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G = C9.A4order 108 = 22·33

The central extension by C9 of A4

Aliases: C9.A4, C22⋊C27, (C2×C6).C9, (C2×C18).C3, C3.(C3.A4), SmallGroup(108,3)

Series: Derived Chief Lower central Upper central

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

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

Smallest permutation representation of C9.A4
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)
(2 45)(3 46)(5 48)(6 49)(8 51)(9 52)(11 54)(12 28)(14 30)(15 31)(17 33)(18 34)(20 36)(21 37)(23 39)(24 40)(26 42)(27 43)
(1 44)(3 46)(4 47)(6 49)(7 50)(9 52)(10 53)(12 28)(13 29)(15 31)(16 32)(18 34)(19 35)(21 37)(22 38)(24 40)(25 41)(27 43)
(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)```

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

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

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

C9.A4 is a maximal subgroup of   C9.S4  A4×C27  C27⋊A4  C62.C9  C42⋊C27  C24⋊C27
C9.A4 is a maximal quotient of   Q8⋊C27  C27.A4  C42⋊C27  C24⋊C27

36 conjugacy classes

 class 1 2 3A 3B 6A 6B 9A ··· 9F 18A ··· 18F 27A ··· 27R order 1 2 3 3 6 6 9 ··· 9 18 ··· 18 27 ··· 27 size 1 3 1 1 3 3 1 ··· 1 3 ··· 3 4 ··· 4

36 irreducible representations

 dim 1 1 1 1 3 3 3 type + + image C1 C3 C9 C27 A4 C3.A4 C9.A4 kernel C9.A4 C2×C18 C2×C6 C22 C9 C3 C1 # reps 1 2 6 18 1 2 6

Matrix representation of C9.A4 in GL3(𝔽109) generated by

 27 0 0 0 27 0 0 0 27
,
 1 0 0 22 108 0 27 0 108
,
 108 0 0 0 108 0 82 0 1
,
 22 107 0 65 87 1 11 82 0
`G:=sub<GL(3,GF(109))| [27,0,0,0,27,0,0,0,27],[1,22,27,0,108,0,0,0,108],[108,0,82,0,108,0,0,0,1],[22,65,11,107,87,82,0,1,0] >;`

C9.A4 in GAP, Magma, Sage, TeX

`C_9.A_4`
`% in TeX`

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

`G:=SmallGroup(108,3);`
`// by ID`

`G=gap.SmallGroup(108,3);`
`# by ID`

`G:=PCGroup([5,-3,-3,-3,-2,2,15,36,1083,2029]);`
`// Polycyclic`

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

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