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## G = C72⋊Q8order 392 = 23·72

### The semidirect product of C72 and Q8 acting faithfully

Aliases: C72⋊Q8, C72⋊C4.2C2, C7⋊D7.2C22, SmallGroup(392,38)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C72 — C7⋊D7 — C72⋊Q8
 Chief series C1 — C72 — C7⋊D7 — C72⋊C4 — C72⋊Q8
 Lower central C72 — C7⋊D7 — C72⋊Q8
 Upper central C1

Generators and relations for C72⋊Q8
G = < a,b,c,d | a7=b7=c4=1, d2=c2, ab=ba, cac-1=a4b2, dad-1=a3b-1, cbc-1=a2b3, dbd-1=a3b4, dcd-1=c-1 >

49C2
4C7
4C7
49C4
49C4
49C4
28D7
28D7
49Q8

Character table of C72⋊Q8

 class 1 2 4A 4B 4C 7A 7B 7C 7D 7E 7F size 1 49 98 98 98 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 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 linear of order 2 ρ4 1 1 1 -1 -1 1 1 1 1 1 1 linear of order 2 ρ5 2 -2 0 0 0 2 2 2 2 2 2 symplectic lifted from Q8, Schur index 2 ρ6 8 0 0 0 0 3ζ75+ζ74+ζ73+3ζ72 ζ76+3ζ74+3ζ73+ζ7 1 1 1 3ζ76+ζ75+ζ72+3ζ7 orthogonal faithful ρ7 8 0 0 0 0 3ζ76+ζ75+ζ72+3ζ7 3ζ75+ζ74+ζ73+3ζ72 1 1 1 ζ76+3ζ74+3ζ73+ζ7 orthogonal faithful ρ8 8 0 0 0 0 1 1 3ζ76+ζ75+ζ72+3ζ7 3ζ75+ζ74+ζ73+3ζ72 ζ76+3ζ74+3ζ73+ζ7 1 orthogonal faithful ρ9 8 0 0 0 0 1 1 3ζ75+ζ74+ζ73+3ζ72 ζ76+3ζ74+3ζ73+ζ7 3ζ76+ζ75+ζ72+3ζ7 1 orthogonal faithful ρ10 8 0 0 0 0 1 1 ζ76+3ζ74+3ζ73+ζ7 3ζ76+ζ75+ζ72+3ζ7 3ζ75+ζ74+ζ73+3ζ72 1 orthogonal faithful ρ11 8 0 0 0 0 ζ76+3ζ74+3ζ73+ζ7 3ζ76+ζ75+ζ72+3ζ7 1 1 1 3ζ75+ζ74+ζ73+3ζ72 orthogonal faithful

Permutation representations of C72⋊Q8
On 28 points - transitive group 28T58
Generators in S28
```(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)
(8 12 9 13 10 14 11)(15 16 17 18 19 20 21)(22 24 26 28 23 25 27)
(1 9)(2 11 7 14)(3 13 6 12)(4 8 5 10)(15 27 16 26)(17 25 21 28)(18 24 20 22)(19 23)
(1 19)(2 17 7 21)(3 15 6 16)(4 20 5 18)(8 24 10 22)(9 23)(11 28 14 25)(12 27 13 26)```

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

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

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

`G:=TransitiveGroup(28,58);`

Matrix representation of C72⋊Q8 in GL8(𝔽29)

 28 1 0 0 0 0 0 0 9 19 0 0 0 0 0 0 24 10 11 4 0 0 0 0 9 6 25 25 0 0 0 0 20 23 0 0 25 25 0 0 5 19 0 0 4 11 0 0 24 10 0 0 0 0 11 4 9 6 0 0 0 0 25 25
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 23 10 4 11 0 0 0 0 18 28 18 28 0 0 0 0 21 23 0 0 25 25 0 0 6 19 0 0 4 11 0 0 18 28 0 0 0 0 18 28 0 0 0 0 0 0 1 0
,
 0 0 28 1 0 0 0 0 18 28 27 18 0 0 0 0 0 0 1 0 0 0 0 0 10 1 1 0 0 0 0 0 0 0 28 0 0 0 1 0 18 28 28 0 0 0 18 28 0 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0
,
 0 0 0 0 28 1 0 0 11 1 0 0 27 18 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 28 0 0 0 10 1 0 0 28 0 0 0 0 0 1 0 1 0 0 0 18 28 18 28 1 0 0 0

`G:=sub<GL(8,GF(29))| [28,9,24,9,20,5,24,9,1,19,10,6,23,19,10,6,0,0,11,25,0,0,0,0,0,0,4,25,0,0,0,0,0,0,0,0,25,4,0,0,0,0,0,0,25,11,0,0,0,0,0,0,0,0,11,25,0,0,0,0,0,0,4,25],[1,0,23,18,21,6,18,0,0,1,10,28,23,19,28,0,0,0,4,18,0,0,0,0,0,0,11,28,0,0,0,0,0,0,0,0,25,4,0,0,0,0,0,0,25,11,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,28,0],[0,18,0,10,0,18,0,0,0,28,0,1,0,28,0,0,28,27,1,1,28,28,1,1,1,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,18,0,0,0,0,0,0,0,28,0,0],[0,11,0,0,0,10,0,18,0,1,0,0,0,1,0,28,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,28,28,27,1,1,28,28,1,1,1,18,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;`

C72⋊Q8 in GAP, Magma, Sage, TeX

`C_7^2\rtimes Q_8`
`% in TeX`

`G:=Group("C7^2:Q8");`
`// GroupNames label`

`G:=SmallGroup(392,38);`
`// by ID`

`G=gap.SmallGroup(392,38);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-7,7,20,61,26,1763,3048,253,5004,3309,2114]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^7=b^7=c^4=1,d^2=c^2,a*b=b*a,c*a*c^-1=a^4*b^2,d*a*d^-1=a^3*b^-1,c*b*c^-1=a^2*b^3,d*b*d^-1=a^3*b^4,d*c*d^-1=c^-1>;`
`// generators/relations`

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