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## G = C72⋊2C8order 392 = 23·72

### The semidirect product of C72 and C8 acting via C8/C2=C4

Aliases: C722C8, (C7×C14).C4, C2.(C72⋊C4), C7⋊Dic7.1C2, SmallGroup(392,17)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C72 — C72⋊2C8
 Chief series C1 — C72 — C7×C14 — C7⋊Dic7 — C72⋊2C8
 Lower central C72 — C72⋊2C8
 Upper central C1 — C2

Generators and relations for C722C8
G = < a,b,c | a7=b7=c8=1, ab=ba, cac-1=a3b-1, cbc-1=a3b4 >

2C7
2C7
2C7
2C7
49C4
2C14
2C14
2C14
2C14
49C8
14Dic7
14Dic7
14Dic7
14Dic7

Smallest permutation representation of C722C8
On 56 points
Generators in S56
```(1 48 31 17 13 49 37)(2 50 18 41 38 14 32)(3 39 51 15 19 25 42)(4 26 16 40 43 20 52)(5 44 27 21 9 53 33)(6 54 22 45 34 10 28)(7 35 55 11 23 29 46)(8 30 12 36 47 24 56)
(1 49 17 48 37 13 31)(2 32 14 38 41 18 50)(3 25 15 39 42 19 51)(4 52 20 43 40 16 26)(5 53 21 44 33 9 27)(6 28 10 34 45 22 54)(7 29 11 35 46 23 55)(8 56 24 47 36 12 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 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)```

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

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

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

32 conjugacy classes

 class 1 2 4A 4B 7A ··· 7L 8A 8B 8C 8D 14A ··· 14L order 1 2 4 4 7 ··· 7 8 8 8 8 14 ··· 14 size 1 1 49 49 4 ··· 4 49 49 49 49 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 4 4 type + + + - image C1 C2 C4 C8 C72⋊C4 C72⋊2C8 kernel C72⋊2C8 C7⋊Dic7 C7×C14 C72 C2 C1 # reps 1 1 2 4 12 12

Matrix representation of C722C8 in GL4(𝔽113) generated by

 34 25 0 0 88 88 0 0 0 0 79 112 0 0 1 0
,
 0 1 0 0 112 79 0 0 0 0 0 1 0 0 112 79
,
 0 0 1 0 0 0 0 1 98 55 0 0 0 15 0 0
`G:=sub<GL(4,GF(113))| [34,88,0,0,25,88,0,0,0,0,79,1,0,0,112,0],[0,112,0,0,1,79,0,0,0,0,0,112,0,0,1,79],[0,0,98,0,0,0,55,15,1,0,0,0,0,1,0,0] >;`

C722C8 in GAP, Magma, Sage, TeX

`C_7^2\rtimes_2C_8`
`% in TeX`

`G:=Group("C7^2:2C8");`
`// GroupNames label`

`G:=SmallGroup(392,17);`
`// by ID`

`G=gap.SmallGroup(392,17);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-7,7,10,26,1763,488,5004,4209]);`
`// Polycyclic`

`G:=Group<a,b,c|a^7=b^7=c^8=1,a*b=b*a,c*a*c^-1=a^3*b^-1,c*b*c^-1=a^3*b^4>;`
`// generators/relations`

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