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G = D7≀C2order 392 = 23·72

Wreath product of D7 by C2

Aliases: D7C2, C72⋊D4, D72⋊C2, C72⋊C4⋊C2, C7⋊D7.1C22, SmallGroup(392,37)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C72 — C7⋊D7 — D7≀C2
 Chief series C1 — C72 — C7⋊D7 — D72 — D7≀C2
 Lower central C72 — C7⋊D7 — D7≀C2
 Upper central C1

Generators and relations for D7≀C2
G = < a,b,c,d | a7=b7=c4=d2=1, ab=ba, cac-1=b, dad=cbc-1=a-1, bd=db, dcd=c-1 >

14C2
14C2
49C2
2C7
2C7
4C7
49C4
49C22
49C22
2D7
2D7
14D7
14C14
14D7
14C14
28D7
49D4
14D14
14D14

Character table of D7≀C2

 class 1 2A 2B 2C 4 7A 7B 7C 7D 7E 7F 7G 7H 7I 14A 14B 14C 14D 14E 14F size 1 14 14 49 98 4 4 4 4 4 4 8 8 8 28 28 28 28 28 28 ρ1 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 linear of order 2 ρ3 1 -1 1 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 -1 -1 -1 linear of order 2 ρ5 2 0 0 -2 0 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ6 4 -2 0 0 0 2ζ76+2ζ7 2ζ74+2ζ73 ζ76+ζ7+2 ζ75+ζ72+2 ζ74+ζ73+2 2ζ75+2ζ72 -ζ76-ζ7-1 -ζ75-ζ72-1 -ζ74-ζ73-1 0 0 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 0 orthogonal faithful ρ7 4 -2 0 0 0 2ζ75+2ζ72 2ζ76+2ζ7 ζ75+ζ72+2 ζ74+ζ73+2 ζ76+ζ7+2 2ζ74+2ζ73 -ζ75-ζ72-1 -ζ74-ζ73-1 -ζ76-ζ7-1 0 0 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 0 orthogonal faithful ρ8 4 0 2 0 0 ζ75+ζ72+2 ζ76+ζ7+2 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 ζ74+ζ73+2 -ζ74-ζ73-1 -ζ76-ζ7-1 -ζ75-ζ72-1 ζ74+ζ73 ζ76+ζ7 0 0 0 ζ75+ζ72 orthogonal faithful ρ9 4 -2 0 0 0 2ζ74+2ζ73 2ζ75+2ζ72 ζ74+ζ73+2 ζ76+ζ7+2 ζ75+ζ72+2 2ζ76+2ζ7 -ζ74-ζ73-1 -ζ76-ζ7-1 -ζ75-ζ72-1 0 0 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 0 orthogonal faithful ρ10 4 0 -2 0 0 ζ74+ζ73+2 ζ75+ζ72+2 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 ζ76+ζ7+2 -ζ76-ζ7-1 -ζ75-ζ72-1 -ζ74-ζ73-1 -ζ76-ζ7 -ζ75-ζ72 0 0 0 -ζ74-ζ73 orthogonal faithful ρ11 4 0 -2 0 0 ζ75+ζ72+2 ζ76+ζ7+2 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 ζ74+ζ73+2 -ζ74-ζ73-1 -ζ76-ζ7-1 -ζ75-ζ72-1 -ζ74-ζ73 -ζ76-ζ7 0 0 0 -ζ75-ζ72 orthogonal faithful ρ12 4 2 0 0 0 2ζ75+2ζ72 2ζ76+2ζ7 ζ75+ζ72+2 ζ74+ζ73+2 ζ76+ζ7+2 2ζ74+2ζ73 -ζ75-ζ72-1 -ζ74-ζ73-1 -ζ76-ζ7-1 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 0 orthogonal faithful ρ13 4 0 2 0 0 ζ76+ζ7+2 ζ74+ζ73+2 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 ζ75+ζ72+2 -ζ75-ζ72-1 -ζ74-ζ73-1 -ζ76-ζ7-1 ζ75+ζ72 ζ74+ζ73 0 0 0 ζ76+ζ7 orthogonal faithful ρ14 4 0 -2 0 0 ζ76+ζ7+2 ζ74+ζ73+2 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 ζ75+ζ72+2 -ζ75-ζ72-1 -ζ74-ζ73-1 -ζ76-ζ7-1 -ζ75-ζ72 -ζ74-ζ73 0 0 0 -ζ76-ζ7 orthogonal faithful ρ15 4 2 0 0 0 2ζ74+2ζ73 2ζ75+2ζ72 ζ74+ζ73+2 ζ76+ζ7+2 ζ75+ζ72+2 2ζ76+2ζ7 -ζ74-ζ73-1 -ζ76-ζ7-1 -ζ75-ζ72-1 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 0 orthogonal faithful ρ16 4 2 0 0 0 2ζ76+2ζ7 2ζ74+2ζ73 ζ76+ζ7+2 ζ75+ζ72+2 ζ74+ζ73+2 2ζ75+2ζ72 -ζ76-ζ7-1 -ζ75-ζ72-1 -ζ74-ζ73-1 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 0 orthogonal faithful ρ17 4 0 2 0 0 ζ74+ζ73+2 ζ75+ζ72+2 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 ζ76+ζ7+2 -ζ76-ζ7-1 -ζ75-ζ72-1 -ζ74-ζ73-1 ζ76+ζ7 ζ75+ζ72 0 0 0 ζ74+ζ73 orthogonal faithful ρ18 8 0 0 0 0 -2ζ76-2ζ7-2 -2ζ74-2ζ73-2 -2ζ75-2ζ72-2 -2ζ74-2ζ73-2 -2ζ76-2ζ7-2 -2ζ75-2ζ72-2 ζ75+2ζ74+2ζ73+ζ72+2 2ζ76+ζ74+ζ73+2ζ7+2 ζ76+2ζ75+2ζ72+ζ7+2 0 0 0 0 0 0 orthogonal faithful ρ19 8 0 0 0 0 -2ζ75-2ζ72-2 -2ζ76-2ζ7-2 -2ζ74-2ζ73-2 -2ζ76-2ζ7-2 -2ζ75-2ζ72-2 -2ζ74-2ζ73-2 2ζ76+ζ74+ζ73+2ζ7+2 ζ76+2ζ75+2ζ72+ζ7+2 ζ75+2ζ74+2ζ73+ζ72+2 0 0 0 0 0 0 orthogonal faithful ρ20 8 0 0 0 0 -2ζ74-2ζ73-2 -2ζ75-2ζ72-2 -2ζ76-2ζ7-2 -2ζ75-2ζ72-2 -2ζ74-2ζ73-2 -2ζ76-2ζ7-2 ζ76+2ζ75+2ζ72+ζ7+2 ζ75+2ζ74+2ζ73+ζ72+2 2ζ76+ζ74+ζ73+2ζ7+2 0 0 0 0 0 0 orthogonal faithful

Permutation representations of D7≀C2
On 14 points - transitive group 14T20
Generators in S14
```(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)
(1 2 3 4 5 6 7)(8 14 13 12 11 10 9)
(1 8)(2 9 7 14)(3 10 6 13)(4 11 5 12)
(1 8)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)```

`G:=sub<Sym(14)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14), (1,2,3,4,5,6,7)(8,14,13,12,11,10,9), (1,8)(2,9,7,14)(3,10,6,13)(4,11,5,12), (1,8)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)>;`

`G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14), (1,2,3,4,5,6,7)(8,14,13,12,11,10,9), (1,8)(2,9,7,14)(3,10,6,13)(4,11,5,12), (1,8)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9) );`

`G=PermutationGroup([(1,2,3,4,5,6,7),(8,9,10,11,12,13,14)], [(1,2,3,4,5,6,7),(8,14,13,12,11,10,9)], [(1,8),(2,9,7,14),(3,10,6,13),(4,11,5,12)], [(1,8),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9)])`

`G:=TransitiveGroup(14,20);`

On 28 points - transitive group 28T53
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 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 21 20 19 18 17 16)(22 28 27 26 25 24 23)
(1 27 8 15)(2 28 14 21)(3 22 13 20)(4 23 12 19)(5 24 11 18)(6 25 10 17)(7 26 9 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), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,21,20,19,18,17,16)(22,28,27,26,25,24,23), (1,27,8,15)(2,28,14,21)(3,22,13,20)(4,23,12,19)(5,24,11,18)(6,25,10,17)(7,26,9,16), (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), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,21,20,19,18,17,16)(22,28,27,26,25,24,23), (1,27,8,15)(2,28,14,21)(3,22,13,20)(4,23,12,19)(5,24,11,18)(6,25,10,17)(7,26,9,16), (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)], [(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,21,20,19,18,17,16),(22,28,27,26,25,24,23)], [(1,27,8,15),(2,28,14,21),(3,22,13,20),(4,23,12,19),(5,24,11,18),(6,25,10,17),(7,26,9,16)], [(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(28,53);`

On 28 points - transitive group 28T54
Generators in S28
```(15 16 17 18 19 20 21)(22 23 24 25 26 27 28)
(1 11 12 9 13 10 14)(2 6 3 7 8 4 5)
(1 24 4 15)(2 17 10 22)(3 19 9 27)(5 16 14 23)(6 18 13 28)(7 20 12 26)(8 21 11 25)
(1 4)(2 12)(3 13)(5 11)(6 9)(7 10)(8 14)(16 21)(17 20)(18 19)(22 26)(23 25)(27 28)```

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

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

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

`G:=TransitiveGroup(28,54);`

On 28 points - transitive group 28T55
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 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 21 20 19 18 17 16)(22 28 27 26 25 24 23)
(1 22)(2 23 7 28)(3 24 6 27)(4 25 5 26)(8 17 11 20)(9 18 10 19)(12 21 14 16)(13 15)
(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), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,21,20,19,18,17,16)(22,28,27,26,25,24,23), (1,22)(2,23,7,28)(3,24,6,27)(4,25,5,26)(8,17,11,20)(9,18,10,19)(12,21,14,16)(13,15), (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), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,21,20,19,18,17,16)(22,28,27,26,25,24,23), (1,22)(2,23,7,28)(3,24,6,27)(4,25,5,26)(8,17,11,20)(9,18,10,19)(12,21,14,16)(13,15), (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)], [(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,21,20,19,18,17,16),(22,28,27,26,25,24,23)], [(1,22),(2,23,7,28),(3,24,6,27),(4,25,5,26),(8,17,11,20),(9,18,10,19),(12,21,14,16),(13,15)], [(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(28,55);`

On 28 points - transitive group 28T57
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 3 5 7 2 4 6)(8 12 9 13 10 14 11)(15 20 18 16 21 19 17)(22 25 28 24 27 23 26)
(1 22)(2 26 7 25)(3 23 6 28)(4 27 5 24)(8 20 11 19)(9 15 10 17)(12 21 14 18)(13 16)
(1 16)(2 15)(3 21)(4 20)(5 19)(6 18)(7 17)(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), (1,3,5,7,2,4,6)(8,12,9,13,10,14,11)(15,20,18,16,21,19,17)(22,25,28,24,27,23,26), (1,22)(2,26,7,25)(3,23,6,28)(4,27,5,24)(8,20,11,19)(9,15,10,17)(12,21,14,18)(13,16), (1,16)(2,15)(3,21)(4,20)(5,19)(6,18)(7,17)(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), (1,3,5,7,2,4,6)(8,12,9,13,10,14,11)(15,20,18,16,21,19,17)(22,25,28,24,27,23,26), (1,22)(2,26,7,25)(3,23,6,28)(4,27,5,24)(8,20,11,19)(9,15,10,17)(12,21,14,18)(13,16), (1,16)(2,15)(3,21)(4,20)(5,19)(6,18)(7,17)(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)], [(1,3,5,7,2,4,6),(8,12,9,13,10,14,11),(15,20,18,16,21,19,17),(22,25,28,24,27,23,26)], [(1,22),(2,26,7,25),(3,23,6,28),(4,27,5,24),(8,20,11,19),(9,15,10,17),(12,21,14,18),(13,16)], [(1,16),(2,15),(3,21),(4,20),(5,19),(6,18),(7,17),(8,27),(9,26),(10,25),(11,24),(12,23),(13,22),(14,28)])`

`G:=TransitiveGroup(28,57);`

Polynomial with Galois group D7≀C2 over ℚ
actionf(x)Disc(f)
14T20x14+210x12-3164x11+63455x10-534016x9+7977046x8-27661364x7+1002627612x6+6022284016x5-28570776528x4+138886748224x3-3146649429952x2+4701085568256x-596180527260162108·57·724·197·292·4112·892·1812·4092·17212·50232·162292

Matrix representation of D7≀C2 in GL4(𝔽29) generated by

 21 1 0 0 23 26 0 0 26 28 25 28 9 22 2 22
,
 21 1 0 0 23 26 0 0 1 0 22 1 4 28 27 25
,
 0 0 21 1 25 28 19 25 0 19 1 0 1 11 8 0
,
 0 0 21 1 25 28 19 25 0 0 1 0 1 0 8 0
`G:=sub<GL(4,GF(29))| [21,23,26,9,1,26,28,22,0,0,25,2,0,0,28,22],[21,23,1,4,1,26,0,28,0,0,22,27,0,0,1,25],[0,25,0,1,0,28,19,11,21,19,1,8,1,25,0,0],[0,25,0,1,0,28,0,0,21,19,1,8,1,25,0,0] >;`

D7≀C2 in GAP, Magma, Sage, TeX

`D_7\wr C_2`
`% in TeX`

`G:=Group("D7wrC2");`
`// GroupNames label`

`G:=SmallGroup(392,37);`
`// by ID`

`G=gap.SmallGroup(392,37);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-7,7,61,963,568,253,109,2114]);`
`// Polycyclic`

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

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