direct product, non-abelian, supersoluble, monomial
Aliases: C2×He5⋊C2, C52⋊3D10, He5⋊3C22, (C5×C10)⋊2D5, (C2×He5)⋊2C2, C10.5(C5⋊D5), C5.2(C2×C5⋊D5), SmallGroup(500,33)
Series: Derived ►Chief ►Lower central ►Upper central
| He5 — C2×He5⋊C2 |
Generators and relations for C2×He5⋊C2
G = < a,b,c,d,e | a2=b5=c5=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, ebe=b-1c-1, cd=dc, ce=ec, ede=d-1 >
Subgroups: 411 in 75 conjugacy classes, 21 normal (7 characteristic)
C1, C2, C2, C22, C5, C5, D5, C10, C10, D10, C2×C10, C52, C5×D5, C5×C10, D5×C10, He5, He5⋊C2, C2×He5, C2×He5⋊C2
Quotients: C1, C2, C22, D5, D10, C5⋊D5, C2×C5⋊D5, He5⋊C2, C2×He5⋊C2
►On 50 points
Generators in S50
(1 10)(2 8)(3 6)(4 9)(5 7)(11 41)(12 42)(13 43)(14 44)(15 45)(16 23)(17 24)(18 25)(19 21)(20 22)(26 34)(27 35)(28 31)(29 32)(30 33)(36 46)(37 47)(38 48)(39 49)(40 50)
(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 2 5 4 3)(6 10 8 7 9)(11 15 14 13 12)(16 17 18 19 20)(21 22 23 24 25)(26 29 27 30 28)(31 34 32 35 33)(36 38 40 37 39)(41 45 44 43 42)(46 48 50 47 49)
(1 24 30 48 41)(2 25 28 50 45)(3 23 27 46 42)(4 22 29 49 43)(5 21 26 47 44)(6 16 35 36 12)(7 19 34 37 14)(8 18 31 40 15)(9 20 32 39 13)(10 17 33 38 11)
(1 11)(2 15)(3 12)(4 13)(5 14)(6 42)(7 44)(8 45)(9 43)(10 41)(16 46)(17 48)(18 50)(19 47)(20 49)(21 37)(22 39)(23 36)(24 38)(25 40)(26 34)(27 35)(28 31)(29 32)(30 33)
G:=sub<Sym(50)| (1,10)(2,8)(3,6)(4,9)(5,7)(11,41)(12,42)(13,43)(14,44)(15,45)(16,23)(17,24)(18,25)(19,21)(20,22)(26,34)(27,35)(28,31)(29,32)(30,33)(36,46)(37,47)(38,48)(39,49)(40,50), (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), (1,2,5,4,3)(6,10,8,7,9)(11,15,14,13,12)(16,17,18,19,20)(21,22,23,24,25)(26,29,27,30,28)(31,34,32,35,33)(36,38,40,37,39)(41,45,44,43,42)(46,48,50,47,49), (1,24,30,48,41)(2,25,28,50,45)(3,23,27,46,42)(4,22,29,49,43)(5,21,26,47,44)(6,16,35,36,12)(7,19,34,37,14)(8,18,31,40,15)(9,20,32,39,13)(10,17,33,38,11), (1,11)(2,15)(3,12)(4,13)(5,14)(6,42)(7,44)(8,45)(9,43)(10,41)(16,46)(17,48)(18,50)(19,47)(20,49)(21,37)(22,39)(23,36)(24,38)(25,40)(26,34)(27,35)(28,31)(29,32)(30,33)>;
G:=Group( (1,10)(2,8)(3,6)(4,9)(5,7)(11,41)(12,42)(13,43)(14,44)(15,45)(16,23)(17,24)(18,25)(19,21)(20,22)(26,34)(27,35)(28,31)(29,32)(30,33)(36,46)(37,47)(38,48)(39,49)(40,50), (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), (1,2,5,4,3)(6,10,8,7,9)(11,15,14,13,12)(16,17,18,19,20)(21,22,23,24,25)(26,29,27,30,28)(31,34,32,35,33)(36,38,40,37,39)(41,45,44,43,42)(46,48,50,47,49), (1,24,30,48,41)(2,25,28,50,45)(3,23,27,46,42)(4,22,29,49,43)(5,21,26,47,44)(6,16,35,36,12)(7,19,34,37,14)(8,18,31,40,15)(9,20,32,39,13)(10,17,33,38,11), (1,11)(2,15)(3,12)(4,13)(5,14)(6,42)(7,44)(8,45)(9,43)(10,41)(16,46)(17,48)(18,50)(19,47)(20,49)(21,37)(22,39)(23,36)(24,38)(25,40)(26,34)(27,35)(28,31)(29,32)(30,33) );
G=PermutationGroup([[(1,10),(2,8),(3,6),(4,9),(5,7),(11,41),(12,42),(13,43),(14,44),(15,45),(16,23),(17,24),(18,25),(19,21),(20,22),(26,34),(27,35),(28,31),(29,32),(30,33),(36,46),(37,47),(38,48),(39,49),(40,50)], [(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)], [(1,2,5,4,3),(6,10,8,7,9),(11,15,14,13,12),(16,17,18,19,20),(21,22,23,24,25),(26,29,27,30,28),(31,34,32,35,33),(36,38,40,37,39),(41,45,44,43,42),(46,48,50,47,49)], [(1,24,30,48,41),(2,25,28,50,45),(3,23,27,46,42),(4,22,29,49,43),(5,21,26,47,44),(6,16,35,36,12),(7,19,34,37,14),(8,18,31,40,15),(9,20,32,39,13),(10,17,33,38,11)], [(1,11),(2,15),(3,12),(4,13),(5,14),(6,42),(7,44),(8,45),(9,43),(10,41),(16,46),(17,48),(18,50),(19,47),(20,49),(21,37),(22,39),(23,36),(24,38),(25,40),(26,34),(27,35),(28,31),(29,32),(30,33)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 5A | 5B | 5C | 5D | 5E | ··· | 5P | 10A | 10B | 10C | 10D | 10E | ··· | 10P | 10Q | ··· | 10X |
| order | 1 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | ··· | 10 |
| size | 1 | 1 | 25 | 25 | 1 | 1 | 1 | 1 | 10 | ··· | 10 | 1 | 1 | 1 | 1 | 10 | ··· | 10 | 25 | ··· | 25 |
44 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 5 | 5 |
| type | + | + | + | + | + | ||
| image | C1 | C2 | C2 | D5 | D10 | He5⋊C2 | C2×He5⋊C2 |
| kernel | C2×He5⋊C2 | He5⋊C2 | C2×He5 | C5×C10 | C52 | C2 | C1 |
| # reps | 1 | 2 | 1 | 12 | 12 | 8 | 8 |
Matrix representation of C2×He5⋊C2 ►in GL5(𝔽11)
| 10 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 10 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 5 |
| 3 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 |
| 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 10 | 0 | 0 |
| 0 | 10 | 0 | 0 | 0 |
G:=sub<GL(5,GF(11))| [10,0,0,0,0,0,10,0,0,0,0,0,10,0,0,0,0,0,10,0,0,0,0,0,10],[9,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,5],[3,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[0,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0],[10,0,0,0,0,0,0,0,0,10,0,0,0,10,0,0,0,10,0,0,0,10,0,0,0] >;
C2×He5⋊C2 in GAP, Magma, Sage, TeX
C_2\times {\rm He}_5\rtimes C_2 % in TeX
G:=Group("C2xHe5:C2"); // GroupNames label
G:=SmallGroup(500,33);
// by ID
G=gap.SmallGroup(500,33);
# by ID
G:=PCGroup([5,-2,-2,-5,-5,-5,242,1603,613]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^5=c^5=d^5=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,e*b*e=b^-1*c^-1,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations