metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D14⋊4D4, C22⋊2D28, C23.15D14, C7⋊1C22≀C2, (C2×C4)⋊1D14, (C2×C14)⋊1D4, C2.7(D4×D7), D14⋊C4⋊4C2, (C2×D28)⋊2C2, C22⋊C4⋊2D7, C14.5(C2×D4), C2.7(C2×D28), (C2×C28)⋊1C22, (C23×D7)⋊1C2, (C2×C14).23C23, (C2×Dic7)⋊1C22, (C22×D7)⋊1C22, C22.41(C22×D7), (C22×C14).12C22, (C2×C7⋊D4)⋊1C2, (C7×C22⋊C4)⋊3C2, SmallGroup(224,77)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊D28
G = < a,b,c,d | a2=b2=c28=d2=1, cac-1=dad=ab=ba, bc=cb, bd=db, dcd=c-1 >
Subgroups: 710 in 130 conjugacy classes, 37 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C14, C22⋊C4, C22⋊C4, C2×D4, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C22≀C2, D28, C2×Dic7, C7⋊D4, C2×C28, C22×D7, C22×D7, C22×D7, C22×C14, D14⋊C4, C7×C22⋊C4, C2×D28, C2×C7⋊D4, C23×D7, C22⋊D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, D28, C22×D7, C2×D28, D4×D7, C22⋊D28
►On 56 points
(2 49)(4 51)(6 53)(8 55)(10 29)(12 31)(14 33)(16 35)(18 37)(20 39)(22 41)(24 43)(26 45)(28 47)
(1 48)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(20 39)(21 40)(22 41)(23 42)(24 43)(25 44)(26 45)(27 46)(28 47)
(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 47)(2 46)(3 45)(4 44)(5 43)(6 42)(7 41)(8 40)(9 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(16 32)(17 31)(18 30)(19 29)(20 56)(21 55)(22 54)(23 53)(24 52)(25 51)(26 50)(27 49)(28 48)
G:=sub<Sym(56)| (2,49)(4,51)(6,53)(8,55)(10,29)(12,31)(14,33)(16,35)(18,37)(20,39)(22,41)(24,43)(26,45)(28,47), (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,40)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47), (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), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,56)(21,55)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(28,48)>;
G:=Group( (2,49)(4,51)(6,53)(8,55)(10,29)(12,31)(14,33)(16,35)(18,37)(20,39)(22,41)(24,43)(26,45)(28,47), (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,40)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47), (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), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,56)(21,55)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(28,48) );
G=PermutationGroup([[(2,49),(4,51),(6,53),(8,55),(10,29),(12,31),(14,33),(16,35),(18,37),(20,39),(22,41),(24,43),(26,45),(28,47)], [(1,48),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,55),(9,56),(10,29),(11,30),(12,31),(13,32),(14,33),(15,34),(16,35),(17,36),(18,37),(19,38),(20,39),(21,40),(22,41),(23,42),(24,43),(25,44),(26,45),(27,46),(28,47)], [(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)], [(1,47),(2,46),(3,45),(4,44),(5,43),(6,42),(7,41),(8,40),(9,39),(10,38),(11,37),(12,36),(13,35),(14,34),(15,33),(16,32),(17,31),(18,30),(19,29),(20,56),(21,55),(22,54),(23,53),(24,52),(25,51),(26,50),(27,49),(28,48)]])
C22⋊D28 is a maximal subgroup of
C23⋊D28 C24.27D14 C23⋊3D28 C42⋊8D14 C42⋊9D14 C42⋊10D14 C42⋊12D14 D4×D28 D4⋊5D28 C42⋊17D14 D7×C22≀C2 C24⋊3D14 C24.34D14 C14.372+ 1+4 C14.382+ 1+4 D28⋊19D4 C14.482+ 1+4 C4⋊C4⋊26D14 D28⋊21D4 C14.532+ 1+4 C14.562+ 1+4 C14.1202+ 1+4 C14.1212+ 1+4 C4⋊C4⋊28D14 C14.612+ 1+4 C14.682+ 1+4 C42⋊18D14 D28⋊10D4 C42⋊20D14 C42⋊22D14 C42⋊23D14 C42⋊24D14 C42⋊25D14
C22⋊D28 is a maximal quotient of
(C2×Dic7)⋊Q8 (C2×C4)⋊9D28 D14⋊C4⋊C4 (C2×C28)⋊5D4 (C2×Dic7)⋊3D4 (C2×C4).20D28 D28.31D4 D28⋊13D4 D28.32D4 D28⋊14D4 Dic14⋊14D4 C22⋊Dic28 C23⋊D28 C23.5D28 D28.1D4 D28⋊1D4 D28.4D4 D28.5D4 D4⋊D28 D4.6D28 D4⋊3D28 D4.D28 Q8⋊2D28 D14⋊4Q16 Q8.D28 D28⋊4D4 D4⋊4D28 M4(2)⋊D14 D4.9D28 D4.10D28 C24.47D14 C23.44D28 C23.45D28 C23⋊2D28 C23.16D28
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 14 | 14 | 14 | 14 | 28 | 4 | 4 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | D28 | D4×D7 |
| kernel | C22⋊D28 | D14⋊C4 | C7×C22⋊C4 | C2×D28 | C2×C7⋊D4 | C23×D7 | D14 | C2×C14 | C22⋊C4 | C2×C4 | C23 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 2 | 3 | 6 | 3 | 12 | 6 |
Matrix representation of C22⋊D28 ►in GL4(𝔽29) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 22 | 24 | 0 | 0 |
| 25 | 26 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 24 | 17 | 0 | 0 |
| 2 | 5 | 0 | 0 |
| 0 | 0 | 0 | 28 |
| 0 | 0 | 28 | 0 |
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,28],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[22,25,0,0,24,26,0,0,0,0,0,1,0,0,1,0],[24,2,0,0,17,5,0,0,0,0,0,28,0,0,28,0] >;
C22⋊D28 in GAP, Magma, Sage, TeX
C_2^2\rtimes D_{28} % in TeX
G:=Group("C2^2:D28"); // GroupNames label
G:=SmallGroup(224,77);
// by ID
G=gap.SmallGroup(224,77);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,218,188,50,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^28=d^2=1,c*a*c^-1=d*a*d=a*b=b*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations