direct product, metabelian, nilpotent (class 2), monomial, 5-elementary
Aliases: C2xHe5, C52:2C10, C10.1C52, (C5xC10):C5, C5.1(C5xC10), SmallGroup(250,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2xHe5
G = < a,b,c,d | a2=b5=c5=d5=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Quotients: C1, C2, C5, C10, C52, C5xC10, He5, C2xHe5
Smallest permutation representation of C2xHe5
►On 50 points
Generators in S50
(1 11)(2 12)(3 13)(4 14)(5 15)(6 39)(7 40)(8 36)(9 37)(10 38)(16 32)(17 33)(18 34)(19 35)(20 31)(21 44)(22 45)(23 41)(24 42)(25 43)(26 50)(27 46)(28 47)(29 48)(30 49)
(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)
(1 43 29 33 39)(2 44 30 34 40)(3 45 26 35 36)(4 41 27 31 37)(5 42 28 32 38)(6 11 25 48 17)(7 12 21 49 18)(8 13 22 50 19)(9 14 23 46 20)(10 15 24 47 16)
(1 2 36 27 38)(3 31 5 43 44)(4 28 33 34 26)(6 7 19 23 16)(8 46 10 11 12)(9 24 48 49 22)(13 20 15 25 21)(14 47 17 18 50)(29 30 45 37 42)(32 39 40 35 41)
G:=sub<Sym(50)| (1,11)(2,12)(3,13)(4,14)(5,15)(6,39)(7,40)(8,36)(9,37)(10,38)(16,32)(17,33)(18,34)(19,35)(20,31)(21,44)(22,45)(23,41)(24,42)(25,43)(26,50)(27,46)(28,47)(29,48)(30,49), (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), (1,43,29,33,39)(2,44,30,34,40)(3,45,26,35,36)(4,41,27,31,37)(5,42,28,32,38)(6,11,25,48,17)(7,12,21,49,18)(8,13,22,50,19)(9,14,23,46,20)(10,15,24,47,16), (1,2,36,27,38)(3,31,5,43,44)(4,28,33,34,26)(6,7,19,23,16)(8,46,10,11,12)(9,24,48,49,22)(13,20,15,25,21)(14,47,17,18,50)(29,30,45,37,42)(32,39,40,35,41)>;
G:=Group( (1,11)(2,12)(3,13)(4,14)(5,15)(6,39)(7,40)(8,36)(9,37)(10,38)(16,32)(17,33)(18,34)(19,35)(20,31)(21,44)(22,45)(23,41)(24,42)(25,43)(26,50)(27,46)(28,47)(29,48)(30,49), (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), (1,43,29,33,39)(2,44,30,34,40)(3,45,26,35,36)(4,41,27,31,37)(5,42,28,32,38)(6,11,25,48,17)(7,12,21,49,18)(8,13,22,50,19)(9,14,23,46,20)(10,15,24,47,16), (1,2,36,27,38)(3,31,5,43,44)(4,28,33,34,26)(6,7,19,23,16)(8,46,10,11,12)(9,24,48,49,22)(13,20,15,25,21)(14,47,17,18,50)(29,30,45,37,42)(32,39,40,35,41) );
G=PermutationGroup([[(1,11),(2,12),(3,13),(4,14),(5,15),(6,39),(7,40),(8,36),(9,37),(10,38),(16,32),(17,33),(18,34),(19,35),(20,31),(21,44),(22,45),(23,41),(24,42),(25,43),(26,50),(27,46),(28,47),(29,48),(30,49)], [(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)], [(1,43,29,33,39),(2,44,30,34,40),(3,45,26,35,36),(4,41,27,31,37),(5,42,28,32,38),(6,11,25,48,17),(7,12,21,49,18),(8,13,22,50,19),(9,14,23,46,20),(10,15,24,47,16)], [(1,2,36,27,38),(3,31,5,43,44),(4,28,33,34,26),(6,7,19,23,16),(8,46,10,11,12),(9,24,48,49,22),(13,20,15,25,21),(14,47,17,18,50),(29,30,45,37,42),(32,39,40,35,41)]])
C2xHe5 is a maximal subgroup of
He5:5C4 He5:6C4
58 conjugacy classes
| class | 1 | 2 | 5A | 5B | 5C | 5D | 5E | ··· | 5AB | 10A | 10B | 10C | 10D | 10E | ··· | 10AB |
| order | 1 | 2 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 10 | ··· | 10 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 5 | ··· | 5 | 1 | 1 | 1 | 1 | 5 | ··· | 5 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 5 | 5 |
| type | + | + | ||||
| image | C1 | C2 | C5 | C10 | He5 | C2xHe5 |
| kernel | C2xHe5 | He5 | C5xC10 | C52 | C2 | C1 |
| # reps | 1 | 1 | 24 | 24 | 4 | 4 |
Matrix representation of C2xHe5 ►in GL6(F11)
| 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 1 | 1 | 7 | 2 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 2 | 8 | 2 | 6 | 7 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 2 | 8 | 2 | 6 | 7 |
G:=sub<GL(6,GF(11))| [10,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,4,0,0,0,2,0,1,0,0,0,8,0,1,9,0,0,2,0,7,0,5,0,6,0,2,0,0,4,7],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,4,0,0,0,2,0,0,0,0,0,8,0,0,1,0,0,2,0,0,0,1,0,6,0,0,0,0,1,7] >;
C2xHe5 in GAP, Magma, Sage, TeX
C_2\times {\rm He}_5 % in TeX
G:=Group("C2xHe5"); // GroupNames label
G:=SmallGroup(250,10);
// by ID
G=gap.SmallGroup(250,10);
# by ID
G:=PCGroup([4,-2,-5,-5,-5,366]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^5=c^5=d^5=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations
Export