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G = C2×He5  order 250 = 2·53

Direct product of C2 and He5

direct product, metabelian, nilpotent (class 2), monomial, 5-elementary

Aliases: C2×He5, C52⋊2C10, C10.1C52, (C5×C10)⋊C5, C5.1(C5×C10), SmallGroup(250,10)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5 — C2×He5
Chief series
C1 — C5 — C52 — He5 — C2×He5
Lower central
C1 — C5 — C2×He5
Upper central
C1 — C10 — C2×He5

Generators and relations for C2×He5
 G = < a,b,c,d | a2=b5=c5=d5=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

C1
C2
C5
5C5
5C5
5C5
5C5
5C5
5C5
C10
5C10
5C10
5C10
5C10
5C10
5C10
C52
C52
C52
C52
C52
C52
C5×C10
C5×C10
C5×C10
C5×C10
C5×C10
C5×C10
He5
C2×He5

Smallest permutation representation of C2×He5
►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)]])
 

C2×He5 is a maximal subgroup of   He5⋊5C4  He5⋊6C4

58 conjugacy classes

class 1  2 5A5B5C5D5E···5AB10A10B10C10D10E···10AB
order1255555···51010101010···10
size1111115···511115···5

58 irreducible representations

dim111155
type++
imageC1C2C5C10He5C2×He5
kernelC2×He5He5C5×C10C52C2C1
# reps11242444

Matrix representation of C2×He5 ►in GL6(𝔽11)

1000000
010000
001000
000100
000010
000001
,
100000
041172
000900
000050
000004
028267
,
100000
040000
004000
000400
000040
000004
,
300000
040000
000100
000010
000001
028267

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] >;
 

C2×He5 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

Subgroup lattice of C2×He5 in TeX

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