metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D56, C7⋊1D8, C8⋊1D7, C56⋊1C2, D28⋊1C2, C2.4D28, C4.9D14, C14.2D4, C28.9C22, sometimes denoted D112 or Dih56 or Dih112, SmallGroup(112,6)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D56
G = < a,b | a56=b2=1, bab=a-1 >
Smallest permutation representation of D56
►On 56 points
Generators in S56
(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 56)(2 55)(3 54)(4 53)(5 52)(6 51)(7 50)(8 49)(9 48)(10 47)(11 46)(12 45)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)
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), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)>;
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), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29) );
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)], [(1,56),(2,55),(3,54),(4,53),(5,52),(6,51),(7,50),(8,49),(9,48),(10,47),(11,46),(12,45),(13,44),(14,43),(15,42),(16,41),(17,40),(18,39),(19,38),(20,37),(21,36),(22,35),(23,34),(24,33),(25,32),(26,31),(27,30),(28,29)]])
D56 is a maximal subgroup of
D112 C112⋊C2 C7⋊D16 C7⋊SD32 D56⋊7C2 C8⋊D14 D7×D8 D56⋊C2 Q8.D14 D56⋊C3 C3⋊D56 D168
D56 is a maximal quotient of
D112 C112⋊C2 Dic56 C56⋊1C4 C2.D56 C3⋊D56 D168
31 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 28A | ··· | 28F | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
31 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D4 | D7 | D8 | D14 | D28 | D56 |
| kernel | D56 | C56 | D28 | C14 | C8 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 3 | 2 | 3 | 6 | 12 |
Matrix representation of D56 ►in GL2(𝔽113) generated by
| 49 | 44 |
| 25 | 34 |
| 21 | 106 |
| 79 | 92 |
G:=sub<GL(2,GF(113))| [49,25,44,34],[21,79,106,92] >;
D56 in GAP, Magma, Sage, TeX
D_{56} % in TeX
G:=Group("D56"); // GroupNames label
G:=SmallGroup(112,6);
// by ID
G=gap.SmallGroup(112,6);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-7,61,66,182,42,2404]);
// Polycyclic
G:=Group<a,b|a^56=b^2=1,b*a*b=a^-1>;
// generators/relations
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