direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4×D7, C4⋊1D14, C28⋊C22, D28⋊3C2, C22⋊1D14, D14⋊2C22, C14.5C23, Dic7⋊1C22, C7⋊2(C2×D4), (C4×D7)⋊1C2, (C7×D4)⋊2C2, C7⋊D4⋊1C2, (C2×C14)⋊C22, (C22×D7)⋊2C2, C2.6(C22×D7), SmallGroup(112,31)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×D7
G = < a,b,c,d | a4=b2=c7=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 220 in 54 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, D4, D4, C23, D7, D7, C14, C14, C2×D4, Dic7, C28, D14, D14, D14, C2×C14, C4×D7, D28, C7⋊D4, C7×D4, C22×D7, D4×D7
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22×D7, D4×D7
Character table of D4×D7
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 7A | 7B | 7C | 14A | 14B | 14C | 14D | 14E | 14F | 14G | 14H | 14I | 28A | 28B | 28C | |
| size | 1 | 1 | 2 | 2 | 7 | 7 | 14 | 14 | 2 | 14 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 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 | 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 | -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 | -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 | -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 | -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 | 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 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ9 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ76-ζ7 | -ζ74-ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | orthogonal lifted from D14 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | orthogonal lifted from D7 |
| ρ13 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | ζ75+ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | orthogonal lifted from D14 |
| ρ14 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ75-ζ72 | -ζ76-ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | orthogonal lifted from D14 |
| ρ15 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ76-ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | orthogonal lifted from D14 |
| ρ16 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | ζ74+ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | orthogonal lifted from D14 |
| ρ17 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | ζ76+ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | orthogonal lifted from D14 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | orthogonal lifted from D7 |
| ρ19 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ74-ζ73 | -ζ75-ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | orthogonal lifted from D14 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ75-ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | orthogonal lifted from D14 |
| ρ21 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | orthogonal lifted from D7 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ74-ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | orthogonal lifted from D14 |
| ρ23 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | -2ζ76-2ζ7 | -2ζ75-2ζ72 | -2ζ74-2ζ73 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ24 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | -2ζ75-2ζ72 | -2ζ74-2ζ73 | -2ζ76-2ζ7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | -2ζ74-2ζ73 | -2ζ76-2ζ7 | -2ζ75-2ζ72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 28 points - transitive group 28T18
(1 20 13 27)(2 21 14 28)(3 15 8 22)(4 16 9 23)(5 17 10 24)(6 18 11 25)(7 19 12 26)
(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 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 17)(18 21)(19 20)(22 24)(25 28)(26 27)
G:=sub<Sym(28)| (1,20,13,27)(2,21,14,28)(3,15,8,22)(4,16,9,23)(5,17,10,24)(6,18,11,25)(7,19,12,26), (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), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,17)(18,21)(19,20)(22,24)(25,28)(26,27)>;
G:=Group( (1,20,13,27)(2,21,14,28)(3,15,8,22)(4,16,9,23)(5,17,10,24)(6,18,11,25)(7,19,12,26), (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), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,17)(18,21)(19,20)(22,24)(25,28)(26,27) );
G=PermutationGroup([[(1,20,13,27),(2,21,14,28),(3,15,8,22),(4,16,9,23),(5,17,10,24),(6,18,11,25),(7,19,12,26)], [(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)], [(1,7),(2,6),(3,5),(8,10),(11,14),(12,13),(15,17),(18,21),(19,20),(22,24),(25,28),(26,27)]])
G:=TransitiveGroup(28,18);
D4×D7 is a maximal subgroup of
D8⋊D7 D56⋊C2 D4⋊6D14 D4⋊8D14 C28⋊D6 D6⋊D14
D4×D7 is a maximal quotient of
C22⋊Dic14 Dic7⋊4D4 C22⋊D28 D14.D4 D14⋊D4 Dic7.D4 C28⋊Q8 D28⋊C4 D14.5D4 C4⋊D28 D14⋊Q8 D8⋊D7 D8⋊3D7 D56⋊C2 SD16⋊D7 SD16⋊3D7 Q16⋊D7 Q8.D14 C23⋊D14 C28⋊2D4 Dic7⋊D4 C28⋊D4 C28⋊D6 D6⋊D14
Matrix representation of D4×D7 ►in GL4(𝔽29) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 17 |
| 0 | 0 | 5 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 24 | 28 |
| 22 | 1 | 0 | 0 |
| 27 | 25 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 25 | 28 | 0 | 0 |
| 15 | 4 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,28,5,0,0,17,1],[1,0,0,0,0,1,0,0,0,0,1,24,0,0,0,28],[22,27,0,0,1,25,0,0,0,0,1,0,0,0,0,1],[25,15,0,0,28,4,0,0,0,0,28,0,0,0,0,28] >;
D4×D7 in GAP, Magma, Sage, TeX
D_4\times D_7
% in TeX
G:=Group("D4xD7"); // GroupNames label
G:=SmallGroup(112,31);
// by ID
G=gap.SmallGroup(112,31);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-7,97,2404]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^7=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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