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## G = C9⋊A4order 108 = 22·33

### The semidirect product of C9 and A4 acting via A4/C22=C3

Aliases: C9⋊A4, C2213- 1+2, (C3×A4).C3, (C2×C18)⋊2C3, C3.A41C3, C3.3(C3×A4), (C2×C6).2C32, SmallGroup(108,19)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C9⋊A4
 Chief series C1 — C22 — C2×C6 — C3×A4 — C9⋊A4
 Lower central C22 — C2×C6 — C9⋊A4
 Upper central C1 — C3 — C9

Generators and relations for C9⋊A4
G = < a,b,c,d | a9=b2=c2=d3=1, ab=ba, ac=ca, dad-1=a7, dbd-1=bc=cb, dcd-1=b >

Character table of C9⋊A4

 class 1 2 3A 3B 3C 3D 6A 6B 9A 9B 9C 9D 9E 9F 18A 18B 18C 18D 18E 18F size 1 3 1 1 12 12 3 3 3 3 12 12 12 12 3 3 3 3 3 3 ρ1 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 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ3 1 1 1 1 ζ3 ζ32 1 1 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 linear of order 3 ρ4 1 1 1 1 ζ32 ζ3 1 1 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 linear of order 3 ρ5 1 1 1 1 ζ32 ζ3 1 1 ζ32 ζ3 ζ3 ζ32 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ6 1 1 1 1 ζ3 ζ32 1 1 ζ3 ζ32 ζ32 ζ3 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ7 1 1 1 1 ζ3 ζ32 1 1 ζ32 ζ3 1 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ8 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ9 1 1 1 1 ζ32 ζ3 1 1 ζ3 ζ32 1 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ10 3 -1 3 3 0 0 -1 -1 3 3 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from A4 ρ11 3 3 -3-3√-3/2 -3+3√-3/2 0 0 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from 3- 1+2 ρ12 3 3 -3+3√-3/2 -3-3√-3/2 0 0 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from 3- 1+2 ρ13 3 -1 3 3 0 0 -1 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 ζ6 ζ65 ζ65 ζ6 ζ65 ζ6 complex lifted from C3×A4 ρ14 3 -1 3 3 0 0 -1 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 ζ65 ζ6 ζ6 ζ65 ζ6 ζ65 complex lifted from C3×A4 ρ15 3 -1 -3-3√-3/2 -3+3√-3/2 0 0 ζ6 ζ65 0 0 0 0 0 0 2ζ98 2ζ9 2ζ97 2ζ92 2ζ94 2ζ95 complex faithful ρ16 3 -1 -3-3√-3/2 -3+3√-3/2 0 0 ζ6 ζ65 0 0 0 0 0 0 2ζ95 2ζ94 2ζ9 2ζ98 2ζ97 2ζ92 complex faithful ρ17 3 -1 -3+3√-3/2 -3-3√-3/2 0 0 ζ65 ζ6 0 0 0 0 0 0 2ζ94 2ζ95 2ζ98 2ζ9 2ζ92 2ζ97 complex faithful ρ18 3 -1 -3+3√-3/2 -3-3√-3/2 0 0 ζ65 ζ6 0 0 0 0 0 0 2ζ97 2ζ92 2ζ95 2ζ94 2ζ98 2ζ9 complex faithful ρ19 3 -1 -3-3√-3/2 -3+3√-3/2 0 0 ζ6 ζ65 0 0 0 0 0 0 2ζ92 2ζ97 2ζ94 2ζ95 2ζ9 2ζ98 complex faithful ρ20 3 -1 -3+3√-3/2 -3-3√-3/2 0 0 ζ65 ζ6 0 0 0 0 0 0 2ζ9 2ζ98 2ζ92 2ζ97 2ζ95 2ζ94 complex faithful

Smallest permutation representation of C9⋊A4
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 15)(2 16)(3 17)(4 18)(5 10)(6 11)(7 12)(8 13)(9 14)(19 35)(20 36)(21 28)(22 29)(23 30)(24 31)(25 32)(26 33)(27 34)
(1 25)(2 26)(3 27)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 36)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)
(2 5 8)(3 9 6)(10 23 33)(11 27 31)(12 22 29)(13 26 36)(14 21 34)(15 25 32)(16 20 30)(17 24 28)(18 19 35)```

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

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

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

C9⋊A4 is a maximal subgroup of
D9⋊A4  C62.25C32  A4×3- 1+2  C62.9C32  C42⋊3- 1+2  C24⋊3- 1+2  C2423- 1+2  C2443- 1+2
C9⋊A4 is a maximal quotient of
C18.A4  C62.11C32  C62.12C32  C62.16C32  C42⋊3- 1+2  C24⋊3- 1+2  C2423- 1+2  C2443- 1+2

Matrix representation of C9⋊A4 in GL3(𝔽19) generated by

 11 4 10 13 1 9 6 15 7
,
 0 0 1 18 18 18 1 0 0
,
 0 1 0 1 0 0 18 18 18
,
 1 0 0 0 0 1 18 18 18
`G:=sub<GL(3,GF(19))| [11,13,6,4,1,15,10,9,7],[0,18,1,0,18,0,1,18,0],[0,1,18,1,0,18,0,0,18],[1,0,18,0,0,18,0,1,18] >;`

C9⋊A4 in GAP, Magma, Sage, TeX

`C_9\rtimes A_4`
`% in TeX`

`G:=Group("C9:A4");`
`// GroupNames label`

`G:=SmallGroup(108,19);`
`// by ID`

`G=gap.SmallGroup(108,19);`
`# by ID`

`G:=PCGroup([5,-3,-3,-3,-2,2,121,36,1083,2029]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^9=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^7,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;`
`// generators/relations`

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