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## G = C7⋊D7order 98 = 2·72

### The semidirect product of C7 and D7 acting via D7/C7=C2

Aliases: C7⋊D7, C722C2, SmallGroup(98,4)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C7⋊D7
G = < a,b,c | a7=b7=c2=1, ab=ba, cac=a-1, cbc=b-1 >

Character table of C7⋊D7

 class 1 2 7A 7B 7C 7D 7E 7F 7G 7H 7I 7J 7K 7L 7M 7N 7O 7P 7Q 7R 7S 7T 7U 7V 7W 7X size 1 49 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 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 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 linear of order 2 ρ3 2 0 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 2 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 2 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 2 ζ74+ζ73 orthogonal lifted from D7 ρ4 2 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 2 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 2 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 2 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 orthogonal lifted from D7 ρ5 2 0 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 2 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 2 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 2 ζ76+ζ7 ζ76+ζ7 orthogonal lifted from D7 ρ6 2 0 2 2 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 2 orthogonal lifted from D7 ρ7 2 0 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 2 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 2 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 2 ζ75+ζ72 orthogonal lifted from D7 ρ8 2 0 2 2 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 2 orthogonal lifted from D7 ρ9 2 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 2 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 2 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 2 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 orthogonal lifted from D7 ρ10 2 0 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 2 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 2 ζ74+ζ73 ζ75+ζ72 2 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 orthogonal lifted from D7 ρ11 2 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 2 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 2 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 2 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 orthogonal lifted from D7 ρ12 2 0 ζ76+ζ7 ζ75+ζ72 2 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 2 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 2 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 orthogonal lifted from D7 ρ13 2 0 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 2 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 2 ζ76+ζ7 ζ74+ζ73 2 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 orthogonal lifted from D7 ρ14 2 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 2 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 2 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 2 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 orthogonal lifted from D7 ρ15 2 0 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 2 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 2 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 2 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 orthogonal lifted from D7 ρ16 2 0 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 2 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 2 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 2 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 orthogonal lifted from D7 ρ17 2 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 2 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 2 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 2 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D7 ρ18 2 0 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 2 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 2 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 2 ζ74+ζ73 ζ74+ζ73 orthogonal lifted from D7 ρ19 2 0 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 2 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 2 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 2 ζ75+ζ72 ζ75+ζ72 orthogonal lifted from D7 ρ20 2 0 ζ74+ζ73 ζ76+ζ7 2 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 2 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 2 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 orthogonal lifted from D7 ρ21 2 0 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 2 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 2 ζ75+ζ72 ζ76+ζ7 2 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D7 ρ22 2 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 2 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 2 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 2 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 orthogonal lifted from D7 ρ23 2 0 ζ75+ζ72 ζ74+ζ73 2 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 2 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 2 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 orthogonal lifted from D7 ρ24 2 0 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 2 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 2 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 2 ζ76+ζ7 orthogonal lifted from D7 ρ25 2 0 2 2 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 2 orthogonal lifted from D7 ρ26 2 0 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 2 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 2 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 2 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 orthogonal lifted from D7

Smallest permutation representation of C7⋊D7
On 49 points
Generators in S49
```(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)
(1 31 9 17 25 46 41)(2 32 10 18 26 47 42)(3 33 11 19 27 48 36)(4 34 12 20 28 49 37)(5 35 13 21 22 43 38)(6 29 14 15 23 44 39)(7 30 8 16 24 45 40)
(1 41)(2 40)(3 39)(4 38)(5 37)(6 36)(7 42)(8 26)(9 25)(10 24)(11 23)(12 22)(13 28)(14 27)(15 19)(16 18)(20 21)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 49)```

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

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

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

C7⋊D7 is a maximal subgroup of   C72⋊C4  D72  C75F7  C7⋊F7  C72⋊C6  C7⋊D21  C7⋊D35
C7⋊D7 is a maximal quotient of   C7⋊Dic7  C7⋊D21  C7⋊D35

Matrix representation of C7⋊D7 in GL4(𝔽29) generated by

 22 1 0 0 16 10 0 0 0 0 22 19 0 0 10 10
,
 22 1 0 0 16 10 0 0 0 0 0 1 0 0 28 7
,
 10 28 0 0 12 19 0 0 0 0 10 7 0 0 19 19
`G:=sub<GL(4,GF(29))| [22,16,0,0,1,10,0,0,0,0,22,10,0,0,19,10],[22,16,0,0,1,10,0,0,0,0,0,28,0,0,1,7],[10,12,0,0,28,19,0,0,0,0,10,19,0,0,7,19] >;`

C7⋊D7 in GAP, Magma, Sage, TeX

`C_7\rtimes D_7`
`% in TeX`

`G:=Group("C7:D7");`
`// GroupNames label`

`G:=SmallGroup(98,4);`
`// by ID`

`G=gap.SmallGroup(98,4);`
`# by ID`

`G:=PCGroup([3,-2,-7,-7,73,758]);`
`// Polycyclic`

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

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