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G = C2×5- 1+2  order 250 = 2·53

Direct product of C2 and 5- 1+2

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

Aliases: C2×5- 1+2, C50⋊C5, C25⋊2C10, C52.C10, C10.2C52, (C5×C10).C5, C5.2(C5×C10), SmallGroup(250,11)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5 — C2×5- 1+2
Chief series
C1 — C5 — C52 — 5- 1+2 — C2×5- 1+2
Lower central
C1 — C5 — C2×5- 1+2
Upper central
C1 — C10 — C2×5- 1+2

Generators and relations for C2×5- 1+2
 G = < a,b,c | a2=b25=c5=1, ab=ba, ac=ca, cbc-1=b6 >

C1
C2
C5
5C5
C10
5C10
C25
C25
C25
C52
C25
C25
C50
C50
C50
C50
C50
C5×C10
5- 1+2
C2×5- 1+2

Smallest permutation representation of C2×5- 1+2
►On 50 points
Generators in S50
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(20 39)(21 40)(22 41)(23 42)(24 43)(25 44)

(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 6 11 16 21)(3 23 18 13 8)(4 19 9 24 14)(5 15 25 10 20)(27 47 42 37 32)(28 43 33 48 38)(29 39 49 34 44)(30 35 40 45 50)
 
G:=sub<Sym(50)| (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,40)(22,41)(23,42)(24,43)(25,44), (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), (1,6,11,16,21)(3,23,18,13,8)(4,19,9,24,14)(5,15,25,10,20)(27,47,42,37,32)(28,43,33,48,38)(29,39,49,34,44)(30,35,40,45,50)>;
 
G:=Group( (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,40)(22,41)(23,42)(24,43)(25,44), (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), (1,6,11,16,21)(3,23,18,13,8)(4,19,9,24,14)(5,15,25,10,20)(27,47,42,37,32)(28,43,33,48,38)(29,39,49,34,44)(30,35,40,45,50) );
 
G=PermutationGroup([[(1,45),(2,46),(3,47),(4,48),(5,49),(6,50),(7,26),(8,27),(9,28),(10,29),(11,30),(12,31),(13,32),(14,33),(15,34),(16,35),(17,36),(18,37),(19,38),(20,39),(21,40),(22,41),(23,42),(24,43),(25,44)], [(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)], [(1,6,11,16,21),(3,23,18,13,8),(4,19,9,24,14),(5,15,25,10,20),(27,47,42,37,32),(28,43,33,48,38),(29,39,49,34,44),(30,35,40,45,50)]])
 

C2×5- 1+2 is a maximal subgroup of   C50.C10

58 conjugacy classes

class 1  2 5A5B5C5D5E5F5G5H10A10B10C10D10E10F10G10H25A···25T50A···50T
order1255555555101010101010101025···2550···50
size1111115555111155555···55···5

58 irreducible representations

dim11111155
type++
imageC1C2C5C5C10C105- 1+2C2×5- 1+2
kernelC2×5- 1+25- 1+2C50C5×C10C25C52C2C1
# reps1120420444

Matrix representation of C2×5- 1+2 ►in GL5(𝔽101)

1000000
0100000
0010000
0001000
0000100
,
084000
00100
000950
000036
10000
,
360000
084000
009500
000870
00001

G:=sub<GL(5,GF(101))| [100,0,0,0,0,0,100,0,0,0,0,0,100,0,0,0,0,0,100,0,0,0,0,0,100],[0,0,0,0,1,84,0,0,0,0,0,1,0,0,0,0,0,95,0,0,0,0,0,36,0],[36,0,0,0,0,0,84,0,0,0,0,0,95,0,0,0,0,0,87,0,0,0,0,0,1] >;
 

C2×5- 1+2 in GAP, Magma, Sage, TeX 

C_2\times 5_-^{1+2}
 
% in TeX
 
G:=Group("C2xES-(5,1)");
 
// GroupNames label
 
G:=SmallGroup(250,11);
 
// by ID
 
G=gap.SmallGroup(250,11);
 
# by ID
 
G:=PCGroup([4,-2,-5,-5,-5,205,1266]);
 
// Polycyclic
 
G:=Group<a,b,c|a^2=b^25=c^5=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^6>;
 
// generators/relations
 

Export

Subgroup lattice of C2×5- 1+2 in TeX

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