metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊3D14, D56⋊6C2, Q8⋊2D14, C56⋊3C22, SD16⋊1D7, D4.3D14, D14.7D4, D28⋊2C22, C28.5C23, Dic7.9D4, D4⋊D7⋊3C2, (D4×D7)⋊3C2, C7⋊C8⋊2C22, Q8⋊D7⋊2C2, C8⋊D7⋊1C2, C7⋊3(C8⋊C22), C2.19(D4×D7), Q8⋊2D7⋊1C2, (C7×SD16)⋊1C2, C14.31(C2×D4), (C7×Q8)⋊2C22, C4.5(C22×D7), (C7×D4).3C22, (C4×D7).2C22, SmallGroup(224,109)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D56⋊C2
G = < a,b,c | a56=b2=c2=1, bab=a-1, cac=a43, bc=cb >
Subgroups: 374 in 68 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, D4, D4, Q8, C23, D7, C14, C14, M4(2), D8, SD16, SD16, C2×D4, C4○D4, Dic7, C28, C28, D14, D14, C2×C14, C8⋊C22, C7⋊C8, C56, C4×D7, C4×D7, D28, D28, C7⋊D4, C7×D4, C7×Q8, C22×D7, C8⋊D7, D56, D4⋊D7, Q8⋊D7, C7×SD16, D4×D7, Q8⋊2D7, D56⋊C2
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C8⋊C22, C22×D7, D4×D7, D56⋊C2
►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)
(1 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 56)(23 55)(24 54)(25 53)(26 52)(27 51)(28 50)(29 49)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)
(2 44)(3 31)(4 18)(6 48)(7 35)(8 22)(10 52)(11 39)(12 26)(14 56)(15 43)(16 30)(19 47)(20 34)(23 51)(24 38)(27 55)(28 42)(32 46)(36 50)(40 54)
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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,56)(23,55)(24,54)(25,53)(26,52)(27,51)(28,50)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40), (2,44)(3,31)(4,18)(6,48)(7,35)(8,22)(10,52)(11,39)(12,26)(14,56)(15,43)(16,30)(19,47)(20,34)(23,51)(24,38)(27,55)(28,42)(32,46)(36,50)(40,54)>;
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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,56)(23,55)(24,54)(25,53)(26,52)(27,51)(28,50)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40), (2,44)(3,31)(4,18)(6,48)(7,35)(8,22)(10,52)(11,39)(12,26)(14,56)(15,43)(16,30)(19,47)(20,34)(23,51)(24,38)(27,55)(28,42)(32,46)(36,50)(40,54) );
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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(22,56),(23,55),(24,54),(25,53),(26,52),(27,51),(28,50),(29,49),(30,48),(31,47),(32,46),(33,45),(34,44),(35,43),(36,42),(37,41),(38,40)], [(2,44),(3,31),(4,18),(6,48),(7,35),(8,22),(10,52),(11,39),(12,26),(14,56),(15,43),(16,30),(19,47),(20,34),(23,51),(24,38),(27,55),(28,42),(32,46),(36,50),(40,54)]])
D56⋊C2 is a maximal subgroup of
D28.29D4 D8⋊10D14 D8⋊15D14 D7×C8⋊C22 D8⋊5D14 D56⋊C22 C56.C23
D56⋊C2 is a maximal quotient of
C4⋊C4.D14 D4.2Dic14 (D4×D7)⋊C4 D4⋊D28 C7⋊C8⋊D4 C56⋊1C4⋊C2 D4⋊D7⋊C4 D28⋊3D4 Dic7.Q16 Q8⋊C4⋊D7 Q8⋊(C4×D7) D14.Q16 D28⋊4D4 C7⋊(C8⋊D4) Q8⋊D7⋊C4 D28.12D4 C56⋊3Q8 C8⋊(C4×D7) D14.4SD16 C56⋊7D4 C4.Q8⋊D7 D56⋊9C4 D28⋊Q8 D28.Q8 Dic7⋊5SD16 SD16⋊Dic7 (C7×D4).D4 D14⋊6SD16 D28⋊7D4 C56⋊8D4 C56⋊9D4
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 14D | 14E | 14F | 28A | 28B | 28C | 28D | 28E | 28F | 56A | ··· | 56F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | 28 | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 4 | 14 | 28 | 28 | 2 | 4 | 14 | 2 | 2 | 2 | 4 | 28 | 2 | 2 | 2 | 8 | 8 | 8 | 4 | 4 | 4 | 8 | 8 | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | D14 | C8⋊C22 | D4×D7 | D56⋊C2 |
| kernel | D56⋊C2 | C8⋊D7 | D56 | D4⋊D7 | Q8⋊D7 | C7×SD16 | D4×D7 | Q8⋊2D7 | Dic7 | D14 | SD16 | C8 | D4 | Q8 | C7 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 3 | 6 |
Matrix representation of D56⋊C2 ►in GL4(𝔽113) generated by
| 98 | 7 | 98 | 7 |
| 106 | 86 | 106 | 86 |
| 64 | 53 | 0 | 0 |
| 60 | 70 | 0 | 0 |
| 34 | 1 | 0 | 0 |
| 88 | 79 | 0 | 0 |
| 79 | 112 | 79 | 112 |
| 25 | 34 | 25 | 34 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 112 | 0 | 112 | 0 |
| 0 | 112 | 0 | 112 |
G:=sub<GL(4,GF(113))| [98,106,64,60,7,86,53,70,98,106,0,0,7,86,0,0],[34,88,79,25,1,79,112,34,0,0,79,25,0,0,112,34],[1,0,112,0,0,1,0,112,0,0,112,0,0,0,0,112] >;
D56⋊C2 in GAP, Magma, Sage, TeX
D_{56}\rtimes C_2 % in TeX
G:=Group("D56:C2"); // GroupNames label
G:=SmallGroup(224,109);
// by ID
G=gap.SmallGroup(224,109);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,362,116,86,297,159,69,6917]);
// Polycyclic
G:=Group<a,b,c|a^56=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^43,b*c=c*b>;
// generators/relations