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## G = C42⋊D9order 288 = 25·32

### The semidirect product of C42 and D9 acting via D9/C3=S3

Aliases: C42⋊D9, (C4×C12).S3, C42⋊C92C2, (C2×C6).2S4, C3.(C42⋊S3), C22.(C3.S4), SmallGroup(288,67)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C42⋊C9 — C42⋊D9
 Chief series C1 — C22 — C42 — C4×C12 — C42⋊C9 — C42⋊D9
 Lower central C42⋊C9 — C42⋊D9
 Upper central C1

Generators and relations for C42⋊D9
G = < a,b,c,d | a4=b4=c9=d2=1, ab=ba, cac-1=dbd=a-1b-1, ad=da, cbc-1=a, dcd=c-1 >

3C2
36C2
3C4
3C4
18C22
18C4
3C6
12S3
16C9
9D4
9Q8
18C2×C4
18C8
18D4
3C12
3C12
6D6
6Dic3
16D9
3Dic6
3D12

Character table of C42⋊D9

 class 1 2A 2B 3 4A 4B 4C 4D 6 8A 8B 9A 9B 9C 12A 12B 12C 12D size 1 3 36 2 3 3 6 36 6 36 36 32 32 32 6 6 6 6 ρ1 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 linear of order 2 ρ3 2 2 0 2 2 2 2 0 2 0 0 -1 -1 -1 2 2 2 2 orthogonal lifted from S3 ρ4 2 2 0 -1 2 2 2 0 -1 0 0 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -1 -1 -1 -1 orthogonal lifted from D9 ρ5 2 2 0 -1 2 2 2 0 -1 0 0 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -1 -1 -1 -1 orthogonal lifted from D9 ρ6 2 2 0 -1 2 2 2 0 -1 0 0 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -1 -1 -1 -1 orthogonal lifted from D9 ρ7 3 3 -1 3 -1 -1 -1 -1 3 1 1 0 0 0 -1 -1 -1 -1 orthogonal lifted from S4 ρ8 3 3 1 3 -1 -1 -1 1 3 -1 -1 0 0 0 -1 -1 -1 -1 orthogonal lifted from S4 ρ9 3 -1 -1 3 -1-2i -1+2i 1 1 -1 i -i 0 0 0 -1-2i 1 1 -1+2i complex lifted from C42⋊S3 ρ10 3 -1 1 3 -1+2i -1-2i 1 -1 -1 i -i 0 0 0 -1+2i 1 1 -1-2i complex lifted from C42⋊S3 ρ11 3 -1 1 3 -1-2i -1+2i 1 -1 -1 -i i 0 0 0 -1-2i 1 1 -1+2i complex lifted from C42⋊S3 ρ12 3 -1 -1 3 -1+2i -1-2i 1 1 -1 -i i 0 0 0 -1+2i 1 1 -1-2i complex lifted from C42⋊S3 ρ13 6 -2 0 6 2 2 -2 0 -2 0 0 0 0 0 2 -2 -2 2 orthogonal lifted from C42⋊S3 ρ14 6 6 0 -3 -2 -2 -2 0 -3 0 0 0 0 0 1 1 1 1 orthogonal lifted from C3.S4 ρ15 6 -2 0 -3 2 2 -2 0 1 0 0 0 0 0 -1 1+2√3 1-2√3 -1 orthogonal faithful ρ16 6 -2 0 -3 2 2 -2 0 1 0 0 0 0 0 -1 1-2√3 1+2√3 -1 orthogonal faithful ρ17 6 -2 0 -3 -2-4i -2+4i 2 0 1 0 0 0 0 0 1+2i -1 -1 1-2i complex faithful ρ18 6 -2 0 -3 -2+4i -2-4i 2 0 1 0 0 0 0 0 1-2i -1 -1 1+2i complex faithful

Smallest permutation representation of C42⋊D9
On 36 points
Generators in S36
```(1 34 19 10)(3 12 21 36)(4 28 22 13)(6 15 24 30)(7 31 25 16)(9 18 27 33)
(1 10 19 34)(2 35 20 11)(4 13 22 28)(5 29 23 14)(7 16 25 31)(8 32 26 17)
(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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 36)(8 35)(9 34)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)```

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

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

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

Matrix representation of C42⋊D9 in GL5(𝔽73)

 1 0 0 0 0 0 1 0 0 0 0 0 27 0 0 0 0 0 72 0 0 0 0 0 27
,
 1 0 0 0 0 0 1 0 0 0 0 0 72 0 0 0 0 0 27 0 0 0 0 0 27
,
 28 70 0 0 0 3 31 0 0 0 0 0 0 13 0 0 0 0 0 72 0 0 28 0 0
,
 28 70 0 0 0 42 45 0 0 0 0 0 0 0 13 0 0 0 72 0 0 0 45 0 0

`G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,27,0,0,0,0,0,72,0,0,0,0,0,27],[1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,27,0,0,0,0,0,27],[28,3,0,0,0,70,31,0,0,0,0,0,0,0,28,0,0,13,0,0,0,0,0,72,0],[28,42,0,0,0,70,45,0,0,0,0,0,0,0,45,0,0,0,72,0,0,0,13,0,0] >;`

C42⋊D9 in GAP, Magma, Sage, TeX

`C_4^2\rtimes D_9`
`% in TeX`

`G:=Group("C4^2:D9");`
`// GroupNames label`

`G:=SmallGroup(288,67);`
`// by ID`

`G=gap.SmallGroup(288,67);`
`# by ID`

`G:=PCGroup([7,-2,-3,-3,-2,2,-2,2,141,92,254,1011,514,360,634,3476,102,9077,3036,5305]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=c^9=d^2=1,a*b=b*a,c*a*c^-1=d*b*d=a^-1*b^-1,a*d=d*a,c*b*c^-1=a,d*c*d=c^-1>;`
`// generators/relations`

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