metabelian, soluble, monomial, A-group
Aliases: C72⋊C4, C7⋊D7.C2, SmallGroup(196,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C72 — C72⋊C4 |
Generators and relations for C72⋊C4
G = < a,b,c | a7=b7=c4=1, ab=ba, cac-1=a3b-1, cbc-1=a3b4 >
Character table of C72⋊C4
| class | 1 | 2 | 4A | 4B | 7A | 7B | 7C | 7D | 7E | 7F | 7G | 7H | 7I | 7J | 7K | 7L | |
| size | 1 | 49 | 49 | 49 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | -1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | -1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 4 | 0 | 0 | 0 | -ζ74-ζ73-1 | -ζ74-ζ73-1 | -ζ76-ζ7-1 | 2ζ75+2ζ72 | -ζ76-ζ7-1 | 2ζ74+2ζ73 | -ζ75-ζ72-1 | -ζ75-ζ72-1 | 2ζ76+2ζ7 | ζ75+ζ72+2 | ζ74+ζ73+2 | ζ76+ζ7+2 | orthogonal faithful |
| ρ6 | 4 | 0 | 0 | 0 | 2ζ74+2ζ73 | ζ75+ζ72+2 | ζ74+ζ73+2 | -ζ74-ζ73-1 | 2ζ76+2ζ7 | -ζ76-ζ7-1 | ζ76+ζ7+2 | 2ζ75+2ζ72 | -ζ75-ζ72-1 | -ζ76-ζ7-1 | -ζ75-ζ72-1 | -ζ74-ζ73-1 | orthogonal faithful |
| ρ7 | 4 | 0 | 0 | 0 | -ζ76-ζ7-1 | -ζ76-ζ7-1 | -ζ75-ζ72-1 | ζ75+ζ72+2 | -ζ75-ζ72-1 | ζ74+ζ73+2 | -ζ74-ζ73-1 | -ζ74-ζ73-1 | ζ76+ζ7+2 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | orthogonal faithful |
| ρ8 | 4 | 0 | 0 | 0 | -ζ74-ζ73-1 | -ζ74-ζ73-1 | -ζ76-ζ7-1 | ζ76+ζ7+2 | -ζ76-ζ7-1 | ζ75+ζ72+2 | -ζ75-ζ72-1 | -ζ75-ζ72-1 | ζ74+ζ73+2 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | orthogonal faithful |
| ρ9 | 4 | 0 | 0 | 0 | 2ζ76+2ζ7 | ζ74+ζ73+2 | ζ76+ζ7+2 | -ζ76-ζ7-1 | 2ζ75+2ζ72 | -ζ75-ζ72-1 | ζ75+ζ72+2 | 2ζ74+2ζ73 | -ζ74-ζ73-1 | -ζ75-ζ72-1 | -ζ74-ζ73-1 | -ζ76-ζ7-1 | orthogonal faithful |
| ρ10 | 4 | 0 | 0 | 0 | ζ76+ζ7+2 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | -ζ75-ζ72-1 | ζ75+ζ72+2 | -ζ74-ζ73-1 | 2ζ76+2ζ7 | ζ74+ζ73+2 | -ζ76-ζ7-1 | -ζ74-ζ73-1 | -ζ76-ζ7-1 | -ζ75-ζ72-1 | orthogonal faithful |
| ρ11 | 4 | 0 | 0 | 0 | -ζ76-ζ7-1 | -ζ76-ζ7-1 | -ζ75-ζ72-1 | 2ζ74+2ζ73 | -ζ75-ζ72-1 | 2ζ76+2ζ7 | -ζ74-ζ73-1 | -ζ74-ζ73-1 | 2ζ75+2ζ72 | ζ74+ζ73+2 | ζ76+ζ7+2 | ζ75+ζ72+2 | orthogonal faithful |
| ρ12 | 4 | 0 | 0 | 0 | ζ74+ζ73+2 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | -ζ76-ζ7-1 | ζ76+ζ7+2 | -ζ75-ζ72-1 | 2ζ74+2ζ73 | ζ75+ζ72+2 | -ζ74-ζ73-1 | -ζ75-ζ72-1 | -ζ74-ζ73-1 | -ζ76-ζ7-1 | orthogonal faithful |
| ρ13 | 4 | 0 | 0 | 0 | 2ζ75+2ζ72 | ζ76+ζ7+2 | ζ75+ζ72+2 | -ζ75-ζ72-1 | 2ζ74+2ζ73 | -ζ74-ζ73-1 | ζ74+ζ73+2 | 2ζ76+2ζ7 | -ζ76-ζ7-1 | -ζ74-ζ73-1 | -ζ76-ζ7-1 | -ζ75-ζ72-1 | orthogonal faithful |
| ρ14 | 4 | 0 | 0 | 0 | ζ75+ζ72+2 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | -ζ74-ζ73-1 | ζ74+ζ73+2 | -ζ76-ζ7-1 | 2ζ75+2ζ72 | ζ76+ζ7+2 | -ζ75-ζ72-1 | -ζ76-ζ7-1 | -ζ75-ζ72-1 | -ζ74-ζ73-1 | orthogonal faithful |
| ρ15 | 4 | 0 | 0 | 0 | -ζ75-ζ72-1 | -ζ75-ζ72-1 | -ζ74-ζ73-1 | 2ζ76+2ζ7 | -ζ74-ζ73-1 | 2ζ75+2ζ72 | -ζ76-ζ7-1 | -ζ76-ζ7-1 | 2ζ74+2ζ73 | ζ76+ζ7+2 | ζ75+ζ72+2 | ζ74+ζ73+2 | orthogonal faithful |
| ρ16 | 4 | 0 | 0 | 0 | -ζ75-ζ72-1 | -ζ75-ζ72-1 | -ζ74-ζ73-1 | ζ74+ζ73+2 | -ζ74-ζ73-1 | ζ76+ζ7+2 | -ζ76-ζ7-1 | -ζ76-ζ7-1 | ζ75+ζ72+2 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | orthogonal faithful |
►On 14 points - transitive group 14T12
Generators in S14
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)
(1 6 4 2 7 5 3)(8 14 13 12 11 10 9)
(1 8)(2 11 7 12)(3 14 6 9)(4 10 5 13)
G:=sub<Sym(14)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14), (1,6,4,2,7,5,3)(8,14,13,12,11,10,9), (1,8)(2,11,7,12)(3,14,6,9)(4,10,5,13)>;
G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14), (1,6,4,2,7,5,3)(8,14,13,12,11,10,9), (1,8)(2,11,7,12)(3,14,6,9)(4,10,5,13) );
G=PermutationGroup([[(1,2,3,4,5,6,7),(8,9,10,11,12,13,14)], [(1,6,4,2,7,5,3),(8,14,13,12,11,10,9)], [(1,8),(2,11,7,12),(3,14,6,9),(4,10,5,13)]])
G:=TransitiveGroup(14,12);
►On 28 points - transitive group 28T35
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 6 4 2 7 5 3)(8 13 11 9 14 12 10)(15 21 20 19 18 17 16)(22 28 27 26 25 24 23)
(1 27 12 15)(2 23 11 19)(3 26 10 16)(4 22 9 20)(5 25 8 17)(6 28 14 21)(7 24 13 18)
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), (1,6,4,2,7,5,3)(8,13,11,9,14,12,10)(15,21,20,19,18,17,16)(22,28,27,26,25,24,23), (1,27,12,15)(2,23,11,19)(3,26,10,16)(4,22,9,20)(5,25,8,17)(6,28,14,21)(7,24,13,18)>;
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), (1,6,4,2,7,5,3)(8,13,11,9,14,12,10)(15,21,20,19,18,17,16)(22,28,27,26,25,24,23), (1,27,12,15)(2,23,11,19)(3,26,10,16)(4,22,9,20)(5,25,8,17)(6,28,14,21)(7,24,13,18) );
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)], [(1,6,4,2,7,5,3),(8,13,11,9,14,12,10),(15,21,20,19,18,17,16),(22,28,27,26,25,24,23)], [(1,27,12,15),(2,23,11,19),(3,26,10,16),(4,22,9,20),(5,25,8,17),(6,28,14,21),(7,24,13,18)]])
G:=TransitiveGroup(28,35);
C72⋊C4 is a maximal subgroup of
C72⋊C8 D7≀C2 C72⋊Q8
C72⋊C4 is a maximal quotient of C72⋊2C8
| action | f(x) | Disc(f) |
|---|---|---|
| 14T12 | x14+28x12-189x11+756x10-4004x9+15953x8-48856x7+129262x6-251559x5+330764x4-272986x3-123305x2+739662x-577916 | 22·518·738·192·1732·7012·21112·171592·9818430376572 |
Matrix representation of C72⋊C4 ►in GL4(𝔽29) generated by
| 10 | 1 | 0 | 0 |
| 27 | 26 | 0 | 0 |
| 20 | 27 | 4 | 10 |
| 14 | 1 | 14 | 28 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 21 | 27 | 4 | 10 |
| 4 | 1 | 14 | 28 |
| 0 | 0 | 10 | 1 |
| 4 | 1 | 23 | 25 |
| 25 | 8 | 28 | 0 |
| 17 | 26 | 10 | 0 |
G:=sub<GL(4,GF(29))| [10,27,20,14,1,26,27,1,0,0,4,14,0,0,10,28],[1,0,21,4,0,1,27,1,0,0,4,14,0,0,10,28],[0,4,25,17,0,1,8,26,10,23,28,10,1,25,0,0] >;
C72⋊C4 in GAP, Magma, Sage, TeX
C_7^2\rtimes C_4
% in TeX
G:=Group("C7^2:C4"); // GroupNames label
G:=SmallGroup(196,8);
// by ID
G=gap.SmallGroup(196,8);
# by ID
G:=PCGroup([4,-2,-2,-7,7,8,530,150,1603,1351]);
// Polycyclic
G:=Group<a,b,c|a^7=b^7=c^4=1,a*b=b*a,c*a*c^-1=a^3*b^-1,c*b*c^-1=a^3*b^4>;
// generators/relations
Export
Subgroup lattice of C72⋊C4 in TeX
Character table of C72⋊C4 in TeX