direct product, abelian, monomial, 2-elementary
Aliases: C2xC26, SmallGroup(52,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2xC26 |
| C1 — C2xC26 |
| C1 — C2xC26 |
Generators and relations for C2xC26
G = < a,b | a2=b26=1, ab=ba >
Quotients: C1, C2, C22, C13, C26, C2xC26
Smallest permutation representation of C2xC26
►Regular action on 52 points
Generators in S52
(1 51)(2 52)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)
(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)
G:=sub<Sym(52)| (1,51)(2,52)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50), (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)>;
G:=Group( (1,51)(2,52)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50), (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) );
G=PermutationGroup([[(1,51),(2,52),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,33),(10,34),(11,35),(12,36),(13,37),(14,38),(15,39),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,49),(26,50)], [(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)]])
C2xC26 is a maximal subgroup of
C13:D4 C13:A4
52 conjugacy classes
| class | 1 | 2A | 2B | 2C | 13A | ··· | 13L | 26A | ··· | 26AJ |
| order | 1 | 2 | 2 | 2 | 13 | ··· | 13 | 26 | ··· | 26 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C13 | C26 |
| kernel | C2xC26 | C26 | C22 | C2 |
| # reps | 1 | 3 | 12 | 36 |
Matrix representation of C2xC26 ►in GL2(F53) generated by
| 52 | 0 |
| 0 | 52 |
| 10 | 0 |
| 0 | 38 |
G:=sub<GL(2,GF(53))| [52,0,0,52],[10,0,0,38] >;
C2xC26 in GAP, Magma, Sage, TeX
C_2\times C_{26} % in TeX
G:=Group("C2xC26"); // GroupNames label
G:=SmallGroup(52,5);
// by ID
G=gap.SmallGroup(52,5);
# by ID
G:=PCGroup([3,-2,-2,-13]);
// Polycyclic
G:=Group<a,b|a^2=b^26=1,a*b=b*a>;
// generators/relations
Export