metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8:2D7, C8:2D14, D4:2D14, C56:4C22, D14.6D4, C28.2C23, Dic7.8D4, D28.1C22, Dic14:1C22, D4:D7:2C2, (C7xD8):4C2, (D4xD7):2C2, C7:C8:1C22, C8:D7:3C2, C56:C2:3C2, C7:2(C8:C22), D4.D7:1C2, C2.16(D4xD7), D4:2D7:1C2, C14.28(C2xD4), (C7xD4):2C22, C4.2(C22xD7), (C4xD7).1C22, SmallGroup(224,106)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8:D7
G = < a,b,c,d | a8=b2=c7=d2=1, bab=a-1, ac=ca, dad=a5, bc=cb, bd=db, dcd=c-1 >
Subgroups: 350 in 68 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2xC4, D4, D4, Q8, C23, D7, C14, C14, M4(2), D8, D8, SD16, C2xD4, C4oD4, Dic7, Dic7, C28, D14, D14, C2xC14, C8:C22, C7:C8, C56, Dic14, C4xD7, D28, C2xDic7, C7:D4, C7xD4, C22xD7, C8:D7, C56:C2, D4:D7, D4.D7, C7xD8, D4xD7, D4:2D7, D8:D7
Quotients: C1, C2, C22, D4, C23, D7, C2xD4, D14, C8:C22, C22xD7, D4xD7, D8:D7
►On 56 points
(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)
(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 46)(3 43)(4 48)(5 45)(6 42)(7 47)(8 44)(9 40)(10 37)(11 34)(12 39)(13 36)(14 33)(15 38)(16 35)(17 31)(18 28)(19 25)(20 30)(21 27)(22 32)(23 29)(24 26)(50 54)(52 56)
G:=sub<Sym(56)| (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), (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)(2,46)(3,43)(4,48)(5,45)(6,42)(7,47)(8,44)(9,40)(10,37)(11,34)(12,39)(13,36)(14,33)(15,38)(16,35)(17,31)(18,28)(19,25)(20,30)(21,27)(22,32)(23,29)(24,26)(50,54)(52,56)>;
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)(25,26,27,28,29,30,31,32)(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), (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)(2,46)(3,43)(4,48)(5,45)(6,42)(7,47)(8,44)(9,40)(10,37)(11,34)(12,39)(13,36)(14,33)(15,38)(16,35)(17,31)(18,28)(19,25)(20,30)(21,27)(22,32)(23,29)(24,26)(50,54)(52,56) );
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),(25,26,27,28,29,30,31,32),(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)], [(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),(2,46),(3,43),(4,48),(5,45),(6,42),(7,47),(8,44),(9,40),(10,37),(11,34),(12,39),(13,36),(14,33),(15,38),(16,35),(17,31),(18,28),(19,25),(20,30),(21,27),(22,32),(23,29),(24,26),(50,54),(52,56)]])
D8:D7 is a maximal subgroup of
D8:13D14 D8:10D14 D8:11D14 D7xC8:C22 SD16:D14 D8:5D14 D8:6D14
D8:D7 is a maximal quotient of
D4.D7:C4 D4:Dic14 Dic14:2D4 D4.Dic14 C4:C4.D14 C28:Q8:C2 (D4xD7):C4 D4:(C4xD7) D4.6D28 D14.SD16 C8:Dic7:C2 C7:C8:1D4 D4:3D28 C7:C8:D4 D4:D7:C4 D28.D4 Dic14:2Q8 C56:4Q8 C56:(C2xC4) D14.2Q16 C2.D8:D7 C8:3D28 C56:C2:C4 D28.2Q8 Dic7:D8 D8:Dic7 (C2xD8).D7 C56:11D4 D28:D4 Dic14:D4 C56:12D4
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 14D | ··· | 14I | 28A | 28B | 28C | 56A | ··· | 56F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | 28 | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 4 | 4 | 14 | 28 | 2 | 14 | 28 | 2 | 2 | 2 | 4 | 28 | 2 | 2 | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | C8:C22 | D4xD7 | D8:D7 |
| kernel | D8:D7 | C8:D7 | C56:C2 | D4:D7 | D4.D7 | C7xD8 | D4xD7 | D4:2D7 | Dic7 | D14 | D8 | C8 | D4 | C7 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 6 | 1 | 3 | 6 |
Matrix representation of D8:D7 ►in GL4(F113) generated by
| 0 | 0 | 43 | 49 |
| 0 | 0 | 71 | 5 |
| 17 | 82 | 103 | 95 |
| 75 | 101 | 18 | 10 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 51 | 39 | 112 | 0 |
| 74 | 0 | 0 | 112 |
| 34 | 1 | 0 | 0 |
| 111 | 103 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 112 | 24 |
| 103 | 112 | 0 | 0 |
| 99 | 10 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(113))| [0,0,17,75,0,0,82,101,43,71,103,18,49,5,95,10],[1,0,51,74,0,1,39,0,0,0,112,0,0,0,0,112],[34,111,0,0,1,103,0,0,0,0,0,112,0,0,1,24],[103,99,0,0,112,10,0,0,0,0,0,1,0,0,1,0] >;
D8:D7 in GAP, Magma, Sage, TeX
D_8\rtimes D_7
% in TeX
G:=Group("D8:D7"); // GroupNames label
G:=SmallGroup(224,106);
// by ID
G=gap.SmallGroup(224,106);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,362,116,297,159,69,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^7=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^5,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations