direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D7×D8, C8⋊4D14, D56⋊4C2, D4⋊1D14, C56⋊2C22, D14.12D4, D28⋊1C22, C28.1C23, Dic7.3D4, C7⋊2(C2×D8), D4⋊D7⋊1C2, (D4×D7)⋊1C2, (C8×D7)⋊1C2, (C7×D8)⋊2C2, C7⋊C8⋊5C22, C2.15(D4×D7), C14.27(C2×D4), (C7×D4)⋊1C22, C4.1(C22×D7), (C4×D7).7C22, SmallGroup(224,105)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D7×D8
G = < a,b,c,d | a7=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 446 in 76 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, D4, D4, C23, D7, D7, C14, C14, C2×C8, D8, D8, C2×D4, Dic7, C28, D14, D14, C2×C14, C2×D8, C7⋊C8, C56, C4×D7, D28, C7⋊D4, C7×D4, C22×D7, C8×D7, D56, D4⋊D7, C7×D8, D4×D7, D7×D8
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, C2×D8, C22×D7, D4×D7, D7×D8
►On 56 points
(1 26 16 53 35 24 41)(2 27 9 54 36 17 42)(3 28 10 55 37 18 43)(4 29 11 56 38 19 44)(5 30 12 49 39 20 45)(6 31 13 50 40 21 46)(7 32 14 51 33 22 47)(8 25 15 52 34 23 48)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)
(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 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)
(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(17 23)(18 22)(19 21)(25 27)(28 32)(29 31)(33 37)(34 36)(38 40)(42 48)(43 47)(44 46)(50 56)(51 55)(52 54)
G:=sub<Sym(56)| (1,26,16,53,35,24,41)(2,27,9,54,36,17,42)(3,28,10,55,37,18,43)(4,29,11,56,38,19,44)(5,30,12,49,39,20,45)(6,31,13,50,40,21,46)(7,32,14,51,33,22,47)(8,25,15,52,34,23,48), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,25)(24,26), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56), (2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(17,23)(18,22)(19,21)(25,27)(28,32)(29,31)(33,37)(34,36)(38,40)(42,48)(43,47)(44,46)(50,56)(51,55)(52,54)>;
G:=Group( (1,26,16,53,35,24,41)(2,27,9,54,36,17,42)(3,28,10,55,37,18,43)(4,29,11,56,38,19,44)(5,30,12,49,39,20,45)(6,31,13,50,40,21,46)(7,32,14,51,33,22,47)(8,25,15,52,34,23,48), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,25)(24,26), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56), (2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(17,23)(18,22)(19,21)(25,27)(28,32)(29,31)(33,37)(34,36)(38,40)(42,48)(43,47)(44,46)(50,56)(51,55)(52,54) );
G=PermutationGroup([[(1,26,16,53,35,24,41),(2,27,9,54,36,17,42),(3,28,10,55,37,18,43),(4,29,11,56,38,19,44),(5,30,12,49,39,20,45),(6,31,13,50,40,21,46),(7,32,14,51,33,22,47),(8,25,15,52,34,23,48)], [(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,36),(10,37),(11,38),(12,39),(13,40),(14,33),(15,34),(16,35),(17,27),(18,28),(19,29),(20,30),(21,31),(22,32),(23,25),(24,26)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56)], [(2,8),(3,7),(4,6),(9,15),(10,14),(11,13),(17,23),(18,22),(19,21),(25,27),(28,32),(29,31),(33,37),(34,36),(38,40),(42,48),(43,47),(44,46),(50,56),(51,55),(52,54)]])
D7×D8 is a maximal subgroup of
D8⋊D14 D112⋊C2 D8⋊13D14 D8⋊15D14 D8⋊5D14
D7×D8 is a maximal quotient of
Dic7⋊4D8 Dic7.D8 Dic7.SD16 D4⋊D28 D14.D8 D14⋊D8 D28⋊3D4 Dic7⋊5D8 C56⋊2Q8 D14.5D8 C8⋊7D28 D28⋊2Q8 D8⋊D14 D16⋊3D7 D112⋊C2 SD32⋊D7 SD32⋊3D7 Q32⋊D7 Q32⋊3D7 Dic7⋊D8 C56⋊5D4 D28⋊D4 C56⋊6D4
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | 14B | 14C | 14D | ··· | 14I | 28A | 28B | 28C | 56A | ··· | 56F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | 28 | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 4 | 4 | 7 | 7 | 28 | 28 | 2 | 14 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 2 | 2 | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D8 | D14 | D14 | D4×D7 | D7×D8 |
| kernel | D7×D8 | C8×D7 | D56 | D4⋊D7 | C7×D8 | D4×D7 | Dic7 | D14 | D8 | D7 | C8 | D4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 3 | 4 | 3 | 6 | 3 | 6 |
Matrix representation of D7×D8 ►in GL4(𝔽113) generated by
| 0 | 1 | 0 | 0 |
| 112 | 24 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 51 | 9 |
| 0 | 0 | 25 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 32 | 112 |
G:=sub<GL(4,GF(113))| [0,112,0,0,1,24,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,112,0,0,0,0,51,25,0,0,9,0],[1,0,0,0,0,1,0,0,0,0,1,32,0,0,0,112] >;
D7×D8 in GAP, Magma, Sage, TeX
D_7\times D_8
% in TeX
G:=Group("D7xD8"); // GroupNames label
G:=SmallGroup(224,105);
// by ID
G=gap.SmallGroup(224,105);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,116,297,159,69,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^2=c^8=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