metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊2D7, C56⋊2C2, C7⋊1SD16, C4.8D14, C2.3D28, C14.1D4, D28.1C2, Dic14⋊1C2, C28.8C22, SmallGroup(112,5)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊C2
G = < a,b | a56=b2=1, bab=a27 >
Smallest permutation representation of C56⋊C2
►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)
(2 28)(3 55)(4 26)(5 53)(6 24)(7 51)(8 22)(9 49)(10 20)(11 47)(12 18)(13 45)(14 16)(15 43)(17 41)(19 39)(21 37)(23 35)(25 33)(27 31)(30 56)(32 54)(34 52)(36 50)(38 48)(40 46)(42 44)
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), (2,28)(3,55)(4,26)(5,53)(6,24)(7,51)(8,22)(9,49)(10,20)(11,47)(12,18)(13,45)(14,16)(15,43)(17,41)(19,39)(21,37)(23,35)(25,33)(27,31)(30,56)(32,54)(34,52)(36,50)(38,48)(40,46)(42,44)>;
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), (2,28)(3,55)(4,26)(5,53)(6,24)(7,51)(8,22)(9,49)(10,20)(11,47)(12,18)(13,45)(14,16)(15,43)(17,41)(19,39)(21,37)(23,35)(25,33)(27,31)(30,56)(32,54)(34,52)(36,50)(38,48)(40,46)(42,44) );
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)], [(2,28),(3,55),(4,26),(5,53),(6,24),(7,51),(8,22),(9,49),(10,20),(11,47),(12,18),(13,45),(14,16),(15,43),(17,41),(19,39),(21,37),(23,35),(25,33),(27,31),(30,56),(32,54),(34,52),(36,50),(38,48),(40,46),(42,44)]])
C56⋊C2 is a maximal subgroup of
D56⋊7C2 C8⋊D14 C8.D14 D8⋊D7 D7×SD16 SD16⋊3D7 Q16⋊D7 C56⋊C6 C6.D28 C21⋊SD16 C8⋊D21
C56⋊C2 is a maximal quotient of
C28.44D4 C8⋊Dic7 C2.D56 C6.D28 C21⋊SD16 C8⋊D21
31 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 28A | ··· | 28F | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 28 | 2 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
31 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | D4 | D7 | SD16 | D14 | D28 | C56⋊C2 |
| kernel | C56⋊C2 | C56 | Dic14 | D28 | C14 | C8 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 3 | 6 | 12 |
Matrix representation of C56⋊C2 ►in GL4(𝔽113) generated by
| 13 | 100 | 0 | 0 |
| 13 | 13 | 0 | 0 |
| 0 | 0 | 1 | 112 |
| 0 | 0 | 11 | 103 |
| 1 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 103 | 1 |
| 0 | 0 | 14 | 10 |
G:=sub<GL(4,GF(113))| [13,13,0,0,100,13,0,0,0,0,1,11,0,0,112,103],[1,0,0,0,0,112,0,0,0,0,103,14,0,0,1,10] >;
C56⋊C2 in GAP, Magma, Sage, TeX
C_{56}\rtimes C_2 % in TeX
G:=Group("C56:C2"); // GroupNames label
G:=SmallGroup(112,5);
// by ID
G=gap.SmallGroup(112,5);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-7,61,26,182,42,2404]);
// Polycyclic
G:=Group<a,b|a^56=b^2=1,b*a*b=a^27>;
// generators/relations
Export