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## G = C9⋊S4order 216 = 23·33

### The semidirect product of C9 and S4 acting via S4/A4=C2

Aliases: C9⋊S4, A4⋊D9, (C9×A4)⋊2C2, (C2×C18)⋊2S3, C3.A41S3, C3.1(C3⋊S4), (C3×A4).2S3, C221(C9⋊S3), (C2×C6).2(C3⋊S3), SmallGroup(216,93)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C9×A4 — C9⋊S4
 Chief series C1 — C22 — C2×C6 — C2×C18 — C9×A4 — C9⋊S4
 Lower central C9×A4 — C9⋊S4
 Upper central C1

Generators and relations for C9⋊S4
G = < a,b,c,d,e | a9=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >

3C2
54C2
4C3
4C3
4C3
27C4
27C22
3C6
18S3
36S3
36S3
36S3
4C9
4C9
4C32
27D4
9Dic3
9D6
3C18
6D9
12C3⋊S3
12D9
12D9
9S4
9S4
9S4
3D18
3Dic9

Character table of C9⋊S4

 class 1 2A 2B 3A 3B 3C 3D 4 6 9A 9B 9C 9D 9E 9F 9G 9H 9I 18A 18B 18C size 1 3 54 2 8 8 8 54 6 2 2 2 8 8 8 8 8 8 6 6 6 ρ1 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 linear of order 2 ρ3 2 2 0 2 -1 -1 -1 0 2 -1 -1 -1 2 -1 -1 -1 2 2 -1 -1 -1 orthogonal lifted from S3 ρ4 2 2 0 2 -1 -1 -1 0 2 2 2 2 -1 -1 -1 -1 -1 -1 2 2 2 orthogonal lifted from S3 ρ5 2 2 0 2 -1 -1 -1 0 2 -1 -1 -1 -1 2 2 2 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 2 0 2 2 2 2 0 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ7 2 2 0 -1 -1 -1 2 0 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 orthogonal lifted from D9 ρ8 2 2 0 -1 2 -1 -1 0 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 orthogonal lifted from D9 ρ9 2 2 0 -1 2 -1 -1 0 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 orthogonal lifted from D9 ρ10 2 2 0 -1 -1 -1 2 0 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 orthogonal lifted from D9 ρ11 2 2 0 -1 -1 2 -1 0 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ95+ζ94 ζ97+ζ92 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 orthogonal lifted from D9 ρ12 2 2 0 -1 -1 -1 2 0 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 orthogonal lifted from D9 ρ13 2 2 0 -1 2 -1 -1 0 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 orthogonal lifted from D9 ρ14 2 2 0 -1 -1 2 -1 0 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ97+ζ92 ζ98+ζ9 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 orthogonal lifted from D9 ρ15 2 2 0 -1 -1 2 -1 0 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ98+ζ9 ζ95+ζ94 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 orthogonal lifted from D9 ρ16 3 -1 1 3 0 0 0 -1 -1 3 3 3 0 0 0 0 0 0 -1 -1 -1 orthogonal lifted from S4 ρ17 3 -1 -1 3 0 0 0 1 -1 3 3 3 0 0 0 0 0 0 -1 -1 -1 orthogonal lifted from S4 ρ18 6 -2 0 6 0 0 0 0 -2 -3 -3 -3 0 0 0 0 0 0 1 1 1 orthogonal lifted from C3⋊S4 ρ19 6 -2 0 -3 0 0 0 0 1 3ζ97+3ζ92 3ζ95+3ζ94 3ζ98+3ζ9 0 0 0 0 0 0 -ζ95-ζ94 -ζ97-ζ92 -ζ98-ζ9 orthogonal faithful ρ20 6 -2 0 -3 0 0 0 0 1 3ζ98+3ζ9 3ζ97+3ζ92 3ζ95+3ζ94 0 0 0 0 0 0 -ζ97-ζ92 -ζ98-ζ9 -ζ95-ζ94 orthogonal faithful ρ21 6 -2 0 -3 0 0 0 0 1 3ζ95+3ζ94 3ζ98+3ζ9 3ζ97+3ζ92 0 0 0 0 0 0 -ζ98-ζ9 -ζ95-ζ94 -ζ97-ζ92 orthogonal faithful

Smallest permutation representation of C9⋊S4
On 36 points
Generators in S36
```(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 12)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 10)(9 11)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 28)(26 29)(27 30)
(1 27)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)
(1 7 4)(2 8 5)(3 9 6)(10 22 31)(11 23 32)(12 24 33)(13 25 34)(14 26 35)(15 27 36)(16 19 28)(17 20 29)(18 21 30)
(2 9)(3 8)(4 7)(5 6)(10 32)(11 31)(12 30)(13 29)(14 28)(15 36)(16 35)(17 34)(18 33)(19 26)(20 25)(21 24)(22 23)```

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

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

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

C9⋊S4 is a maximal subgroup of   D9×S4
C9⋊S4 is a maximal quotient of   C18.5S4  C18.6S4  A4⋊Dic9

Matrix representation of C9⋊S4 in GL5(𝔽37)

 31 20 0 0 0 17 11 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 36 36 36 0 0 1 0 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 36 36 36 0 0 0 0 1 0 0 0 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 36 36 36 0 0 0 1 0
,
 1 0 0 0 0 36 36 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

`G:=sub<GL(5,GF(37))| [31,17,0,0,0,20,11,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,36,1,0,0,0,36,0,0,0,1,36,0],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,36,0,1,0,0,36,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,36,0,0,0,0,36,1,0,0,0,36,0],[1,36,0,0,0,0,36,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;`

C9⋊S4 in GAP, Magma, Sage, TeX

`C_9\rtimes S_4`
`% in TeX`

`G:=Group("C9:S4");`
`// GroupNames label`

`G:=SmallGroup(216,93);`
`// by ID`

`G=gap.SmallGroup(216,93);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-2,2,49,542,122,867,3244,1630,1949,2927]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^9=b^2=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,d*b*d^-1=e*b*e=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e=d^-1>;`
`// generators/relations`

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