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## G = C62.14C32order 324 = 22·34

### 5th non-split extension by C62 of C32 acting via C32/C3=C3

Aliases: C62.14C32, (C3×C9)⋊2A4, (C6×C18)⋊2C3, C32⋊A41C3, (C2×C6).2He3, C3.4(C32⋊A4), C32.11(C3×A4), C221(He3⋊C3), SmallGroup(324,50)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.14C32
 Chief series C1 — C22 — C2×C6 — C62 — C32⋊A4 — C62.14C32
 Lower central C22 — C2×C6 — C62 — C62.14C32
 Upper central C1 — C3 — C32 — C3×C9

Generators and relations for C62.14C32
G = < a,b,c,d | a6=b6=c3=1, d3=b2, cac-1=ab=ba, ad=da, cbc-1=a3b4, bd=db, dcd-1=a2b2c >

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 3 3 3 3 3 3 type + + image C1 C3 C3 A4 He3 C3×A4 He3⋊C3 C32⋊A4 C62.14C32 kernel C62.14C32 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 C62.14C32 in GL3(𝔽19) generated by

 18 0 0 0 8 0 0 0 7
,
 12 0 0 0 7 0 0 0 12
,
 0 1 0 0 0 1 1 0 0
,
 16 0 0 0 16 0 0 0 17
`G:=sub<GL(3,GF(19))| [18,0,0,0,8,0,0,0,7],[12,0,0,0,7,0,0,0,12],[0,0,1,1,0,0,0,1,0],[16,0,0,0,16,0,0,0,17] >;`

C62.14C32 in GAP, Magma, Sage, TeX

`C_6^2._{14}C_3^2`
`% in TeX`

`G:=Group("C6^2.14C3^2");`
`// GroupNames label`

`G:=SmallGroup(324,50);`
`// by ID`

`G=gap.SmallGroup(324,50);`
`# by ID`

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

`G:=Group<a,b,c,d|a^6=b^6=c^3=1,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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