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## G = C72⋊C8order 392 = 23·72

### The semidirect product of C72 and C8 acting faithfully

Aliases: C72⋊C8, C7⋊D7.C4, C72⋊C4.1C2, SmallGroup(392,36)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C72 — C72⋊C8
 Chief series C1 — C72 — C7⋊D7 — C72⋊C4 — C72⋊C8
 Lower central C72 — C72⋊C8
 Upper central C1

Generators and relations for C72⋊C8
G = < a,b,c | a7=b7=c8=1, ab=ba, cac-1=a-1b2, cbc-1=ab4 >

49C2
4C7
4C7
49C4
28D7
28D7
49C8

Character table of C72⋊C8

 class 1 2 4A 4B 7A 7B 7C 7D 7E 7F 8A 8B 8C 8D size 1 49 49 49 8 8 8 8 8 8 49 49 49 49 ρ1 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 linear of order 2 ρ3 1 1 -1 -1 1 1 1 1 1 1 i i -i -i linear of order 4 ρ4 1 1 -1 -1 1 1 1 1 1 1 -i -i i i linear of order 4 ρ5 1 -1 i -i 1 1 1 1 1 1 ζ87 ζ83 ζ85 ζ8 linear of order 8 ρ6 1 -1 -i i 1 1 1 1 1 1 ζ8 ζ85 ζ83 ζ87 linear of order 8 ρ7 1 -1 i -i 1 1 1 1 1 1 ζ83 ζ87 ζ8 ζ85 linear of order 8 ρ8 1 -1 -i i 1 1 1 1 1 1 ζ85 ζ8 ζ87 ζ83 linear of order 8 ρ9 8 0 0 0 -2ζ76-2ζ7-2 ζ75+2ζ74+2ζ73+ζ72+2 2ζ76+ζ74+ζ73+2ζ7+2 -2ζ74-2ζ73-2 -2ζ75-2ζ72-2 ζ76+2ζ75+2ζ72+ζ7+2 0 0 0 0 orthogonal faithful ρ10 8 0 0 0 2ζ76+ζ74+ζ73+2ζ7+2 -2ζ76-2ζ7-2 -2ζ75-2ζ72-2 ζ75+2ζ74+2ζ73+ζ72+2 ζ76+2ζ75+2ζ72+ζ7+2 -2ζ74-2ζ73-2 0 0 0 0 orthogonal faithful ρ11 8 0 0 0 ζ75+2ζ74+2ζ73+ζ72+2 -2ζ74-2ζ73-2 -2ζ76-2ζ7-2 ζ76+2ζ75+2ζ72+ζ7+2 2ζ76+ζ74+ζ73+2ζ7+2 -2ζ75-2ζ72-2 0 0 0 0 orthogonal faithful ρ12 8 0 0 0 ζ76+2ζ75+2ζ72+ζ7+2 -2ζ75-2ζ72-2 -2ζ74-2ζ73-2 2ζ76+ζ74+ζ73+2ζ7+2 ζ75+2ζ74+2ζ73+ζ72+2 -2ζ76-2ζ7-2 0 0 0 0 orthogonal faithful ρ13 8 0 0 0 -2ζ74-2ζ73-2 ζ76+2ζ75+2ζ72+ζ7+2 ζ75+2ζ74+2ζ73+ζ72+2 -2ζ75-2ζ72-2 -2ζ76-2ζ7-2 2ζ76+ζ74+ζ73+2ζ7+2 0 0 0 0 orthogonal faithful ρ14 8 0 0 0 -2ζ75-2ζ72-2 2ζ76+ζ74+ζ73+2ζ7+2 ζ76+2ζ75+2ζ72+ζ7+2 -2ζ76-2ζ7-2 -2ζ74-2ζ73-2 ζ75+2ζ74+2ζ73+ζ72+2 0 0 0 0 orthogonal faithful

Permutation representations of C72⋊C8
On 28 points - transitive group 28T56
Generators in S28
```(1 17 24 8 12 28 13)(2 9 14 25 21 18 5)(3 19 26 10 6 22 15)
(1 24 12 13 17 8 28)(2 25 5 14 18 9 21)(3 6 19 22 26 15 10)(4 11 16 27 23 20 7)
(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)```

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

`G:=Group( (1,17,24,8,12,28,13)(2,9,14,25,21,18,5)(3,19,26,10,6,22,15), (1,24,12,13,17,8,28)(2,25,5,14,18,9,21)(3,6,19,22,26,15,10)(4,11,16,27,23,20,7), (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) );`

`G=PermutationGroup([[(1,17,24,8,12,28,13),(2,9,14,25,21,18,5),(3,19,26,10,6,22,15)], [(1,24,12,13,17,8,28),(2,25,5,14,18,9,21),(3,6,19,22,26,15,10),(4,11,16,27,23,20,7)], [(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)]])`

`G:=TransitiveGroup(28,56);`

Matrix representation of C72⋊C8 in GL8(𝔽113)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 24 112 112 24 0 0 0 0 73 77 0 0 10 89 0 0 21 98 0 0 24 112 0 0 40 36 0 0 0 0 10 89 92 15 0 0 0 0 24 112
,
 112 1 0 0 0 0 0 0 87 25 0 0 0 0 0 0 36 88 89 10 0 0 0 0 50 78 103 103 0 0 0 0 36 98 0 0 24 112 0 0 15 0 0 0 1 0 0 0 25 36 0 0 0 0 89 10 41 73 0 0 0 0 103 103
,
 0 0 0 0 112 1 0 0 21 98 0 0 111 24 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 15 0 0 0 0 0 0 1 15 0 0 0 0 0 0 0 98 0 0 0 88 1 0 0 98 0 0 0

`G:=sub<GL(8,GF(113))| [1,0,0,24,73,21,40,92,0,1,0,112,77,98,36,15,0,0,0,112,0,0,0,0,0,0,1,24,0,0,0,0,0,0,0,0,10,24,0,0,0,0,0,0,89,112,0,0,0,0,0,0,0,0,10,24,0,0,0,0,0,0,89,112],[112,87,36,50,36,15,25,41,1,25,88,78,98,0,36,73,0,0,89,103,0,0,0,0,0,0,10,103,0,0,0,0,0,0,0,0,24,1,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,89,103,0,0,0,0,0,0,10,103],[0,21,0,0,0,0,0,88,0,98,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,112,111,1,1,15,15,98,98,1,24,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;`

C72⋊C8 in GAP, Magma, Sage, TeX

`C_7^2\rtimes C_8`
`% in TeX`

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

`G:=SmallGroup(392,36);`
`// by ID`

`G=gap.SmallGroup(392,36);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-7,7,10,26,6243,888,253,9604,2509,2114]);`
`// Polycyclic`

`G:=Group<a,b,c|a^7=b^7=c^8=1,a*b=b*a,c*a*c^-1=a^-1*b^2,c*b*c^-1=a*b^4>;`
`// generators/relations`

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