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## G = C4○D4⋊A4order 192 = 26·3

### 1st semidirect product of C4○D4 and A4 acting via A4/C22=C3

Aliases: C4○D41A4, C22⋊(C4.A4), Q8.1(C2×A4), Q8⋊A43C2, (C22×C4).9A4, C4.1(C22⋊A4), C23.23(C2×A4), (C22×Q8).7C6, (C22×C4○D4)⋊3C3, C2.3(C2×C22⋊A4), SmallGroup(192,1507)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C22×Q8 — C4○D4⋊A4
 Chief series C1 — C2 — Q8 — C22×Q8 — Q8⋊A4 — C4○D4⋊A4
 Lower central C22×Q8 — C4○D4⋊A4
 Upper central C1 — C4

Generators and relations for C4○D4⋊A4
G = < a,b,c,d,e,f | a4=c2=d2=e2=f3=1, b2=a2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=a2b, bd=db, be=eb, fbf-1=a-1bc, cd=dc, ce=ec, fcf-1=a-1b, fdf-1=de=ed, fef-1=d >

Subgroups: 599 in 175 conjugacy classes, 18 normal (10 characteristic)
C1, C2, C2 [×6], C3, C4, C4 [×5], C22, C22 [×18], C6, C2×C4 [×24], D4 [×16], Q8 [×4], Q8 [×4], C23, C23 [×10], C12, A4, C22×C4, C22×C4 [×13], C2×D4 [×12], C2×Q8 [×4], C4○D4 [×4], C4○D4 [×20], C24, SL2(𝔽3) [×4], C2×A4, C23×C4, C22×D4, C22×Q8, C2×C4○D4 [×8], C4×A4, C4.A4 [×4], C22×C4○D4, Q8⋊A4, C4○D4⋊A4
Quotients: C1, C2, C3, C6, A4 [×5], C2×A4 [×5], C4.A4, C22⋊A4, C2×C22⋊A4, C4○D4⋊A4

Character table of C4○D4⋊A4

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 12A 12B 12C 12D size 1 1 3 3 6 6 6 6 16 16 1 1 3 3 6 6 6 6 16 16 16 16 16 16 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 ζ3 ζ32 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ4 1 1 1 1 -1 -1 -1 -1 ζ32 ζ3 -1 -1 -1 -1 1 1 1 1 ζ32 ζ3 ζ65 ζ6 ζ65 ζ6 linear of order 6 ρ5 1 1 1 1 -1 -1 -1 -1 ζ3 ζ32 -1 -1 -1 -1 1 1 1 1 ζ3 ζ32 ζ6 ζ65 ζ6 ζ65 linear of order 6 ρ6 1 1 1 1 1 1 1 1 ζ32 ζ3 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ7 2 -2 2 -2 0 0 0 0 -1 -1 -2i 2i 2i -2i 0 0 0 0 1 1 -i -i i i complex lifted from C4.A4 ρ8 2 -2 2 -2 0 0 0 0 -1 -1 2i -2i -2i 2i 0 0 0 0 1 1 i i -i -i complex lifted from C4.A4 ρ9 2 -2 2 -2 0 0 0 0 ζ6 ζ65 -2i 2i 2i -2i 0 0 0 0 ζ32 ζ3 ζ43ζ3 ζ43ζ32 ζ4ζ3 ζ4ζ32 complex lifted from C4.A4 ρ10 2 -2 2 -2 0 0 0 0 ζ65 ζ6 -2i 2i 2i -2i 0 0 0 0 ζ3 ζ32 ζ43ζ32 ζ43ζ3 ζ4ζ32 ζ4ζ3 complex lifted from C4.A4 ρ11 2 -2 2 -2 0 0 0 0 ζ65 ζ6 2i -2i -2i 2i 0 0 0 0 ζ3 ζ32 ζ4ζ32 ζ4ζ3 ζ43ζ32 ζ43ζ3 complex lifted from C4.A4 ρ12 2 -2 2 -2 0 0 0 0 ζ6 ζ65 2i -2i -2i 2i 0 0 0 0 ζ32 ζ3 ζ4ζ3 ζ4ζ32 ζ43ζ3 ζ43ζ32 complex lifted from C4.A4 ρ13 3 3 -1 -1 1 1 -3 1 0 0 -3 -3 1 1 -1 -1 -1 3 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ14 3 3 3 3 -1 -1 -1 -1 0 0 3 3 3 3 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ15 3 3 -1 -1 -1 3 -1 -1 0 0 3 3 -1 -1 -1 -1 3 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ16 3 3 -1 -1 -3 1 1 1 0 0 -3 -3 1 1 -1 3 -1 -1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ17 3 3 -1 -1 1 1 1 -3 0 0 -3 -3 1 1 3 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ18 3 3 -1 -1 1 -3 1 1 0 0 -3 -3 1 1 -1 -1 3 -1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ19 3 3 -1 -1 -1 -1 -1 3 0 0 3 3 -1 -1 3 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ20 3 3 -1 -1 3 -1 -1 -1 0 0 3 3 -1 -1 -1 3 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ21 3 3 -1 -1 -1 -1 3 -1 0 0 3 3 -1 -1 -1 -1 -1 3 0 0 0 0 0 0 orthogonal lifted from A4 ρ22 3 3 3 3 1 1 1 1 0 0 -3 -3 -3 -3 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ23 6 -6 -2 2 0 0 0 0 0 0 6i -6i 2i -2i 0 0 0 0 0 0 0 0 0 0 complex faithful ρ24 6 -6 -2 2 0 0 0 0 0 0 -6i 6i -2i 2i 0 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C4○D4⋊A4
On 24 points - transitive group 24T302
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 10 3 12)(2 11 4 9)(5 8 7 6)(13 18 15 20)(14 19 16 17)(21 22 23 24)
(1 12)(2 9)(3 10)(4 11)(5 21)(6 22)(7 23)(8 24)(17 19)(18 20)
(1 3)(2 4)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(21 23)(22 24)
(1 24 16)(2 21 13)(3 22 14)(4 23 15)(5 18 10)(6 19 11)(7 20 12)(8 17 9)```

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

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

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

`G:=TransitiveGroup(24,302);`

Matrix representation of C4○D4⋊A4 in GL5(𝔽13)

 5 0 0 0 0 0 5 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 4 3 0 0 0 3 9 0 0 0 0 0 0 1 12 0 0 1 0 12 0 0 0 0 12
,
 11 7 0 0 0 7 2 0 0 0 0 0 1 0 0 0 0 1 0 12 0 0 1 12 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 12 0 1 0 0 12 1 0
,
 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 3 9 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0

`G:=sub<GL(5,GF(13))| [5,0,0,0,0,0,5,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[4,3,0,0,0,3,9,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,12,12,12],[11,7,0,0,0,7,2,0,0,0,0,0,1,1,1,0,0,0,0,12,0,0,0,12,0],[1,0,0,0,0,0,1,0,0,0,0,0,12,12,12,0,0,0,0,1,0,0,0,1,0],[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],[1,3,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0] >;`

C4○D4⋊A4 in GAP, Magma, Sage, TeX

`C_4\circ D_4\rtimes A_4`
`% in TeX`

`G:=Group("C4oD4:A4");`
`// GroupNames label`

`G:=SmallGroup(192,1507);`
`// by ID`

`G=gap.SmallGroup(192,1507);`
`# by ID`

`G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,672,135,262,851,172,1524,285,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^4=c^2=d^2=e^2=f^3=1,b^2=a^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=a^2*b,b*d=d*b,b*e=e*b,f*b*f^-1=a^-1*b*c,c*d=d*c,c*e=e*c,f*c*f^-1=a^-1*b,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;`
`// generators/relations`

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