metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22.2D28, C23.1D14, (C2×Dic7)⋊C4, (C22×D7)⋊C4, C7⋊1(C23⋊C4), C22⋊C4⋊1D7, (C2×C14).27D4, C23.D7⋊1C2, C22.3(C4×D7), C2.4(D14⋊C4), C14.2(C22⋊C4), C22.8(C7⋊D4), (C22×C14).5C22, (C7×C22⋊C4)⋊1C2, (C2×C14).1(C2×C4), (C2×C7⋊D4).1C2, SmallGroup(224,12)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22.2D28
G = < a,b,c,d | a2=b2=c28=1, d2=a, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=ac-1 >
2C2
2C2
2C2
28C2
4C4
4C22
14C22
14C4
28C4
28C22
2C14
2C14
2C14
4D7
7C23
14C2×C4
14D4
14D4
2Dic7
2D14
4Dic7
4D14
4C28
Smallest permutation representation of C22.2D28
►On 56 points
Generators in S56
(1 45)(3 47)(5 49)(7 51)(9 53)(11 55)(13 29)(15 31)(17 33)(19 35)(21 37)(23 39)(25 41)(27 43)
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)
(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 51 45 7)(2 50)(3 5 47 49)(6 46)(8 28)(9 43 53 27)(10 42)(11 25 55 41)(12 24)(13 39 29 23)(14 38)(15 21 31 37)(16 20)(17 35 33 19)(18 34)(22 30)(26 54)(32 36)(40 56)(44 52)
G:=sub<Sym(56)| (1,45)(3,47)(5,49)(7,51)(9,53)(11,55)(13,29)(15,31)(17,33)(19,35)(21,37)(23,39)(25,41)(27,43), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44), (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), (1,51,45,7)(2,50)(3,5,47,49)(6,46)(8,28)(9,43,53,27)(10,42)(11,25,55,41)(12,24)(13,39,29,23)(14,38)(15,21,31,37)(16,20)(17,35,33,19)(18,34)(22,30)(26,54)(32,36)(40,56)(44,52)>;
G:=Group( (1,45)(3,47)(5,49)(7,51)(9,53)(11,55)(13,29)(15,31)(17,33)(19,35)(21,37)(23,39)(25,41)(27,43), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44), (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), (1,51,45,7)(2,50)(3,5,47,49)(6,46)(8,28)(9,43,53,27)(10,42)(11,25,55,41)(12,24)(13,39,29,23)(14,38)(15,21,31,37)(16,20)(17,35,33,19)(18,34)(22,30)(26,54)(32,36)(40,56)(44,52) );
G=PermutationGroup([(1,45),(3,47),(5,49),(7,51),(9,53),(11,55),(13,29),(15,31),(17,33),(19,35),(21,37),(23,39),(25,41),(27,43)], [(1,45),(2,46),(3,47),(4,48),(5,49),(6,50),(7,51),(8,52),(9,53),(10,54),(11,55),(12,56),(13,29),(14,30),(15,31),(16,32),(17,33),(18,34),(19,35),(20,36),(21,37),(22,38),(23,39),(24,40),(25,41),(26,42),(27,43),(28,44)], [(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)], [(1,51,45,7),(2,50),(3,5,47,49),(6,46),(8,28),(9,43,53,27),(10,42),(11,25,55,41),(12,24),(13,39,29,23),(14,38),(15,21,31,37),(16,20),(17,35,33,19),(18,34),(22,30),(26,54),(32,36),(40,56),(44,52)])
41 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 2 | 2 | 28 | 4 | 4 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
41 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D7 | D14 | C4×D7 | D28 | C7⋊D4 | C23⋊C4 | C22.2D28 |
| kernel | C22.2D28 | C23.D7 | C7×C22⋊C4 | C2×C7⋊D4 | C2×Dic7 | C22×D7 | C2×C14 | C22⋊C4 | C23 | C22 | C22 | C22 | C7 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 1 | 6 |
Matrix representation of C22.2D28 ►in GL4(𝔽29) generated by
| 28 | 0 | 3 | 3 |
| 0 | 28 | 12 | 12 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 12 | 14 | 1 | 22 |
| 19 | 27 | 5 | 12 |
| 19 | 1 | 9 | 9 |
| 18 | 18 | 10 | 10 |
| 13 | 15 | 10 | 22 |
| 8 | 16 | 22 | 12 |
| 0 | 0 | 19 | 10 |
| 0 | 0 | 22 | 10 |
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,3,12,1,0,3,12,0,1],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[12,19,19,18,14,27,1,18,1,5,9,10,22,12,9,10],[13,8,0,0,15,16,0,0,10,22,19,22,22,12,10,10] >;
C22.2D28 in GAP, Magma, Sage, TeX
C_2^2._2D_{28} % in TeX
G:=Group("C2^2.2D28"); // GroupNames label
G:=SmallGroup(224,12);
// by ID
G=gap.SmallGroup(224,12);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,121,31,362,297,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^28=1,d^2=a,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a*c^-1>;
// generators/relations