 -1 1 0 0 0 0 0 0 0 0 1 1 -1 -1 orthogonal lifted from D6 ρ11 2 2 2 2 2 2 -1 -2 -2 -2 -2 -2 -2 -1 -1 -1 0 0 0 0 0 0 0 0 1 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ12 2 2 -2 2 2 -2 -1 -2 -2 2 -2 -2 2 1 -1 1 0 0 0 0 0 0 0 0 -1 -1 1 1 symplectic lifted from Dic3, Schur index 2 ρ13 2 -2 0 -2 2 0 2 2i -2i 0 2i -2i 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 -2i 2i complex lifted from M4(2) ρ14 2 -2 0 -2 2 0 2 -2i 2i 0 -2i 2i 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 2i -2i complex lifted from M4(2) ρ15 2 -2 0 -2 2 0 -1 -2i 2i 0 -2i 2i 0 -√-3 1 √-3 0 0 0 0 0 0 0 0 -√3 √3 -i i complex lifted from C4.Dic3 ρ16 2 -2 0 -2 2 0 -1 2i -2i 0 2i -2i 0 -√-3 1 √-3 0 0 0 0 0 0 0 0 √3 -√3 i -i complex lifted from C4.Dic3 ρ17 2 -2 0 -2 2 0 -1 -2i 2i 0 -2i 2i 0 √-3 1 -√-3 0 0 0 0 0 0 0 0 √3 -√3 -i i complex lifted from C4.Dic3 ρ18 2 -2 0 -2 2 0 -1 2i -2i 0 2i -2i 0 √-3 1 -√-3 0 0 0 0 0 0 0 0 -√3 √3 i -i complex lifted from C4.Dic3 ρ19 3 3 -3 -1 -1 1 0 3 3 -3 -1 -1 1 0 0 0 -1 -1 1 -1 1 1 -1 1 0 0 0 0 orthogonal lifted from C2×S4 ρ20 3 3 -3 -1 -1 1 0 3 3 -3 -1 -1 1 0 0 0 1 1 -1 1 -1 -1 1 -1 0 0 0 0 orthogonal lifted from C2×S4 ρ21 3 3 3 -1 -1 -1 0 3 3 3 -1 -1 -1 0 0 0 1 -1 1 -1 1 -1 1 -1 0 0 0 0 orthogonal lifted from S4 ρ22 3 3 3 -1 -1 -1 0 3 3 3 -1 -1 -1 0 0 0 -1 1 -1 1 -1 1 -1 1 0 0 0 0 orthogonal lifted from S4 ρ23 3 3 -3 -1 -1 1 0 -3 -3 3 1 1 -1 0 0 0 i -i i i -i i -i -i 0 0 0 0 complex lifted from A4⋊C4 ρ24 3 3 3 -1 -1 -1 0 -3 -3 -3 1 1 1 0 0 0 i i -i -i i i -i -i 0 0 0 0 complex lifted from A4⋊C4 ρ25 3 3 -3 -1 -1 1 0 -3 -3 3 1 1 -1 0 0 0 -i i -i -i i -i i i 0 0 0 0 complex lifted from A4⋊C4 ρ26 3 3 3 -1 -1 -1 0 -3 -3 -3 1 1 1 0 0 0 -i -i i i -i -i i i 0 0 0 0 complex lifted from A4⋊C4 ρ27 6 -6 0 2 -2 0 0 6i -6i 0 -2i 2i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ28 6 -6 0 2 -2 0 0 -6i 6i 0 2i -2i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of A4⋊M4(2)
On 24 points - transitive group 24T294
Generators in S24
```(1 5)(3 7)(9 13)(10 14)(11 15)(12 16)(17 21)(19 23)
(1 5)(2 6)(3 7)(4 8)(17 21)(18 22)(19 23)(20 24)
(1 22 11)(2 12 23)(3 24 13)(4 14 17)(5 18 15)(6 16 19)(7 20 9)(8 10 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)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)```

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

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

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

`G:=TransitiveGroup(24,294);`

Matrix representation of A4⋊M4(2) in GL5(𝔽73)

 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 72 72 72 0 0 1 0 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 72 72 72 0 0 0 0 1 0 0 0 1 0
,
 64 71 0 0 0 0 8 0 0 0 0 0 1 0 0 0 0 72 72 72 0 0 0 1 0
,
 8 51 0 0 0 5 65 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0
,
 72 47 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1

`G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,72,0,0,0,1,72,0],[1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,72,0,1,0,0,72,1,0],[64,0,0,0,0,71,8,0,0,0,0,0,1,72,0,0,0,0,72,1,0,0,0,72,0],[8,5,0,0,0,51,65,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0],[72,0,0,0,0,47,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;`

A4⋊M4(2) in GAP, Magma, Sage, TeX

`A_4\rtimes M_4(2)`
`% in TeX`

`G:=Group("A4:M4(2)");`
`// GroupNames label`

`G:=SmallGroup(192,968);`
`// by ID`

`G=gap.SmallGroup(192,968);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,28,141,58,1124,4037,285,2358,475]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^8=e^2=1,c*a*c^-1=d*a*d^-1=a*b=b*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e=d^5>;`
`// generators/relations`

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