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## G = C9.S4order 216 = 23·33

### The non-split extension by C9 of S4 acting via S4/A4=C2

Aliases: C9.S4, C22⋊D27, C9.A4⋊C2, (C2×C6).D9, (C2×C18).S3, C3.(C3.S4), SmallGroup(216,21)

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=e2=1, d3=a, ab=ba, ac=ca, ad=da, eae=a-1, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=a-1d2 >

3C2
54C2
27C4
27C22
3C6
18S3
27D4
9D6
9Dic3
3C18
6D9
4C27
3D18
3Dic9
4D27

Character table of C9.S4

 class 1 2A 2B 3 4 6 9A 9B 9C 18A 18B 18C 27A 27B 27C 27D 27E 27F 27G 27H 27I size 1 3 54 2 54 6 2 2 2 6 6 6 8 8 8 8 8 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 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 0 2 2 2 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ4 2 2 0 2 0 2 -1 -1 -1 -1 -1 -1 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 orthogonal lifted from D9 ρ5 2 2 0 2 0 2 -1 -1 -1 -1 -1 -1 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 orthogonal lifted from D9 ρ6 2 2 0 2 0 2 -1 -1 -1 -1 -1 -1 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 orthogonal lifted from D9 ρ7 2 2 0 -1 0 -1 ζ2715+ζ2712 ζ2724+ζ273 ζ2721+ζ276 ζ2721+ζ276 ζ2715+ζ2712 ζ2724+ζ273 ζ2725+ζ272 ζ2714+ζ2713 ζ2722+ζ275 ζ2723+ζ274 ζ2719+ζ278 ζ2726+ζ27 ζ2717+ζ2710 ζ2720+ζ277 ζ2716+ζ2711 orthogonal lifted from D27 ρ8 2 2 0 -1 0 -1 ζ2721+ζ276 ζ2715+ζ2712 ζ2724+ζ273 ζ2724+ζ273 ζ2721+ζ276 ζ2715+ζ2712 ζ2726+ζ27 ζ2720+ζ277 ζ2716+ζ2711 ζ2725+ζ272 ζ2723+ζ274 ζ2714+ζ2713 ζ2722+ζ275 ζ2717+ζ2710 ζ2719+ζ278 orthogonal lifted from D27 ρ9 2 2 0 -1 0 -1 ζ2724+ζ273 ζ2721+ζ276 ζ2715+ζ2712 ζ2715+ζ2712 ζ2724+ζ273 ζ2721+ζ276 ζ2714+ζ2713 ζ2717+ζ2710 ζ2719+ζ278 ζ2726+ζ27 ζ2725+ζ272 ζ2720+ζ277 ζ2716+ζ2711 ζ2722+ζ275 ζ2723+ζ274 orthogonal lifted from D27 ρ10 2 2 0 -1 0 -1 ζ2715+ζ2712 ζ2724+ζ273 ζ2721+ζ276 ζ2721+ζ276 ζ2715+ζ2712 ζ2724+ζ273 ζ2720+ζ277 ζ2722+ζ275 ζ2723+ζ274 ζ2714+ζ2713 ζ2726+ζ27 ζ2717+ζ2710 ζ2719+ζ278 ζ2716+ζ2711 ζ2725+ζ272 orthogonal lifted from D27 ρ11 2 2 0 -1 0 -1 ζ2721+ζ276 ζ2715+ζ2712 ζ2724+ζ273 ζ2724+ζ273 ζ2721+ζ276 ζ2715+ζ2712 ζ2719+ζ278 ζ2725+ζ272 ζ2720+ζ277 ζ2716+ζ2711 ζ2722+ζ275 ζ2723+ζ274 ζ2714+ζ2713 ζ2726+ζ27 ζ2717+ζ2710 orthogonal lifted from D27 ρ12 2 2 0 -1 0 -1 ζ2715+ζ2712 ζ2724+ζ273 ζ2721+ζ276 ζ2721+ζ276 ζ2715+ζ2712 ζ2724+ζ273 ζ2716+ζ2711 ζ2723+ζ274 ζ2714+ζ2713 ζ2722+ζ275 ζ2717+ζ2710 ζ2719+ζ278 ζ2726+ζ27 ζ2725+ζ272 ζ2720+ζ277 orthogonal lifted from D27 ρ13 2 2 0 -1 0 -1 ζ2724+ζ273 ζ2721+ζ276 ζ2715+ζ2712 ζ2715+ζ2712 ζ2724+ζ273 ζ2721+ζ276 ζ2722+ζ275 ζ2719+ζ278 ζ2726+ζ27 ζ2717+ζ2710 ζ2720+ζ277 ζ2716+ζ2711 ζ2725+ζ272 ζ2723+ζ274 ζ2714+ζ2713 orthogonal lifted from D27 ρ14 2 2 0 -1 0 -1 ζ2724+ζ273 ζ2721+ζ276 ζ2715+ζ2712 ζ2715+ζ2712 ζ2724+ζ273 ζ2721+ζ276 ζ2723+ζ274 ζ2726+ζ27 ζ2717+ζ2710 ζ2719+ζ278 ζ2716+ζ2711 ζ2725+ζ272 ζ2720+ζ277 ζ2714+ζ2713 ζ2722+ζ275 orthogonal lifted from D27 ρ15 2 2 0 -1 0 -1 ζ2721+ζ276 ζ2715+ζ2712 ζ2724+ζ273 ζ2724+ζ273 ζ2721+ζ276 ζ2715+ζ2712 ζ2717+ζ2710 ζ2716+ζ2711 ζ2725+ζ272 ζ2720+ζ277 ζ2714+ζ2713 ζ2722+ζ275 ζ2723+ζ274 ζ2719+ζ278 ζ2726+ζ27 orthogonal lifted from D27 ρ16 3 -1 1 3 -1 -1 3 3 3 -1 -1 -1 0 0 0 0 0 0 0 0 0 orthogonal lifted from S4 ρ17 3 -1 -1 3 1 -1 3 3 3 -1 -1 -1 0 0 0 0 0 0 0 0 0 orthogonal lifted from S4 ρ18 6 -2 0 6 0 -2 -3 -3 -3 1 1 1 0 0 0 0 0 0 0 0 0 orthogonal lifted from C3.S4 ρ19 6 -2 0 -3 0 1 3ζ98+3ζ9 3ζ97+3ζ92 3ζ95+3ζ94 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ20 6 -2 0 -3 0 1 3ζ97+3ζ92 3ζ95+3ζ94 3ζ98+3ζ9 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ21 6 -2 0 -3 0 1 3ζ95+3ζ94 3ζ98+3ζ9 3ζ97+3ζ92 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 0 0 0 0 0 0 0 0 0 orthogonal faithful

Smallest permutation representation of C9.S4
On 54 points
Generators in S54
```(1 4 7 10 13 16 19 22 25)(2 5 8 11 14 17 20 23 26)(3 6 9 12 15 18 21 24 27)(28 31 34 37 40 43 46 49 52)(29 32 35 38 41 44 47 50 53)(30 33 36 39 42 45 48 51 54)
(1 46)(2 47)(4 49)(5 50)(7 52)(8 53)(10 28)(11 29)(13 31)(14 32)(16 34)(17 35)(19 37)(20 38)(22 40)(23 41)(25 43)(26 44)
(2 47)(3 48)(5 50)(6 51)(8 53)(9 54)(11 29)(12 30)(14 32)(15 33)(17 35)(18 36)(20 38)(21 39)(23 41)(24 42)(26 44)(27 45)
(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 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54)
(1 46)(2 45)(3 44)(4 43)(5 42)(6 41)(7 40)(8 39)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(20 54)(21 53)(22 52)(23 51)(24 50)(25 49)(26 48)(27 47)```

`G:=sub<Sym(54)| (1,4,7,10,13,16,19,22,25)(2,5,8,11,14,17,20,23,26)(3,6,9,12,15,18,21,24,27)(28,31,34,37,40,43,46,49,52)(29,32,35,38,41,44,47,50,53)(30,33,36,39,42,45,48,51,54), (1,46)(2,47)(4,49)(5,50)(7,52)(8,53)(10,28)(11,29)(13,31)(14,32)(16,34)(17,35)(19,37)(20,38)(22,40)(23,41)(25,43)(26,44), (2,47)(3,48)(5,50)(6,51)(8,53)(9,54)(11,29)(12,30)(14,32)(15,33)(17,35)(18,36)(20,38)(21,39)(23,41)(24,42)(26,44)(27,45), (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)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54), (1,46)(2,45)(3,44)(4,43)(5,42)(6,41)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,54)(21,53)(22,52)(23,51)(24,50)(25,49)(26,48)(27,47)>;`

`G:=Group( (1,4,7,10,13,16,19,22,25)(2,5,8,11,14,17,20,23,26)(3,6,9,12,15,18,21,24,27)(28,31,34,37,40,43,46,49,52)(29,32,35,38,41,44,47,50,53)(30,33,36,39,42,45,48,51,54), (1,46)(2,47)(4,49)(5,50)(7,52)(8,53)(10,28)(11,29)(13,31)(14,32)(16,34)(17,35)(19,37)(20,38)(22,40)(23,41)(25,43)(26,44), (2,47)(3,48)(5,50)(6,51)(8,53)(9,54)(11,29)(12,30)(14,32)(15,33)(17,35)(18,36)(20,38)(21,39)(23,41)(24,42)(26,44)(27,45), (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)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54), (1,46)(2,45)(3,44)(4,43)(5,42)(6,41)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,54)(21,53)(22,52)(23,51)(24,50)(25,49)(26,48)(27,47) );`

`G=PermutationGroup([[(1,4,7,10,13,16,19,22,25),(2,5,8,11,14,17,20,23,26),(3,6,9,12,15,18,21,24,27),(28,31,34,37,40,43,46,49,52),(29,32,35,38,41,44,47,50,53),(30,33,36,39,42,45,48,51,54)], [(1,46),(2,47),(4,49),(5,50),(7,52),(8,53),(10,28),(11,29),(13,31),(14,32),(16,34),(17,35),(19,37),(20,38),(22,40),(23,41),(25,43),(26,44)], [(2,47),(3,48),(5,50),(6,51),(8,53),(9,54),(11,29),(12,30),(14,32),(15,33),(17,35),(18,36),(20,38),(21,39),(23,41),(24,42),(26,44),(27,45)], [(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),(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)], [(1,46),(2,45),(3,44),(4,43),(5,42),(6,41),(7,40),(8,39),(9,38),(10,37),(11,36),(12,35),(13,34),(14,33),(15,32),(16,31),(17,30),(18,29),(19,28),(20,54),(21,53),(22,52),(23,51),(24,50),(25,49),(26,48),(27,47)]])`

C9.S4 is a maximal quotient of   Q8.D27  Q8⋊D27  C18.S4

Matrix representation of C9.S4 in GL5(𝔽109)

 32 82 0 0 0 27 59 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 0 108 0 0 0 0 0 108 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 108 0 0 0 0 0 108
,
 17 7 0 0 0 102 10 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 1 0 0 0 0 108 108 0 0 0 0 0 108 0 0 0 0 0 0 108 0 0 0 108 0

`G:=sub<GL(5,GF(109))| [32,27,0,0,0,82,59,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,108,0,0,0,0,0,108,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,108,0,0,0,0,0,108],[17,102,0,0,0,7,10,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[1,108,0,0,0,0,108,0,0,0,0,0,108,0,0,0,0,0,0,108,0,0,0,108,0] >;`

C9.S4 in GAP, Magma, Sage, TeX

`C_9.S_4`
`% in TeX`

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

`G:=SmallGroup(216,21);`
`// by ID`

`G=gap.SmallGroup(216,21);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-2,2,121,187,542,122,867,3244,1630,1949,2927]);`
`// Polycyclic`

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

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