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## G = C62.A4order 432 = 24·33

### 9th non-split extension by C62 of A4 acting via A4/C22=C3

Aliases: C62.9A4, C2463- 1+2, C24⋊C96C3, C32.(C22⋊A4), (C22×C62).4C3, C222(C32.A4), (C23×C6).14C32, (C2×C6).15(C3×A4), C3.4(C3×C22⋊A4), SmallGroup(432,554)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23×C6 — C62.A4
 Chief series C1 — C22 — C24 — C23×C6 — C24⋊C9 — C62.A4
 Lower central C24 — C23×C6 — C62.A4
 Upper central C1 — C3 — C32

Generators and relations for C62.A4
G = < a,b,c,d,e | a6=b6=c2=d2=1, e3=b2, ab=ba, ac=ca, ad=da, eae-1=ab-1, bc=cb, bd=db, ebe-1=a3b4, ece-1=cd=dc, ede-1=c >

Subgroups: 550 in 176 conjugacy classes, 25 normal (7 characteristic)
C1, C2, C3, C3, C22, C22, C6, C23, C9, C32, C2×C6, C2×C6, C24, C3×C6, C22×C6, 3- 1+2, C3.A4, C62, C62, C23×C6, C23×C6, C2×C62, C32.A4, C24⋊C9, C22×C62, C62.A4
Quotients: C1, C3, C32, A4, 3- 1+2, C3×A4, C22⋊A4, C32.A4, C3×C22⋊A4, C62.A4

Smallest permutation representation of C62.A4
On 36 points
Generators in S36
```(1 20)(2 5 8)(3 19 6 22 9 25)(4 23)(7 26)(10 32)(11 14 17)(12 31 15 34 18 28)(13 35)(16 29)(21 24 27)(30 33 36)
(1 26 4 20 7 23)(2 27 5 21 8 24)(3 9 6)(10 29 13 32 16 35)(11 30 14 33 17 36)(12 18 15)(19 25 22)(28 34 31)
(1 10)(2 11)(4 13)(5 14)(7 16)(8 17)(20 32)(21 33)(23 35)(24 36)(26 29)(27 30)
(2 11)(3 12)(5 14)(6 15)(8 17)(9 18)(19 31)(21 33)(22 34)(24 36)(25 28)(27 30)
(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)```

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

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

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

56 conjugacy classes

 class 1 2A ··· 2E 3A 3B 3C 3D 6A ··· 6AN 9A ··· 9F order 1 2 ··· 2 3 3 3 3 6 ··· 6 9 ··· 9 size 1 3 ··· 3 1 1 3 3 3 ··· 3 48 ··· 48

56 irreducible representations

 dim 1 1 1 3 3 3 3 type + + image C1 C3 C3 A4 3- 1+2 C3×A4 C32.A4 kernel C62.A4 C24⋊C9 C22×C62 C62 C24 C2×C6 C22 # reps 1 6 2 5 2 10 30

Matrix representation of C62.A4 in GL6(𝔽19)

 18 0 0 0 0 0 0 8 0 0 0 0 0 0 7 0 0 0 0 0 0 18 0 0 0 0 0 0 12 0 0 0 0 0 0 11
,
 8 0 0 0 0 0 0 11 0 0 0 0 0 0 8 0 0 0 0 0 0 12 0 0 0 0 0 0 7 0 0 0 0 0 0 12
,
 18 0 0 0 0 0 0 1 0 0 0 0 0 0 18 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 1 0 0 0 7 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 11 0 0

`G:=sub<GL(6,GF(19))| [18,0,0,0,0,0,0,8,0,0,0,0,0,0,7,0,0,0,0,0,0,18,0,0,0,0,0,0,12,0,0,0,0,0,0,11],[8,0,0,0,0,0,0,11,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,7,0,0,0,0,0,0,12],[18,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,0,7,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,0,1,0,0,0,0,0,0,1,0] >;`

C62.A4 in GAP, Magma, Sage, TeX

`C_6^2.A_4`
`% in TeX`

`G:=Group("C6^2.A4");`
`// GroupNames label`

`G:=SmallGroup(432,554);`
`// by ID`

`G=gap.SmallGroup(432,554);`
`# by ID`

`G:=PCGroup([7,-3,-3,-3,-2,2,-2,2,63,169,1515,2839,9077,15882]);`
`// Polycyclic`

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

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