direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D7, C28⋊2C2, D14.C2, C4○Dic7, C2.1D14, Dic7⋊2C2, C14.2C22, C7⋊1(C2×C4), SmallGroup(56,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C4×D7 |
Generators and relations for C4×D7
G = < a,b,c | a4=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Character table of C4×D7
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 14A | 14B | 14C | 28A | 28B | 28C | 28D | 28E | 28F | |
| size | 1 | 1 | 7 | 7 | 1 | 1 | 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 | i | -i | -i | i | 1 | 1 | 1 | -1 | -1 | -1 | -i | -i | i | i | i | -i | linear of order 4 |
| ρ6 | 1 | -1 | -1 | 1 | -i | i | i | -i | 1 | 1 | 1 | -1 | -1 | -1 | i | i | -i | -i | -i | i | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | -i | i | -i | i | 1 | 1 | 1 | -1 | -1 | -1 | i | i | -i | -i | -i | i | linear of order 4 |
| ρ8 | 1 | -1 | 1 | -1 | i | -i | i | -i | 1 | 1 | 1 | -1 | -1 | -1 | -i | -i | i | i | i | -i | linear of order 4 |
| ρ9 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | orthogonal lifted from D7 |
| ρ10 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | orthogonal lifted from D7 |
| ρ11 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ75-ζ72 | orthogonal lifted from D14 |
| ρ12 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ76-ζ7 | orthogonal lifted from D14 |
| ρ13 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | orthogonal lifted from D7 |
| ρ14 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ74-ζ73 | orthogonal lifted from D14 |
| ρ15 | 2 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | ζ4ζ76+ζ4ζ7 | ζ4ζ75+ζ4ζ72 | ζ43ζ76+ζ43ζ7 | ζ43ζ75+ζ43ζ72 | ζ43ζ74+ζ43ζ73 | ζ4ζ74+ζ4ζ73 | complex faithful |
| ρ16 | 2 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | ζ43ζ75+ζ43ζ72 | ζ43ζ74+ζ43ζ73 | ζ4ζ75+ζ4ζ72 | ζ4ζ74+ζ4ζ73 | ζ4ζ76+ζ4ζ7 | ζ43ζ76+ζ43ζ7 | complex faithful |
| ρ17 | 2 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | ζ43ζ76+ζ43ζ7 | ζ43ζ75+ζ43ζ72 | ζ4ζ76+ζ4ζ7 | ζ4ζ75+ζ4ζ72 | ζ4ζ74+ζ4ζ73 | ζ43ζ74+ζ43ζ73 | complex faithful |
| ρ18 | 2 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | ζ4ζ74+ζ4ζ73 | ζ4ζ76+ζ4ζ7 | ζ43ζ74+ζ43ζ73 | ζ43ζ76+ζ43ζ7 | ζ43ζ75+ζ43ζ72 | ζ4ζ75+ζ4ζ72 | complex faithful |
| ρ19 | 2 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | ζ43ζ74+ζ43ζ73 | ζ43ζ76+ζ43ζ7 | ζ4ζ74+ζ4ζ73 | ζ4ζ76+ζ4ζ7 | ζ4ζ75+ζ4ζ72 | ζ43ζ75+ζ43ζ72 | complex faithful |
| ρ20 | 2 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | ζ4ζ75+ζ4ζ72 | ζ4ζ74+ζ4ζ73 | ζ43ζ75+ζ43ζ72 | ζ43ζ74+ζ43ζ73 | ζ43ζ76+ζ43ζ7 | ζ4ζ76+ζ4ζ7 | complex faithful |
►On 28 points - transitive group 28T8
Generators in S28
(1 27 13 20)(2 28 14 21)(3 22 8 15)(4 23 9 16)(5 24 10 17)(6 25 11 18)(7 26 12 19)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 14)(7 13)(15 24)(16 23)(17 22)(18 28)(19 27)(20 26)(21 25)
G:=sub<Sym(28)| (1,27,13,20)(2,28,14,21)(3,22,8,15)(4,23,9,16)(5,24,10,17)(6,25,11,18)(7,26,12,19), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,14)(7,13)(15,24)(16,23)(17,22)(18,28)(19,27)(20,26)(21,25)>;
G:=Group( (1,27,13,20)(2,28,14,21)(3,22,8,15)(4,23,9,16)(5,24,10,17)(6,25,11,18)(7,26,12,19), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,14)(7,13)(15,24)(16,23)(17,22)(18,28)(19,27)(20,26)(21,25) );
G=PermutationGroup([[(1,27,13,20),(2,28,14,21),(3,22,8,15),(4,23,9,16),(5,24,10,17),(6,25,11,18),(7,26,12,19)], [(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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,14),(7,13),(15,24),(16,23),(17,22),(18,28),(19,27),(20,26),(21,25)]])
G:=TransitiveGroup(28,8);
C4×D7 is a maximal subgroup of
C8⋊D7 C4○D28 D4⋊2D7 Q8⋊2D7 D21⋊C4 D70.C2 Dic7⋊2D7
C4×D7 is a maximal quotient of C8⋊D7 Dic7⋊C4 D14⋊C4 D21⋊C4 D70.C2 Dic7⋊2D7
Matrix representation of C4×D7 ►in GL2(𝔽13) generated by
| 8 | 0 |
| 0 | 8 |
| 9 | 4 |
| 4 | 12 |
| 12 | 0 |
| 9 | 1 |
G:=sub<GL(2,GF(13))| [8,0,0,8],[9,4,4,12],[12,9,0,1] >;
C4×D7 in GAP, Magma, Sage, TeX
C_4\times D_7
% in TeX
G:=Group("C4xD7"); // GroupNames label
G:=SmallGroup(56,4);
// by ID
G=gap.SmallGroup(56,4);
# by ID
G:=PCGroup([4,-2,-2,-2,-7,21,771]);
// Polycyclic
G:=Group<a,b,c|a^4=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C4×D7 in TeX
Character table of C4×D7 in TeX