non-abelian, soluble, monomial
Aliases: D7≀C2, C72⋊D4, D72⋊C2, C72⋊C4⋊C2, C7⋊D7.1C22, SmallGroup(392,37)
Series: Derived ►Chief ►Lower central ►Upper central
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 >
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 |
►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 2 3 6 4 7 5)(8 11 9 12 10 13 14)
(1 22 14 16)(2 23 13 15)(3 24 10 21)(4 26 9 19)(5 28 8 17)(6 25 12 20)(7 27 11 18)
(1 10)(2 13)(3 14)(4 11)(5 12)(6 8)(7 9)(16 21)(17 20)(18 19)(22 24)(25 28)(26 27)
G:=sub<Sym(28)| (15,16,17,18,19,20,21)(22,23,24,25,26,27,28), (1,2,3,6,4,7,5)(8,11,9,12,10,13,14), (1,22,14,16)(2,23,13,15)(3,24,10,21)(4,26,9,19)(5,28,8,17)(6,25,12,20)(7,27,11,18), (1,10)(2,13)(3,14)(4,11)(5,12)(6,8)(7,9)(16,21)(17,20)(18,19)(22,24)(25,28)(26,27)>;
G:=Group( (15,16,17,18,19,20,21)(22,23,24,25,26,27,28), (1,2,3,6,4,7,5)(8,11,9,12,10,13,14), (1,22,14,16)(2,23,13,15)(3,24,10,21)(4,26,9,19)(5,28,8,17)(6,25,12,20)(7,27,11,18), (1,10)(2,13)(3,14)(4,11)(5,12)(6,8)(7,9)(16,21)(17,20)(18,19)(22,24)(25,28)(26,27) );
G=PermutationGroup([[(15,16,17,18,19,20,21),(22,23,24,25,26,27,28)], [(1,2,3,6,4,7,5),(8,11,9,12,10,13,14)], [(1,22,14,16),(2,23,13,15),(3,24,10,21),(4,26,9,19),(5,28,8,17),(6,25,12,20),(7,27,11,18)], [(1,10),(2,13),(3,14),(4,11),(5,12),(6,8),(7,9),(16,21),(17,20),(18,19),(22,24),(25,28),(26,27)]])
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
| action | f(x) | Disc(f) |
|---|---|---|
| 14T20 | x14+210x12-3164x11+63455x10-534016x9+7977046x8-27661364x7+1002627612x6+6022284016x5-28570776528x4+138886748224x3-3146649429952x2+4701085568256x-59618052726016 | 2108·57·724·197·292·4112·892·1812·4092·17212·50232·162292 |
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
Export
Subgroup lattice of D7≀C2 in TeX
Character table of D7≀C2 in TeX