Copied to
clipboard

## G = C8.F7order 336 = 24·3·7

### The non-split extension by C8 of F7 acting via F7/C7⋊C3=C2

Aliases: C8.F7, Dic28⋊C3, C56.1C6, Dic14.2C6, C7⋊C31Q16, C71(C3×Q16), C14.3(C3×D4), C2.5(C4⋊F7), C4.10(C2×F7), C4.F7.2C2, C28.10(C2×C6), (C8×C7⋊C3).1C2, (C2×C7⋊C3).3D4, (C4×C7⋊C3).10C22, SmallGroup(336,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C8.F7
 Chief series C1 — C7 — C14 — C28 — C4×C7⋊C3 — C4.F7 — C8.F7
 Lower central C7 — C14 — C28 — C8.F7
 Upper central C1 — C2 — C4 — C8

Generators and relations for C8.F7
G = < a,b,c | a8=b7=1, c6=a4, ab=ba, cac-1=a-1, cbc-1=b5 >

Character table of C8.F7

 class 1 2 3A 3B 4A 4B 4C 6A 6B 7 8A 8B 12A 12B 12C 12D 12E 12F 14 24A 24B 24C 24D 28A 28B 56A 56B 56C 56D size 1 1 7 7 2 28 28 7 7 6 2 2 14 14 28 28 28 28 6 14 14 14 14 6 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 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 1 1 1 1 1 linear of order 2 ρ3 1 1 1 1 1 -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 ρ4 1 1 1 1 1 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 ρ5 1 1 ζ3 ζ32 1 -1 1 ζ3 ζ32 1 -1 -1 ζ32 ζ3 ζ65 ζ32 ζ6 ζ3 1 ζ65 ζ6 ζ65 ζ6 1 1 -1 -1 -1 -1 linear of order 6 ρ6 1 1 ζ32 ζ3 1 1 1 ζ32 ζ3 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 1 ζ32 ζ3 ζ32 ζ3 1 1 1 1 1 1 linear of order 3 ρ7 1 1 ζ32 ζ3 1 -1 -1 ζ32 ζ3 1 1 1 ζ3 ζ32 ζ6 ζ65 ζ65 ζ6 1 ζ32 ζ3 ζ32 ζ3 1 1 1 1 1 1 linear of order 6 ρ8 1 1 ζ3 ζ32 1 1 1 ζ3 ζ32 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 1 ζ3 ζ32 ζ3 ζ32 1 1 1 1 1 1 linear of order 3 ρ9 1 1 ζ32 ζ3 1 1 -1 ζ32 ζ3 1 -1 -1 ζ3 ζ32 ζ32 ζ65 ζ3 ζ6 1 ζ6 ζ65 ζ6 ζ65 1 1 -1 -1 -1 -1 linear of order 6 ρ10 1 1 ζ3 ζ32 1 1 -1 ζ3 ζ32 1 -1 -1 ζ32 ζ3 ζ3 ζ6 ζ32 ζ65 1 ζ65 ζ6 ζ65 ζ6 1 1 -1 -1 -1 -1 linear of order 6 ρ11 1 1 ζ3 ζ32 1 -1 -1 ζ3 ζ32 1 1 1 ζ32 ζ3 ζ65 ζ6 ζ6 ζ65 1 ζ3 ζ32 ζ3 ζ32 1 1 1 1 1 1 linear of order 6 ρ12 1 1 ζ32 ζ3 1 -1 1 ζ32 ζ3 1 -1 -1 ζ3 ζ32 ζ6 ζ3 ζ65 ζ32 1 ζ6 ζ65 ζ6 ζ65 1 1 -1 -1 -1 -1 linear of order 6 ρ13 2 2 2 2 -2 0 0 2 2 2 0 0 -2 -2 0 0 0 0 2 0 0 0 0 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ14 2 -2 2 2 0 0 0 -2 -2 2 √2 -√2 0 0 0 0 0 0 -2 √2 -√2 -√2 √2 0 0 -√2 √2 √2 -√2 symplectic lifted from Q16, Schur index 2 ρ15 2 -2 2 2 0 0 0 -2 -2 2 -√2 √2 0 0 0 0 0 0 -2 -√2 √2 √2 -√2 0 0 √2 -√2 -√2 √2 symplectic lifted from Q16, Schur index 2 ρ16 2 2 -1-√-3 -1+√-3 -2 0 0 -1-√-3 -1+√-3 2 0 0 1-√-3 1+√-3 0 0 0 0 2 0 0 0 0 -2 -2 0 0 0 0 complex lifted from C3×D4 ρ17 2 2 -1+√-3 -1-√-3 -2 0 0 -1+√-3 -1-√-3 2 0 0 1+√-3 1-√-3 0 0 0 0 2 0 0 0 0 -2 -2 0 0 0 0 complex lifted from C3×D4 ρ18 2 -2 -1+√-3 -1-√-3 0 0 0 1-√-3 1+√-3 2 -√2 √2 0 0 0 0 0 0 -2 ζ83ζ3-ζ8ζ3 ζ87ζ32-ζ85ζ32 ζ87ζ3-ζ85ζ3 ζ83ζ32-ζ8ζ32 0 0 √2 -√2 -√2 √2 complex lifted from C3×Q16 ρ19 2 -2 -1-√-3 -1+√-3 0 0 0 1+√-3 1-√-3 2 -√2 √2 0 0 0 0 0 0 -2 ζ83ζ32-ζ8ζ32 ζ87ζ3-ζ85ζ3 ζ87ζ32-ζ85ζ32 ζ83ζ3-ζ8ζ3 0 0 √2 -√2 -√2 √2 complex lifted from C3×Q16 ρ20 2 -2 -1-√-3 -1+√-3 0 0 0 1+√-3 1-√-3 2 √2 -√2 0 0 0 0 0 0 -2 ζ87ζ32-ζ85ζ32 ζ83ζ3-ζ8ζ3 ζ83ζ32-ζ8ζ32 ζ87ζ3-ζ85ζ3 0 0 -√2 √2 √2 -√2 complex lifted from C3×Q16 ρ21 2 -2 -1+√-3 -1-√-3 0 0 0 1-√-3 1+√-3 2 √2 -√2 0 0 0 0 0 0 -2 ζ87ζ3-ζ85ζ3 ζ83ζ32-ζ8ζ32 ζ83ζ3-ζ8ζ3 ζ87ζ32-ζ85ζ32 0 0 -√2 √2 √2 -√2 complex lifted from C3×Q16 ρ22 6 6 0 0 6 0 0 0 0 -1 6 6 0 0 0 0 0 0 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from F7 ρ23 6 6 0 0 6 0 0 0 0 -1 -6 -6 0 0 0 0 0 0 -1 0 0 0 0 -1 -1 1 1 1 1 orthogonal lifted from C2×F7 ρ24 6 6 0 0 -6 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 1 -√7 -√7 √7 √7 orthogonal lifted from C4⋊F7 ρ25 6 6 0 0 -6 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 1 √7 √7 -√7 -√7 orthogonal lifted from C4⋊F7 ρ26 6 -6 0 0 0 0 0 0 0 -1 3√2 -3√2 0 0 0 0 0 0 1 0 0 0 0 √7 -√7 ζ87ζ74+ζ87ζ72+ζ87ζ7+ζ87+ζ85ζ74+ζ85ζ72+ζ85ζ7 ζ87ζ76+ζ87ζ75+ζ87ζ73+ζ85ζ76+ζ85ζ75+ζ85ζ73+ζ85 ζ83ζ76+ζ83ζ75+ζ83ζ73+ζ83+ζ8ζ76+ζ8ζ75+ζ8ζ73 ζ83ζ74+ζ83ζ72+ζ83ζ7+ζ8ζ74+ζ8ζ72+ζ8ζ7+ζ8 symplectic faithful, Schur index 2 ρ27 6 -6 0 0 0 0 0 0 0 -1 -3√2 3√2 0 0 0 0 0 0 1 0 0 0 0 √7 -√7 ζ87ζ76+ζ87ζ75+ζ87ζ73+ζ85ζ76+ζ85ζ75+ζ85ζ73+ζ85 ζ87ζ74+ζ87ζ72+ζ87ζ7+ζ87+ζ85ζ74+ζ85ζ72+ζ85ζ7 ζ83ζ74+ζ83ζ72+ζ83ζ7+ζ8ζ74+ζ8ζ72+ζ8ζ7+ζ8 ζ83ζ76+ζ83ζ75+ζ83ζ73+ζ83+ζ8ζ76+ζ8ζ75+ζ8ζ73 symplectic faithful, Schur index 2 ρ28 6 -6 0 0 0 0 0 0 0 -1 3√2 -3√2 0 0 0 0 0 0 1 0 0 0 0 -√7 √7 ζ83ζ74+ζ83ζ72+ζ83ζ7+ζ8ζ74+ζ8ζ72+ζ8ζ7+ζ8 ζ83ζ76+ζ83ζ75+ζ83ζ73+ζ83+ζ8ζ76+ζ8ζ75+ζ8ζ73 ζ87ζ76+ζ87ζ75+ζ87ζ73+ζ85ζ76+ζ85ζ75+ζ85ζ73+ζ85 ζ87ζ74+ζ87ζ72+ζ87ζ7+ζ87+ζ85ζ74+ζ85ζ72+ζ85ζ7 symplectic faithful, Schur index 2 ρ29 6 -6 0 0 0 0 0 0 0 -1 -3√2 3√2 0 0 0 0 0 0 1 0 0 0 0 -√7 √7 ζ83ζ76+ζ83ζ75+ζ83ζ73+ζ83+ζ8ζ76+ζ8ζ75+ζ8ζ73 ζ83ζ74+ζ83ζ72+ζ83ζ7+ζ8ζ74+ζ8ζ72+ζ8ζ7+ζ8 ζ87ζ74+ζ87ζ72+ζ87ζ7+ζ87+ζ85ζ74+ζ85ζ72+ζ85ζ7 ζ87ζ76+ζ87ζ75+ζ87ζ73+ζ85ζ76+ζ85ζ75+ζ85ζ73+ζ85 symplectic faithful, Schur index 2

Smallest permutation representation of C8.F7
On 112 points
Generators in S112
```(1 13 10 7 3 15 12 5)(2 6 9 16 4 8 11 14)(17 38 112 67 23 32 106 73)(18 74 107 33 24 68 101 39)(19 40 102 69 25 34 108 75)(20 76 109 35 26 70 103 29)(21 30 104 71 27 36 110 65)(22 66 111 37 28 72 105 31)(41 100 58 78 47 94 64 84)(42 85 53 95 48 79 59 89)(43 90 60 80 49 96 54 86)(44 87 55 97 50 81 61 91)(45 92 62 82 51 98 56 88)(46 77 57 99 52 83 63 93)
(1 66 74 48 70 52 44)(2 41 49 67 45 71 75)(3 72 68 42 76 46 50)(4 47 43 73 51 65 69)(5 22 18 95 26 99 91)(6 100 96 23 92 27 19)(7 28 24 89 20 93 97)(8 94 90 17 98 21 25)(9 58 54 32 62 36 40)(10 37 33 59 29 63 55)(11 64 60 38 56 30 34)(12 31 39 53 35 57 61)(13 111 107 79 103 83 87)(14 84 80 112 88 104 108)(15 105 101 85 109 77 81)(16 78 86 106 82 110 102)
(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 85 86 87 88)(89 90 91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110 111 112)```

`G:=sub<Sym(112)| (1,13,10,7,3,15,12,5)(2,6,9,16,4,8,11,14)(17,38,112,67,23,32,106,73)(18,74,107,33,24,68,101,39)(19,40,102,69,25,34,108,75)(20,76,109,35,26,70,103,29)(21,30,104,71,27,36,110,65)(22,66,111,37,28,72,105,31)(41,100,58,78,47,94,64,84)(42,85,53,95,48,79,59,89)(43,90,60,80,49,96,54,86)(44,87,55,97,50,81,61,91)(45,92,62,82,51,98,56,88)(46,77,57,99,52,83,63,93), (1,66,74,48,70,52,44)(2,41,49,67,45,71,75)(3,72,68,42,76,46,50)(4,47,43,73,51,65,69)(5,22,18,95,26,99,91)(6,100,96,23,92,27,19)(7,28,24,89,20,93,97)(8,94,90,17,98,21,25)(9,58,54,32,62,36,40)(10,37,33,59,29,63,55)(11,64,60,38,56,30,34)(12,31,39,53,35,57,61)(13,111,107,79,103,83,87)(14,84,80,112,88,104,108)(15,105,101,85,109,77,81)(16,78,86,106,82,110,102), (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)(77,78,79,80,81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112)>;`

`G:=Group( (1,13,10,7,3,15,12,5)(2,6,9,16,4,8,11,14)(17,38,112,67,23,32,106,73)(18,74,107,33,24,68,101,39)(19,40,102,69,25,34,108,75)(20,76,109,35,26,70,103,29)(21,30,104,71,27,36,110,65)(22,66,111,37,28,72,105,31)(41,100,58,78,47,94,64,84)(42,85,53,95,48,79,59,89)(43,90,60,80,49,96,54,86)(44,87,55,97,50,81,61,91)(45,92,62,82,51,98,56,88)(46,77,57,99,52,83,63,93), (1,66,74,48,70,52,44)(2,41,49,67,45,71,75)(3,72,68,42,76,46,50)(4,47,43,73,51,65,69)(5,22,18,95,26,99,91)(6,100,96,23,92,27,19)(7,28,24,89,20,93,97)(8,94,90,17,98,21,25)(9,58,54,32,62,36,40)(10,37,33,59,29,63,55)(11,64,60,38,56,30,34)(12,31,39,53,35,57,61)(13,111,107,79,103,83,87)(14,84,80,112,88,104,108)(15,105,101,85,109,77,81)(16,78,86,106,82,110,102), (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)(77,78,79,80,81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112) );`

`G=PermutationGroup([[(1,13,10,7,3,15,12,5),(2,6,9,16,4,8,11,14),(17,38,112,67,23,32,106,73),(18,74,107,33,24,68,101,39),(19,40,102,69,25,34,108,75),(20,76,109,35,26,70,103,29),(21,30,104,71,27,36,110,65),(22,66,111,37,28,72,105,31),(41,100,58,78,47,94,64,84),(42,85,53,95,48,79,59,89),(43,90,60,80,49,96,54,86),(44,87,55,97,50,81,61,91),(45,92,62,82,51,98,56,88),(46,77,57,99,52,83,63,93)], [(1,66,74,48,70,52,44),(2,41,49,67,45,71,75),(3,72,68,42,76,46,50),(4,47,43,73,51,65,69),(5,22,18,95,26,99,91),(6,100,96,23,92,27,19),(7,28,24,89,20,93,97),(8,94,90,17,98,21,25),(9,58,54,32,62,36,40),(10,37,33,59,29,63,55),(11,64,60,38,56,30,34),(12,31,39,53,35,57,61),(13,111,107,79,103,83,87),(14,84,80,112,88,104,108),(15,105,101,85,109,77,81),(16,78,86,106,82,110,102)], [(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),(77,78,79,80,81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110,111,112)]])`

Matrix representation of C8.F7 in GL8(𝔽337)

 13 324 0 0 0 0 0 0 13 13 0 0 0 0 0 0 0 0 17 34 34 0 34 0 0 0 0 17 34 34 0 34 0 0 303 303 320 0 0 303 0 0 34 0 0 17 34 34 0 0 303 0 303 303 320 0 0 0 0 303 0 303 303 320
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 336 336 336 336 336 336 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0
,
 206 87 0 0 0 0 0 0 87 131 0 0 0 0 0 0 0 0 50 155 155 0 155 0 0 0 0 182 0 182 182 232 0 0 155 0 0 50 155 155 0 0 0 50 155 155 0 155 0 0 105 105 287 105 287 287 0 0 182 0 182 182 232 0

`G:=sub<GL(8,GF(337))| [13,13,0,0,0,0,0,0,324,13,0,0,0,0,0,0,0,0,17,0,303,34,303,0,0,0,34,17,303,0,0,303,0,0,34,34,320,0,303,0,0,0,0,34,0,17,303,303,0,0,34,0,0,34,320,303,0,0,0,34,303,34,0,320],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,336,1,0,0,0,0,0,0,336,0,1,0,0,0,0,0,336,0,0,1,0,0,0,0,336,0,0,0,1,0,0,0,336,0,0,0,0,1,0,0,336,0,0,0,0,0],[206,87,0,0,0,0,0,0,87,131,0,0,0,0,0,0,0,0,50,0,155,0,105,182,0,0,155,182,0,50,105,0,0,0,155,0,0,155,287,182,0,0,0,182,50,155,105,182,0,0,155,182,155,0,287,232,0,0,0,232,155,155,287,0] >;`

C8.F7 in GAP, Magma, Sage, TeX

`C_8.F_7`
`% in TeX`

`G:=Group("C8.F7");`
`// GroupNames label`

`G:=SmallGroup(336,11);`
`// by ID`

`G=gap.SmallGroup(336,11);`
`# by ID`

`G:=PCGroup([6,-2,-2,-3,-2,-2,-7,144,169,223,867,69,10373,1745]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=b^7=1,c^6=a^4,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^5>;`
`// generators/relations`

Export

׿
×
𝔽