direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C22×D7, C7⋊C23, C14⋊C22, (C2×C14)⋊3C2, SmallGroup(56,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C22×D7 |
Generators and relations for C22×D7
G = < a,b,c,d | a2=b2=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Character table of C22×D7
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 7A | 7B | 7C | 14A | 14B | 14C | 14D | 14E | 14F | 14G | 14H | 14I | |
| size | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 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 | 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 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ6 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | orthogonal lifted from D7 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | orthogonal lifted from D7 |
| ρ11 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ76-ζ7 | ζ75+ζ72 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | ζ74+ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | ζ76+ζ7 | orthogonal lifted from D14 |
| ρ12 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ75-ζ72 | ζ74+ζ73 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | ζ76+ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | ζ75+ζ72 | orthogonal lifted from D14 |
| ρ13 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | -ζ76-ζ7 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ75-ζ72 | ζ76+ζ7 | ζ75+ζ72 | -ζ74-ζ73 | orthogonal lifted from D14 |
| ρ14 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | -ζ75-ζ72 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ74-ζ73 | ζ75+ζ72 | ζ74+ζ73 | -ζ76-ζ7 | orthogonal lifted from D14 |
| ρ15 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | orthogonal lifted from D14 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | -ζ74-ζ73 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ76-ζ7 | ζ74+ζ73 | ζ76+ζ7 | -ζ75-ζ72 | orthogonal lifted from D14 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | orthogonal lifted from D14 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | orthogonal lifted from D14 |
| ρ19 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | orthogonal lifted from D7 |
| ρ20 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ74-ζ73 | ζ76+ζ7 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | ζ75+ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | ζ74+ζ73 | orthogonal lifted from D14 |
►On 28 points - transitive group 28T9
Generators in S28
(1 27)(2 28)(3 22)(4 23)(5 24)(6 25)(7 26)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)
(1 13)(2 14)(3 8)(4 9)(5 10)(6 11)(7 12)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 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 19)(2 18)(3 17)(4 16)(5 15)(6 21)(7 20)(8 24)(9 23)(10 22)(11 28)(12 27)(13 26)(14 25)
G:=sub<Sym(28)| (1,27)(2,28)(3,22)(4,23)(5,24)(6,25)(7,26)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21), (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,21)(7,20)(8,24)(9,23)(10,22)(11,28)(12,27)(13,26)(14,25)>;
G:=Group( (1,27)(2,28)(3,22)(4,23)(5,24)(6,25)(7,26)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21), (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,21)(7,20)(8,24)(9,23)(10,22)(11,28)(12,27)(13,26)(14,25) );
G=PermutationGroup([[(1,27),(2,28),(3,22),(4,23),(5,24),(6,25),(7,26),(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21)], [(1,13),(2,14),(3,8),(4,9),(5,10),(6,11),(7,12),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28)], [(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)], [(1,19),(2,18),(3,17),(4,16),(5,15),(6,21),(7,20),(8,24),(9,23),(10,22),(11,28),(12,27),(13,26),(14,25)]])
G:=TransitiveGroup(28,9);
C22×D7 is a maximal subgroup of
D14⋊C4 D7⋊A4
C22×D7 is a maximal quotient of C4○D28 D4⋊2D7 Q8⋊2D7
Matrix representation of C22×D7 ►in GL3(𝔽29) generated by
| 28 | 0 | 0 |
| 0 | 28 | 0 |
| 0 | 0 | 28 |
| 28 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 28 | 3 |
| 1 | 0 | 0 |
| 0 | 0 | 28 |
| 0 | 28 | 0 |
G:=sub<GL(3,GF(29))| [28,0,0,0,28,0,0,0,28],[28,0,0,0,1,0,0,0,1],[1,0,0,0,0,28,0,1,3],[1,0,0,0,0,28,0,28,0] >;
C22×D7 in GAP, Magma, Sage, TeX
C_2^2\times D_7
% in TeX
G:=Group("C2^2xD7"); // GroupNames label
G:=SmallGroup(56,12);
// by ID
G=gap.SmallGroup(56,12);
# by ID
G:=PCGroup([4,-2,-2,-2,-7,771]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^7=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of C22×D7 in TeX
Character table of C22×D7 in TeX