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## G = C32.F7order 378 = 2·33·7

### The non-split extension by C32 of F7 acting via F7/D7=C3

Aliases: C32.F7, D733- 1+2, C7⋊C93C6, C7⋊C183C3, C3.5(C3×F7), C21.5(C3×C6), (C3×C21).4C6, C21.C32⋊C2, (C32×D7).3C3, (C3×D7).5C32, C73(C2×3- 1+2), SmallGroup(378,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — C32.F7
 Chief series C1 — C7 — C21 — C3×C21 — C21.C32 — C32.F7
 Lower central C7 — C21 — C32.F7
 Upper central C1 — C3 — C32

Generators and relations for C32.F7
G = < a,b,c,d | a3=b3=c7=1, d6=b-1, ab=ba, ac=ca, dad-1=ab-1, bc=cb, bd=db, dcd-1=c5 >

Smallest permutation representation of C32.F7
On 63 points
Generators in S63
```(2 8 5)(3 6 9)(11 17 23)(12 24 18)(14 20 26)(15 27 21)(28 40 34)(30 36 42)(31 43 37)(33 39 45)(47 53 59)(48 60 54)(50 56 62)(51 63 57)
(1 4 7)(2 5 8)(3 6 9)(10 22 16)(11 23 17)(12 24 18)(13 25 19)(14 26 20)(15 27 21)(28 40 34)(29 41 35)(30 42 36)(31 43 37)(32 44 38)(33 45 39)(46 58 52)(47 59 53)(48 60 54)(49 61 55)(50 62 56)(51 63 57)
(1 32 10 49 58 19 41)(2 20 50 33 42 59 11)(3 60 34 21 12 43 51)(4 44 22 61 52 13 35)(5 14 62 45 36 53 23)(6 54 28 15 24 37 63)(7 38 16 55 46 25 29)(8 26 56 39 30 47 17)(9 48 40 27 18 31 57)
(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 55 56 57 58 59 60 61 62 63)```

`G:=sub<Sym(63)| (2,8,5)(3,6,9)(11,17,23)(12,24,18)(14,20,26)(15,27,21)(28,40,34)(30,36,42)(31,43,37)(33,39,45)(47,53,59)(48,60,54)(50,56,62)(51,63,57), (1,4,7)(2,5,8)(3,6,9)(10,22,16)(11,23,17)(12,24,18)(13,25,19)(14,26,20)(15,27,21)(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,44,38)(33,45,39)(46,58,52)(47,59,53)(48,60,54)(49,61,55)(50,62,56)(51,63,57), (1,32,10,49,58,19,41)(2,20,50,33,42,59,11)(3,60,34,21,12,43,51)(4,44,22,61,52,13,35)(5,14,62,45,36,53,23)(6,54,28,15,24,37,63)(7,38,16,55,46,25,29)(8,26,56,39,30,47,17)(9,48,40,27,18,31,57), (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)(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63)>;`

`G:=Group( (2,8,5)(3,6,9)(11,17,23)(12,24,18)(14,20,26)(15,27,21)(28,40,34)(30,36,42)(31,43,37)(33,39,45)(47,53,59)(48,60,54)(50,56,62)(51,63,57), (1,4,7)(2,5,8)(3,6,9)(10,22,16)(11,23,17)(12,24,18)(13,25,19)(14,26,20)(15,27,21)(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,44,38)(33,45,39)(46,58,52)(47,59,53)(48,60,54)(49,61,55)(50,62,56)(51,63,57), (1,32,10,49,58,19,41)(2,20,50,33,42,59,11)(3,60,34,21,12,43,51)(4,44,22,61,52,13,35)(5,14,62,45,36,53,23)(6,54,28,15,24,37,63)(7,38,16,55,46,25,29)(8,26,56,39,30,47,17)(9,48,40,27,18,31,57), (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)(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63) );`

`G=PermutationGroup([[(2,8,5),(3,6,9),(11,17,23),(12,24,18),(14,20,26),(15,27,21),(28,40,34),(30,36,42),(31,43,37),(33,39,45),(47,53,59),(48,60,54),(50,56,62),(51,63,57)], [(1,4,7),(2,5,8),(3,6,9),(10,22,16),(11,23,17),(12,24,18),(13,25,19),(14,26,20),(15,27,21),(28,40,34),(29,41,35),(30,42,36),(31,43,37),(32,44,38),(33,45,39),(46,58,52),(47,59,53),(48,60,54),(49,61,55),(50,62,56),(51,63,57)], [(1,32,10,49,58,19,41),(2,20,50,33,42,59,11),(3,60,34,21,12,43,51),(4,44,22,61,52,13,35),(5,14,62,45,36,53,23),(6,54,28,15,24,37,63),(7,38,16,55,46,25,29),(8,26,56,39,30,47,17),(9,48,40,27,18,31,57)], [(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),(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63)]])`

31 conjugacy classes

 class 1 2 3A 3B 3C 3D 6A 6B 6C 6D 7 9A ··· 9F 18A ··· 18F 21A ··· 21H order 1 2 3 3 3 3 6 6 6 6 7 9 ··· 9 18 ··· 18 21 ··· 21 size 1 7 1 1 3 3 7 7 21 21 6 21 ··· 21 21 ··· 21 6 ··· 6

31 irreducible representations

 dim 1 1 1 1 1 1 3 3 6 6 6 type + + + image C1 C2 C3 C3 C6 C6 3- 1+2 C2×3- 1+2 F7 C3×F7 C32.F7 kernel C32.F7 C21.C32 C7⋊C18 C32×D7 C7⋊C9 C3×C21 D7 C7 C32 C3 C1 # reps 1 1 6 2 6 2 2 2 1 2 6

Matrix representation of C32.F7 in GL6(𝔽127)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 19 0 0 0 0 0 0 19 0 0 0 0 0 0 107 0 0 0 0 0 0 107
,
 19 0 0 0 0 0 0 19 0 0 0 0 0 0 19 0 0 0 0 0 0 19 0 0 0 0 0 0 19 0 0 0 0 0 0 19
,
 37 126 0 0 0 0 38 126 0 0 0 0 0 0 66 25 0 0 0 0 66 0 0 0 0 0 0 0 61 91 0 0 0 0 98 90
,
 0 0 90 1 0 0 0 0 29 37 0 0 0 0 0 0 90 1 0 0 0 0 29 37 59 19 0 0 0 0 43 68 0 0 0 0

`G:=sub<GL(6,GF(127))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,19,0,0,0,0,0,0,19,0,0,0,0,0,0,107,0,0,0,0,0,0,107],[19,0,0,0,0,0,0,19,0,0,0,0,0,0,19,0,0,0,0,0,0,19,0,0,0,0,0,0,19,0,0,0,0,0,0,19],[37,38,0,0,0,0,126,126,0,0,0,0,0,0,66,66,0,0,0,0,25,0,0,0,0,0,0,0,61,98,0,0,0,0,91,90],[0,0,0,0,59,43,0,0,0,0,19,68,90,29,0,0,0,0,1,37,0,0,0,0,0,0,90,29,0,0,0,0,1,37,0,0] >;`

C32.F7 in GAP, Magma, Sage, TeX

`C_3^2.F_7`
`% in TeX`

`G:=Group("C3^2.F7");`
`// GroupNames label`

`G:=SmallGroup(378,11);`
`// by ID`

`G=gap.SmallGroup(378,11);`
`# by ID`

`G:=PCGroup([5,-2,-3,-3,-3,-7,96,187,8104,2709]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^3=b^3=c^7=1,d^6=b^-1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;`
`// generators/relations`

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