direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×D8, D4⋊C14, C8⋊1C14, C56⋊5C2, C14.14D4, C28.17C22, (C7×D4)⋊4C2, C2.3(C7×D4), C4.1(C2×C14), SmallGroup(112,24)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×D8
G = < a,b,c | a7=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C7×D8
►On 56 points
Generators in S56
(1 31 34 55 18 47 14)(2 32 35 56 19 48 15)(3 25 36 49 20 41 16)(4 26 37 50 21 42 9)(5 27 38 51 22 43 10)(6 28 39 52 23 44 11)(7 29 40 53 24 45 12)(8 30 33 54 17 46 13)
(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)
(1 8)(2 7)(3 6)(4 5)(9 10)(11 16)(12 15)(13 14)(17 18)(19 24)(20 23)(21 22)(25 28)(26 27)(29 32)(30 31)(33 34)(35 40)(36 39)(37 38)(41 44)(42 43)(45 48)(46 47)(49 52)(50 51)(53 56)(54 55)
G:=sub<Sym(56)| (1,31,34,55,18,47,14)(2,32,35,56,19,48,15)(3,25,36,49,20,41,16)(4,26,37,50,21,42,9)(5,27,38,51,22,43,10)(6,28,39,52,23,44,11)(7,29,40,53,24,45,12)(8,30,33,54,17,46,13), (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), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,18)(19,24)(20,23)(21,22)(25,28)(26,27)(29,32)(30,31)(33,34)(35,40)(36,39)(37,38)(41,44)(42,43)(45,48)(46,47)(49,52)(50,51)(53,56)(54,55)>;
G:=Group( (1,31,34,55,18,47,14)(2,32,35,56,19,48,15)(3,25,36,49,20,41,16)(4,26,37,50,21,42,9)(5,27,38,51,22,43,10)(6,28,39,52,23,44,11)(7,29,40,53,24,45,12)(8,30,33,54,17,46,13), (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), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,18)(19,24)(20,23)(21,22)(25,28)(26,27)(29,32)(30,31)(33,34)(35,40)(36,39)(37,38)(41,44)(42,43)(45,48)(46,47)(49,52)(50,51)(53,56)(54,55) );
G=PermutationGroup([[(1,31,34,55,18,47,14),(2,32,35,56,19,48,15),(3,25,36,49,20,41,16),(4,26,37,50,21,42,9),(5,27,38,51,22,43,10),(6,28,39,52,23,44,11),(7,29,40,53,24,45,12),(8,30,33,54,17,46,13)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,10),(11,16),(12,15),(13,14),(17,18),(19,24),(20,23),(21,22),(25,28),(26,27),(29,32),(30,31),(33,34),(35,40),(36,39),(37,38),(41,44),(42,43),(45,48),(46,47),(49,52),(50,51),(53,56),(54,55)]])
C7×D8 is a maximal subgroup of
C7⋊D16 D8.D7 D8⋊D7 D8⋊3D7
49 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 7A | ··· | 7F | 8A | 8B | 14A | ··· | 14F | 14G | ··· | 14R | 28A | ··· | 28F | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 4 | 7 | ··· | 7 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 4 | 4 | 2 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 2 | ··· | 2 |
49 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C7 | C14 | C14 | D4 | D8 | C7×D4 | C7×D8 |
| kernel | C7×D8 | C56 | C7×D4 | D8 | C8 | D4 | C14 | C7 | C2 | C1 |
| # reps | 1 | 1 | 2 | 6 | 6 | 12 | 1 | 2 | 6 | 12 |
Matrix representation of C7×D8 ►in GL2(𝔽113) generated by
| 106 | 0 |
| 0 | 106 |
| 31 | 82 |
| 31 | 31 |
| 31 | 82 |
| 82 | 82 |
G:=sub<GL(2,GF(113))| [106,0,0,106],[31,31,82,31],[31,82,82,82] >;
C7×D8 in GAP, Magma, Sage, TeX
C_7\times D_8
% in TeX
G:=Group("C7xD8"); // GroupNames label
G:=SmallGroup(112,24);
// by ID
G=gap.SmallGroup(112,24);
# by ID
G:=PCGroup([5,-2,-2,-7,-2,-2,301,1683,848,58]);
// Polycyclic
G:=Group<a,b,c|a^7=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export