direct product, abelian, monomial, 2-elementary
Aliases: C2×C28, SmallGroup(56,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C28 |
| C1 — C2×C28 |
| C1 — C2×C28 |
Generators and relations for C2×C28
G = < a,b | a2=b28=1, ab=ba >
Smallest permutation representation of C2×C28
►Regular action on 56 points
Generators in S56
(1 47)(2 48)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 43)(26 44)(27 45)(28 46)
(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)
G:=sub<Sym(56)| (1,47)(2,48)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46), (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)>;
G:=Group( (1,47)(2,48)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46), (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) );
G=PermutationGroup([[(1,47),(2,48),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,55),(10,56),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(25,43),(26,44),(27,45),(28,46)], [(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)]])
C2×C28 is a maximal subgroup of
C4.Dic7 Dic7⋊C4 C4⋊Dic7 D14⋊C4 C4○D28
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 7A | ··· | 7F | 14A | ··· | 14R | 28A | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C7 | C14 | C14 | C28 |
| kernel | C2×C28 | C28 | C2×C14 | C14 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 6 | 12 | 6 | 24 |
Matrix representation of C2×C28 ►in GL2(𝔽29) generated by
| 28 | 0 |
| 0 | 28 |
| 24 | 0 |
| 0 | 2 |
G:=sub<GL(2,GF(29))| [28,0,0,28],[24,0,0,2] >;
C2×C28 in GAP, Magma, Sage, TeX
C_2\times C_{28} % in TeX
G:=Group("C2xC28"); // GroupNames label
G:=SmallGroup(56,8);
// by ID
G=gap.SmallGroup(56,8);
# by ID
G:=PCGroup([4,-2,-2,-7,-2,112]);
// Polycyclic
G:=Group<a,b|a^2=b^28=1,a*b=b*a>;
// generators/relations
Export