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## G = C62⋊C9order 324 = 22·34

### 1st semidirect product of C62 and C9 acting via C9/C3=C3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C62⋊C9
 Chief series C1 — C22 — C2×C6 — C62 — C3×C3.A4 — C62⋊C9
 Lower central C22 — C2×C6 — C62⋊C9
 Upper central C1 — C32 — C33

Generators and relations for C62⋊C9
G = < a,b,c | a6=b6=c9=1, ab=ba, cac-1=ab-1, cbc-1=a3b4 >

Subgroups: 205 in 74 conjugacy classes, 24 normal (11 characteristic)
C1, C2, C3, C3, C3, C22, C6, C9, C32, C32, C32, C2×C6, C2×C6, C2×C6, C3×C6, C3×C9, C33, C3.A4, C62, C62, C62, C32×C6, C32⋊C9, C3×C3.A4, C3×C62, C62⋊C9
Quotients: C1, C3, C9, C32, A4, C3×C9, He3, 3- 1+2, C3.A4, C3×A4, C32⋊C9, C3×C3.A4, C32.A4, C32⋊A4, C62⋊C9

Smallest permutation representation of C62⋊C9
On 54 points
Generators in S54
```(1 30 7 36 4 33)(2 52 19 28 39 16)(3 26 43)(5 46 22 31 42 10)(6 20 37)(8 49 25 34 45 13)(9 23 40)(11 53 35)(12 27 18 24 15 21)(14 47 29)(17 50 32)(38 54 44 51 41 48)
(1 21 44)(2 10 45 28 22 49)(3 11 37 29 23 50)(4 24 38)(5 13 39 31 25 52)(6 14 40 32 26 53)(7 27 41)(8 16 42 34 19 46)(9 17 43 35 20 47)(12 51 30)(15 54 33)(18 48 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)(37 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54)```

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

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

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

60 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3N 6A ··· 6Z 9A ··· 9R order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 9 ··· 9 size 1 3 1 ··· 1 3 ··· 3 3 ··· 3 12 ··· 12

60 irreducible representations

 dim 1 1 1 1 3 3 3 3 3 3 3 type + + image C1 C3 C3 C9 A4 He3 3- 1+2 C3.A4 C3×A4 C32.A4 C32⋊A4 kernel C62⋊C9 C3×C3.A4 C3×C62 C62 C33 C2×C6 C2×C6 C32 C32 C3 C3 # reps 1 6 2 18 1 2 4 6 2 12 6

Matrix representation of C62⋊C9 in GL4(𝔽19) generated by

 11 0 0 0 0 8 0 0 0 0 1 0 0 0 0 12
,
 1 0 0 0 0 7 0 0 0 0 12 0 0 0 0 12
,
 6 0 0 0 0 0 1 0 0 0 0 1 0 11 0 0
`G:=sub<GL(4,GF(19))| [11,0,0,0,0,8,0,0,0,0,1,0,0,0,0,12],[1,0,0,0,0,7,0,0,0,0,12,0,0,0,0,12],[6,0,0,0,0,0,0,11,0,1,0,0,0,0,1,0] >;`

C62⋊C9 in GAP, Magma, Sage, TeX

`C_6^2\rtimes C_9`
`% in TeX`

`G:=Group("C6^2:C9");`
`// GroupNames label`

`G:=SmallGroup(324,59);`
`// by ID`

`G=gap.SmallGroup(324,59);`
`# by ID`

`G:=PCGroup([6,-3,-3,-3,-3,-2,2,54,361,4864,8753]);`
`// Polycyclic`

`G:=Group<a,b,c|a^6=b^6=c^9=1,a*b=b*a,c*a*c^-1=a*b^-1,c*b*c^-1=a^3*b^4>;`
`// generators/relations`

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