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## G = C7⋊D4order 56 = 23·7

### The semidirect product of C7 and D4 acting via D4/C22=C2

Aliases: C72D4, C22⋊D7, Dic7⋊C2, D142C2, C2.5D14, C14.5C22, (C2×C14)⋊2C2, SmallGroup(56,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C7⋊D4
 Chief series C1 — C7 — C14 — D14 — C7⋊D4
 Lower central C7 — C14 — C7⋊D4
 Upper central C1 — C2 — C22

Generators and relations for C7⋊D4
G = < a,b,c | a7=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

Character table of C7⋊D4

 class 1 2A 2B 2C 4 7A 7B 7C 14A 14B 14C 14D 14E 14F 14G 14H 14I size 1 1 2 14 14 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 trivial ρ2 1 1 -1 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 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ4 1 1 1 -1 -1 1 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 0 0 0 0 0 0 -2 -2 orthogonal lifted from D4 ρ6 2 2 2 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 orthogonal lifted from D7 ρ7 2 2 2 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D7 ρ8 2 2 2 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 orthogonal lifted from D7 ρ9 2 2 -2 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 ζ76+ζ7 ζ75+ζ72 orthogonal lifted from D14 ρ10 2 2 -2 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D14 ρ11 2 2 -2 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 ζ75+ζ72 ζ74+ζ73 orthogonal lifted from D14 ρ12 2 -2 0 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ76-ζ7 -ζ76+ζ7 -ζ75+ζ72 -ζ74+ζ73 ζ74-ζ73 ζ75-ζ72 ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 complex faithful ρ13 2 -2 0 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ75-ζ72 ζ75-ζ72 -ζ74+ζ73 -ζ76+ζ7 ζ76-ζ7 ζ74-ζ73 -ζ75+ζ72 -ζ74-ζ73 -ζ76-ζ7 complex faithful ρ14 2 -2 0 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ74-ζ73 ζ74-ζ73 -ζ76+ζ7 ζ75-ζ72 -ζ75+ζ72 ζ76-ζ7 -ζ74+ζ73 -ζ76-ζ7 -ζ75-ζ72 complex faithful ρ15 2 -2 0 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ76-ζ7 ζ76-ζ7 ζ75-ζ72 ζ74-ζ73 -ζ74+ζ73 -ζ75+ζ72 -ζ76+ζ7 -ζ75-ζ72 -ζ74-ζ73 complex faithful ρ16 2 -2 0 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ75-ζ72 -ζ75+ζ72 ζ74-ζ73 ζ76-ζ7 -ζ76+ζ7 -ζ74+ζ73 ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 complex faithful ρ17 2 -2 0 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ74-ζ73 -ζ74+ζ73 ζ76-ζ7 -ζ75+ζ72 ζ75-ζ72 -ζ76+ζ7 ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 complex faithful

Permutation representations of C7⋊D4
On 28 points - transitive group 28T6
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)
(1 22 13 15)(2 28 14 21)(3 27 8 20)(4 26 9 19)(5 25 10 18)(6 24 11 17)(7 23 12 16)
(1 15)(2 21)(3 20)(4 19)(5 18)(6 17)(7 16)(8 27)(9 26)(10 25)(11 24)(12 23)(13 22)(14 28)```

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

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

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

`G:=TransitiveGroup(28,6);`

On 28 points - transitive group 28T7
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)
(1 20 13 27)(2 19 14 26)(3 18 8 25)(4 17 9 24)(5 16 10 23)(6 15 11 22)(7 21 12 28)
(2 7)(3 6)(4 5)(8 11)(9 10)(12 14)(15 25)(16 24)(17 23)(18 22)(19 28)(20 27)(21 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), (1,20,13,27)(2,19,14,26)(3,18,8,25)(4,17,9,24)(5,16,10,23)(6,15,11,22)(7,21,12,28), (2,7)(3,6)(4,5)(8,11)(9,10)(12,14)(15,25)(16,24)(17,23)(18,22)(19,28)(20,27)(21,26)>;`

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

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

`G:=TransitiveGroup(28,7);`

C7⋊D4 is a maximal subgroup of
C4○D28  D4×D7  D42D7  Dic7⋊C6  C21⋊D4  C7⋊D12  C217D4  C7⋊S4  C35⋊D4  C7⋊D20  C357D4  C49⋊D4  C722D4  C7⋊D28  C727D4
C7⋊D4 is a maximal quotient of
Dic7⋊C4  D14⋊C4  D4⋊D7  D4.D7  Q8⋊D7  C7⋊Q16  C23.D7  C21⋊D4  C7⋊D12  C217D4  C35⋊D4  C7⋊D20  C357D4  C49⋊D4  C722D4  C7⋊D28  C727D4

Matrix representation of C7⋊D4 in GL2(𝔽29) generated by

 7 1 28 0
,
 9 19 14 20
,
 1 7 0 28
`G:=sub<GL(2,GF(29))| [7,28,1,0],[9,14,19,20],[1,0,7,28] >;`

C7⋊D4 in GAP, Magma, Sage, TeX

`C_7\rtimes D_4`
`% in TeX`

`G:=Group("C7:D4");`
`// GroupNames label`

`G:=SmallGroup(56,7);`
`// by ID`

`G=gap.SmallGroup(56,7);`
`# by ID`

`G:=PCGroup([4,-2,-2,-2,-7,49,771]);`
`// Polycyclic`

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

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