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## G = C6.F7order 252 = 22·32·7

### The non-split extension by C6 of F7 acting via F7/C7⋊C3=C2

Aliases: C6.F7, C211C12, C42.1C6, Dic21⋊C3, C3⋊(C7⋊C12), C7⋊C3⋊Dic3, C7⋊(C3×Dic3), C14.(C3×S3), C2.(C3⋊F7), (C3×C7⋊C3)⋊1C4, (C2×C7⋊C3).S3, (C6×C7⋊C3).1C2, SmallGroup(252,18)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — C6.F7
 Chief series C1 — C7 — C21 — C42 — C6×C7⋊C3 — C6.F7
 Lower central C21 — C6.F7
 Upper central C1 — C2

Generators and relations for C6.F7
G = < a,b,c | a6=b7=1, c6=a3, ab=ba, cac-1=a-1, cbc-1=b5 >

Character table of C6.F7

 class 1 2 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 7 12A 12B 12C 12D 14 21A 21B 42A 42B size 1 1 2 7 7 14 14 21 21 2 7 7 14 14 6 21 21 21 21 6 6 6 6 6 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 linear of order 2 ρ3 1 1 1 ζ32 ζ3 ζ3 ζ32 -1 -1 1 ζ3 ζ32 ζ3 ζ32 1 ζ6 ζ65 ζ65 ζ6 1 1 1 1 1 linear of order 6 ρ4 1 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 ζ32 ζ3 ζ32 ζ3 1 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 linear of order 3 ρ5 1 1 1 ζ3 ζ32 ζ32 ζ3 -1 -1 1 ζ32 ζ3 ζ32 ζ3 1 ζ65 ζ6 ζ6 ζ65 1 1 1 1 1 linear of order 6 ρ6 1 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 ζ3 ζ32 ζ3 ζ32 1 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 linear of order 3 ρ7 1 -1 1 1 1 1 1 i -i -1 -1 -1 -1 -1 1 i -i i -i -1 1 1 -1 -1 linear of order 4 ρ8 1 -1 1 1 1 1 1 -i i -1 -1 -1 -1 -1 1 -i i -i i -1 1 1 -1 -1 linear of order 4 ρ9 1 -1 1 ζ3 ζ32 ζ32 ζ3 -i i -1 ζ6 ζ65 ζ6 ζ65 1 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ3 -1 1 1 -1 -1 linear of order 12 ρ10 1 -1 1 ζ32 ζ3 ζ3 ζ32 i -i -1 ζ65 ζ6 ζ65 ζ6 1 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ32 -1 1 1 -1 -1 linear of order 12 ρ11 1 -1 1 ζ32 ζ3 ζ3 ζ32 -i i -1 ζ65 ζ6 ζ65 ζ6 1 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ32 -1 1 1 -1 -1 linear of order 12 ρ12 1 -1 1 ζ3 ζ32 ζ32 ζ3 i -i -1 ζ6 ζ65 ζ6 ζ65 1 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ3 -1 1 1 -1 -1 linear of order 12 ρ13 2 2 -1 2 2 -1 -1 0 0 -1 2 2 -1 -1 2 0 0 0 0 2 -1 -1 -1 -1 orthogonal lifted from S3 ρ14 2 -2 -1 2 2 -1 -1 0 0 1 -2 -2 1 1 2 0 0 0 0 -2 -1 -1 1 1 symplectic lifted from Dic3, Schur index 2 ρ15 2 -2 -1 -1-√-3 -1+√-3 ζ65 ζ6 0 0 1 1-√-3 1+√-3 ζ3 ζ32 2 0 0 0 0 -2 -1 -1 1 1 complex lifted from C3×Dic3 ρ16 2 2 -1 -1+√-3 -1-√-3 ζ6 ζ65 0 0 -1 -1-√-3 -1+√-3 ζ6 ζ65 2 0 0 0 0 2 -1 -1 -1 -1 complex lifted from C3×S3 ρ17 2 2 -1 -1-√-3 -1+√-3 ζ65 ζ6 0 0 -1 -1+√-3 -1-√-3 ζ65 ζ6 2 0 0 0 0 2 -1 -1 -1 -1 complex lifted from C3×S3 ρ18 2 -2 -1 -1+√-3 -1-√-3 ζ6 ζ65 0 0 1 1+√-3 1-√-3 ζ32 ζ3 2 0 0 0 0 -2 -1 -1 1 1 complex lifted from C3×Dic3 ρ19 6 6 6 0 0 0 0 0 0 6 0 0 0 0 -1 0 0 0 0 -1 -1 -1 -1 -1 orthogonal lifted from F7 ρ20 6 6 -3 0 0 0 0 0 0 -3 0 0 0 0 -1 0 0 0 0 -1 1-√21/2 1+√21/2 1+√21/2 1-√21/2 orthogonal lifted from C3⋊F7 ρ21 6 6 -3 0 0 0 0 0 0 -3 0 0 0 0 -1 0 0 0 0 -1 1+√21/2 1-√21/2 1-√21/2 1+√21/2 orthogonal lifted from C3⋊F7 ρ22 6 -6 -3 0 0 0 0 0 0 3 0 0 0 0 -1 0 0 0 0 1 1-√21/2 1+√21/2 -1-√21/2 -1+√21/2 symplectic faithful, Schur index 2 ρ23 6 -6 -3 0 0 0 0 0 0 3 0 0 0 0 -1 0 0 0 0 1 1+√21/2 1-√21/2 -1+√21/2 -1-√21/2 symplectic faithful, Schur index 2 ρ24 6 -6 6 0 0 0 0 0 0 -6 0 0 0 0 -1 0 0 0 0 1 -1 -1 1 1 symplectic lifted from C7⋊C12, Schur index 2

Smallest permutation representation of C6.F7
On 84 points
Generators in S84
```(1 3 5 7 9 11)(2 12 10 8 6 4)(13 78 28 19 84 34)(14 35 73 20 29 79)(15 80 30 21 74 36)(16 25 75 22 31 81)(17 82 32 23 76 26)(18 27 77 24 33 83)(37 52 63 43 58 69)(38 70 59 44 64 53)(39 54 65 45 60 71)(40 72 49 46 66 55)(41 56 67 47 50 61)(42 62 51 48 68 57)
(1 67 58 17 45 80 28)(2 81 18 68 29 46 59)(3 47 69 82 60 30 19)(4 31 83 48 20 49 70)(5 50 37 32 71 21 84)(6 22 33 51 73 72 38)(7 61 52 23 39 74 34)(8 75 24 62 35 40 53)(9 41 63 76 54 36 13)(10 25 77 42 14 55 64)(11 56 43 26 65 15 78)(12 16 27 57 79 66 44)
(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 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)```

`G:=sub<Sym(84)| (1,3,5,7,9,11)(2,12,10,8,6,4)(13,78,28,19,84,34)(14,35,73,20,29,79)(15,80,30,21,74,36)(16,25,75,22,31,81)(17,82,32,23,76,26)(18,27,77,24,33,83)(37,52,63,43,58,69)(38,70,59,44,64,53)(39,54,65,45,60,71)(40,72,49,46,66,55)(41,56,67,47,50,61)(42,62,51,48,68,57), (1,67,58,17,45,80,28)(2,81,18,68,29,46,59)(3,47,69,82,60,30,19)(4,31,83,48,20,49,70)(5,50,37,32,71,21,84)(6,22,33,51,73,72,38)(7,61,52,23,39,74,34)(8,75,24,62,35,40,53)(9,41,63,76,54,36,13)(10,25,77,42,14,55,64)(11,56,43,26,65,15,78)(12,16,27,57,79,66,44), (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)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)>;`

`G:=Group( (1,3,5,7,9,11)(2,12,10,8,6,4)(13,78,28,19,84,34)(14,35,73,20,29,79)(15,80,30,21,74,36)(16,25,75,22,31,81)(17,82,32,23,76,26)(18,27,77,24,33,83)(37,52,63,43,58,69)(38,70,59,44,64,53)(39,54,65,45,60,71)(40,72,49,46,66,55)(41,56,67,47,50,61)(42,62,51,48,68,57), (1,67,58,17,45,80,28)(2,81,18,68,29,46,59)(3,47,69,82,60,30,19)(4,31,83,48,20,49,70)(5,50,37,32,71,21,84)(6,22,33,51,73,72,38)(7,61,52,23,39,74,34)(8,75,24,62,35,40,53)(9,41,63,76,54,36,13)(10,25,77,42,14,55,64)(11,56,43,26,65,15,78)(12,16,27,57,79,66,44), (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)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84) );`

`G=PermutationGroup([[(1,3,5,7,9,11),(2,12,10,8,6,4),(13,78,28,19,84,34),(14,35,73,20,29,79),(15,80,30,21,74,36),(16,25,75,22,31,81),(17,82,32,23,76,26),(18,27,77,24,33,83),(37,52,63,43,58,69),(38,70,59,44,64,53),(39,54,65,45,60,71),(40,72,49,46,66,55),(41,56,67,47,50,61),(42,62,51,48,68,57)], [(1,67,58,17,45,80,28),(2,81,18,68,29,46,59),(3,47,69,82,60,30,19),(4,31,83,48,20,49,70),(5,50,37,32,71,21,84),(6,22,33,51,73,72,38),(7,61,52,23,39,74,34),(8,75,24,62,35,40,53),(9,41,63,76,54,36,13),(10,25,77,42,14,55,64),(11,56,43,26,65,15,78),(12,16,27,57,79,66,44)], [(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),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84)]])`

Matrix representation of C6.F7 in GL6(𝔽337)

 292 246 246 0 246 0 0 292 246 246 0 246 91 91 46 0 0 91 246 0 0 292 246 246 91 0 91 91 46 0 0 91 0 91 91 46
,
 336 336 336 336 336 336 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 330 14 81 148 221 168 67 134 207 154 323 316 73 20 189 182 203 270 169 162 183 250 317 53 21 88 155 228 175 7 67 140 87 256 249 270

`G:=sub<GL(6,GF(337))| [292,0,91,246,91,0,246,292,91,0,0,91,246,246,46,0,91,0,0,246,0,292,91,91,246,0,0,246,46,91,0,246,91,246,0,46],[336,1,0,0,0,0,336,0,1,0,0,0,336,0,0,1,0,0,336,0,0,0,1,0,336,0,0,0,0,1,336,0,0,0,0,0],[330,67,73,169,21,67,14,134,20,162,88,140,81,207,189,183,155,87,148,154,182,250,228,256,221,323,203,317,175,249,168,316,270,53,7,270] >;`

C6.F7 in GAP, Magma, Sage, TeX

`C_6.F_7`
`% in TeX`

`G:=Group("C6.F7");`
`// GroupNames label`

`G:=SmallGroup(252,18);`
`// by ID`

`G=gap.SmallGroup(252,18);`
`# by ID`

`G:=PCGroup([5,-2,-3,-2,-3,-7,30,483,5404,909]);`
`// Polycyclic`

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

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