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## G = C23.F8order 448 = 26·7

### 2nd non-split extension by C23 of F8 acting via F8/C23=C7

Aliases: C23.2F8, C23.84C23⋊C7, SmallGroup(448,179)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — C23.84C23 — C23.F8
 Chief series C1 — C23 — C23.84C23 — C23.F8
 Lower central C23.84C23 — C23.F8
 Upper central C1

Generators and relations for C23.F8
G = < a,b,c,d,e,f,g | a2=b2=c2=g7=1, d2=ba=ab, e2=gag-1=abc, f2=gcg-1=a, gbg-1=ac=ca, ede-1=ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, fef-1=ce=ec, cf=fc, fdf-1=bcd, gdg-1=abef, geg-1=acd, gfg-1=abe >

7C2
64C7
7C22
28C4
14C2×C4
14C2×C4
14C2×C4
8F8

Character table of C23.F8

 class 1 2 4A 4B 7A 7B 7C 7D 7E 7F size 1 7 28 28 64 64 64 64 64 64 ρ1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 ζ74 ζ76 ζ72 ζ75 ζ7 ζ73 linear of order 7 ρ3 1 1 1 1 ζ72 ζ73 ζ7 ζ76 ζ74 ζ75 linear of order 7 ρ4 1 1 1 1 ζ75 ζ74 ζ76 ζ7 ζ73 ζ72 linear of order 7 ρ5 1 1 1 1 ζ73 ζ7 ζ75 ζ72 ζ76 ζ74 linear of order 7 ρ6 1 1 1 1 ζ7 ζ75 ζ74 ζ73 ζ72 ζ76 linear of order 7 ρ7 1 1 1 1 ζ76 ζ72 ζ73 ζ74 ζ75 ζ7 linear of order 7 ρ8 7 7 -1 -1 0 0 0 0 0 0 orthogonal lifted from F8 ρ9 14 -2 -2i 2i 0 0 0 0 0 0 complex faithful ρ10 14 -2 2i -2i 0 0 0 0 0 0 complex faithful

Smallest permutation representation of C23.F8
On 56 points
Generators in S56
```(1 31)(4 34)(5 35)(6 29)(8 54)(9 55)(11 50)(14 53)(16 25)(17 26)(18 27)(20 22)(36 44)(37 45)(38 46)(40 48)
(1 31)(2 32)(3 33)(5 35)(8 54)(11 50)(12 51)(13 52)(15 24)(17 26)(20 22)(21 23)(37 45)(40 48)(41 49)(42 43)
(2 32)(5 35)(6 29)(7 30)(8 54)(9 55)(10 56)(12 51)(17 26)(18 27)(19 28)(21 23)(37 45)(38 46)(39 47)(41 49)
(2 49 32 41)(3 13 33 52)(4 44 34 36)(5 26)(6 18 29 27)(7 56)(8 45)(9 46 55 38)(10 30)(11 50)(12 21 51 23)(14 25 53 16)(15 42 24 43)(17 35)(19 39)(20 22)(28 47)(37 54)
(1 11)(2 32)(3 42 33 43)(4 14 34 53)(5 37 35 45)(6 27)(7 19 30 28)(8 17 54 26)(9 46)(10 47 56 39)(13 24 52 15)(16 36 25 44)(18 29)(20 48)(22 40)(31 50)(38 55)(41 49)
(1 20 31 22)(2 12)(4 44 34 36)(5 54 35 8)(6 38 29 46)(7 19)(9 18 55 27)(10 39)(11 48 50 40)(13 52)(14 16 53 25)(15 24)(17 45 26 37)(21 49)(23 41)(28 30)(32 51)(47 56)
(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)```

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

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

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

Matrix representation of C23.F8 in GL14(𝔽29)

 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 23 1 0 0 0 0 28 0 0 0 0 0 0 0 13 11 0 0 0 0 0 28 0 0 0 0 26 22 0 0 7 21 11 11 0 0 1 0 0 0 13 11 0 0 7 21 11 11 0 0 0 1 0 0 4 0 0 0 27 23 8 22 0 0 0 0 1 0 1 20 0 0 27 23 8 22 0 0 0 0 0 1
,
 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 17 25 0 0 0 0 0 21 1 0 0 0 0 0 26 17 0 0 0 0 21 0 0 1 0 0 0 0 0 0 13 12 22 8 0 0 0 0 28 0 0 0 0 0 13 9 22 8 0 0 0 0 0 28 0 0 0 0 28 27 2 6 0 0 0 0 0 0 28 0 0 0 28 22 2 6 0 0 0 0 0 0 0 28
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 6 28 17 11 0 21 1 0 0 0 0 0 0 0 16 18 11 17 21 0 0 1 0 0 0 0 0 0 16 17 7 21 11 11 0 0 1 0 0 0 0 0 16 20 7 21 11 11 0 0 0 1 0 0 25 0 0 0 0 0 0 0 0 0 0 0 28 0 28 9 0 0 0 0 0 0 0 0 0 0 0 28
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 21 3 1 16 6 14 4 10 12 0 0 0 0 0 5 8 12 28 15 23 19 4 0 17 0 0 0 0 7 19 24 24 9 21 17 17 0 0 0 12 0 0 19 7 13 6 9 21 17 17 0 0 12 0 0 0 22 0 18 6 14 23 8 12 0 0 0 0 12 0 20 14 19 7 14 23 8 12 0 0 0 0 0 12
,
 0 17 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 6 1 23 18 17 21 10 19 0 12 0 0 0 0 0 16 16 15 21 17 10 10 12 0 0 0 0 0 2 24 14 6 20 1 0 13 0 0 17 0 0 0 1 12 1 17 20 1 0 13 0 0 0 17 0 0 27 24 15 13 4 4 15 2 0 0 0 0 28 0 11 16 4 3 2 27 23 24 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 13 11 21 26 1 0 10 10 12 0 0 0 0 0 5 2 18 0 16 0 10 10 0 12 0 0 0 0 24 27 2 2 2 15 28 28 0 0 1 0 0 0 28 17 2 27 24 23 17 17 0 0 0 28 0 0 23 23 24 2 0 14 26 3 0 0 0 0 0 17 5 14 24 27 5 0 6 6 0 0 0 0 12 0
,
 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 4 23 1 12 18 0 8 27 0 0 0 0 0 3 12 13 11 18 12 8 0 0 27 0 0 0 0 25 1 8 11 0 1 0 0 0 21 1 0 0 0 11 7 0 20 0 1 0 0 21 0 0 1 0 0 21 28 7 11 0 12 0 0 11 11 0 0 1 0 5 9 28 19 0 25 0 0 11 11 0 0 0 1 20 26 24 20 0 27 0 0 8 22 0 0 0 0 6 11 11 10 0 10 0 0 8 22 0 0 0 0

`G:=sub<GL(14,GF(29))| [28,0,0,0,0,0,0,0,0,0,26,13,4,1,0,28,0,0,0,0,0,0,0,0,22,11,0,20,0,0,1,0,0,0,0,0,23,13,0,0,0,0,0,0,0,1,0,0,0,0,1,11,0,0,0,0,0,0,0,0,28,0,0,0,0,0,7,7,27,27,0,0,0,0,0,28,0,0,0,0,21,21,23,23,0,0,0,0,0,0,28,0,0,0,11,11,8,8,0,0,0,0,0,0,0,28,0,0,11,11,22,22,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,0,17,26,0,0,0,0,0,28,0,0,0,0,0,0,25,17,0,0,0,0,0,0,1,0,0,0,0,0,0,0,13,13,28,28,0,0,0,1,0,0,0,0,0,0,12,9,27,22,0,0,0,0,1,0,0,0,0,0,22,22,2,2,0,0,0,0,0,1,0,0,0,0,8,8,6,6,0,0,0,0,0,0,28,0,0,21,0,0,0,0,0,0,0,0,0,0,0,28,21,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,0,0,0,0,0,25,28,0,1,0,0,0,0,0,0,0,0,0,0,0,9,0,0,28,0,0,0,0,0,6,16,16,16,0,0,0,0,0,28,0,0,0,0,28,18,17,20,0,0,0,0,0,0,28,0,0,0,17,11,7,7,0,0,0,0,0,0,0,28,0,0,11,17,21,21,0,0,0,0,0,0,0,0,28,0,0,21,11,11,0,0,0,0,0,0,0,0,0,28,21,0,11,11,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,0,21,5,7,19,22,20,0,28,0,0,0,0,0,0,3,8,19,7,0,14,0,0,0,17,0,0,0,0,1,12,24,13,18,19,0,0,12,0,0,0,0,0,16,28,24,6,6,7,0,0,0,0,0,28,0,0,6,15,9,9,14,14,0,0,0,0,1,0,0,0,14,23,21,21,23,23,0,0,0,0,0,0,0,1,4,19,17,17,8,8,0,0,0,0,0,0,1,0,10,4,17,17,12,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12],[0,12,0,0,0,0,0,0,6,0,2,1,27,11,17,0,0,0,0,0,0,0,1,16,24,12,24,16,0,0,0,28,0,0,0,0,23,16,14,1,15,4,0,0,1,0,0,0,0,0,18,15,6,17,13,3,0,0,0,0,0,1,0,0,17,21,20,20,4,2,0,0,0,0,1,0,0,0,21,17,1,1,4,27,0,0,0,0,0,0,17,0,10,10,0,0,15,23,0,0,0,0,0,0,0,12,19,10,13,13,2,24,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[0,28,0,0,0,0,0,0,13,5,24,28,23,5,1,0,0,0,0,0,0,0,11,2,27,17,23,14,0,0,0,28,0,0,0,0,21,18,2,2,24,24,0,0,28,0,0,0,0,0,26,0,2,27,2,27,0,0,0,0,17,0,0,0,1,16,2,24,0,5,0,0,0,0,0,12,0,0,0,0,15,23,14,0,0,0,0,0,0,0,0,17,10,10,28,17,26,6,0,0,0,0,0,0,17,0,10,10,28,17,3,6,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,17,0],[0,0,0,0,0,0,12,3,25,11,21,5,20,6,0,0,0,0,0,0,4,12,1,7,28,9,26,11,1,0,0,0,0,0,23,13,8,0,7,28,24,11,0,1,0,0,0,0,1,11,11,20,11,19,20,10,0,0,1,0,0,0,12,18,0,0,0,0,0,0,0,0,0,1,0,0,18,12,1,1,12,25,27,10,0,0,0,0,1,0,0,8,0,0,0,0,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,0,21,11,11,8,8,0,0,0,0,0,0,0,27,21,0,11,11,22,22,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0] >;`

C23.F8 in GAP, Magma, Sage, TeX

`C_2^3.F_8`
`% in TeX`

`G:=Group("C2^3.F8");`
`// GroupNames label`

`G:=SmallGroup(448,179);`
`// by ID`

`G=gap.SmallGroup(448,179);`
`# by ID`

`G:=PCGroup([7,-7,-2,2,2,-2,2,2,197,792,590,219,268,983,570,521,80,7844,11765,5494]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^7=1,d^2=b*a=a*b,e^2=g*a*g^-1=a*b*c,f^2=g*c*g^-1=a,g*b*g^-1=a*c=c*a,e*d*e^-1=a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,f*e*f^-1=c*e=e*c,c*f=f*c,f*d*f^-1=b*c*d,g*d*g^-1=a*b*e*f,g*e*g^-1=a*c*d,g*f*g^-1=a*b*e>;`
`// generators/relations`

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