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## G = C62.15C32order 324 = 22·34

### 6th non-split extension by C62 of C32 acting via C32/C3=C3

Aliases: C62.15C32, (C3×C9).2A4, (C6×C18).6C3, (C2×C6).3He3, C3.5(C32⋊A4), C32.12(C3×A4), C32.A4.1C3, C221(C3.He3), SmallGroup(324,51)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.15C32
 Chief series C1 — C22 — C2×C6 — C62 — C32.A4 — C62.15C32
 Lower central C22 — C2×C6 — C62 — C62.15C32
 Upper central C1 — C3 — C32 — C3×C9

Generators and relations for C62.15C32
G = < a,b,c,d | a6=b6=1, c3=d3=b2, cac-1=ab=ba, ad=da, cbc-1=a3b4, bd=db, dcd-1=a2b2c >

Smallest permutation representation of C62.15C32
On 54 points
Generators in S54
```(1 14)(2 8 5)(3 10 9 16 6 13)(4 17)(7 11)(12 18 15)(19 47)(20 26 23)(21 52 27 49 24 46)(22 50)(25 53)(28 43)(29 35 32)(30 39 36 45 33 42)(31 37)(34 40)(38 44 41)(48 54 51)
(1 11 4 14 7 17)(2 12 5 15 8 18)(3 9 6)(10 16 13)(19 53 22 47 25 50)(20 54 23 48 26 51)(21 27 24)(28 40 31 43 34 37)(29 41 32 44 35 38)(30 36 33)(39 45 42)(46 52 49)
(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 28 22 4 31 25 7 34 19)(2 35 26 5 29 20 8 32 23)(3 30 24 6 33 27 9 36 21)(10 39 46 13 42 49 16 45 52)(11 40 47 14 43 50 17 37 53)(12 38 51 15 41 54 18 44 48)```

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

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

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

44 conjugacy classes

 class 1 2 3A 3B 3C 3D 6A ··· 6H 9A ··· 9F 9G ··· 9L 18A ··· 18R order 1 2 3 3 3 3 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 3 1 1 3 3 3 ··· 3 3 ··· 3 36 ··· 36 3 ··· 3

44 irreducible representations

 dim 1 1 1 3 3 3 3 3 3 type + + image C1 C3 C3 A4 He3 C3×A4 C3.He3 C32⋊A4 C62.15C32 kernel C62.15C32 C32.A4 C6×C18 C3×C9 C2×C6 C32 C22 C3 C1 # reps 1 6 2 1 2 2 6 6 18

Matrix representation of C62.15C32 in GL3(𝔽19) generated by

 18 0 0 14 12 0 15 0 11
,
 8 0 0 12 11 0 0 0 8
,
 11 17 0 0 8 1 16 12 0
,
 9 0 0 0 9 0 8 0 4
`G:=sub<GL(3,GF(19))| [18,14,15,0,12,0,0,0,11],[8,12,0,0,11,0,0,0,8],[11,0,16,17,8,12,0,1,0],[9,0,8,0,9,0,0,0,4] >;`

C62.15C32 in GAP, Magma, Sage, TeX

`C_6^2._{15}C_3^2`
`% in TeX`

`G:=Group("C6^2.15C3^2");`
`// GroupNames label`

`G:=SmallGroup(324,51);`
`// by ID`

`G=gap.SmallGroup(324,51);`
`# by ID`

`G:=PCGroup([6,-3,-3,-3,-3,-2,2,162,145,115,1136,4864,8753]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^6=1,c^3=d^3=b^2,c*a*c^-1=a*b=b*a,a*d=d*a,c*b*c^-1=a^3*b^4,b*d=d*b,d*c*d^-1=a^2*b^2*c>;`
`// generators/relations`

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