direct product, abelian, monomial, 2-elementary
Aliases: C2×C26, SmallGroup(52,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C26 |
| C1 — C2×C26 |
| C1 — C2×C26 |
Generators and relations for C2×C26
G = < a,b | a2=b26=1, ab=ba >
Smallest permutation representation of C2×C26
►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)]])
C2×C26 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 | C2×C26 | C26 | C22 | C2 |
| # reps | 1 | 3 | 12 | 36 |
Matrix representation of C2×C26 ►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] >;
C2×C26 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
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