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G = C2×C25⋊C10  order 500 = 22·53

Direct product of C2 and C25⋊C10

direct product, metacyclic, supersoluble, monomial

Aliases: C2×C25⋊C10, C50⋊C10, D50⋊C5, D25⋊C10, C52.D10, 5- 1+2⋊C22, C25⋊(C2×C10), C5.3(D5×C10), C10.6(C5×D5), (C5×C10).4D5, (C2×5- 1+2)⋊C2, SmallGroup(500,31)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C25 — C2×C25⋊C10
Chief series
C1 — C5 — C25 — 5- 1+2 — C25⋊C10 — C2×C25⋊C10
Lower central
C25 — C2×C25⋊C10
Upper central
C1 — C2

Generators and relations for C2×C25⋊C10
 G = < a,b,c | a2=b25=c10=1, ab=ba, ac=ca, cbc-1=b9 >

C1
C2
25C2
25C2
C5
5C5
25C22
C10
5D5
5D5
5C10
25C10
25C10
C25
C52
2C25
2C25
5D10
25C2×C10
C50
D25
D25
C5×C10
2C50
2C50
5C5×D5
5C5×D5
5- 1+2
D50
5D5×C10
C25⋊C10
C2×5- 1+2
C25⋊C10
C2×C25⋊C10

Smallest permutation representation of C2×C25⋊C10
►On 50 points
Generators in S50
(1 48)(2 49)(3 50)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(25 47)

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

44 conjugacy classes

class 1 2A2B2C5A5B5C5D5E5F10A10B10C10D10E10F10G···10N25A···25J50A···50J
order122255555510101010101010···1025···2550···50
size11252522555522555525···2510···1010···10

44 irreducible representations

dim11111110102222
type+++++++
imageC1C2C2C5C10C10C25⋊C10C2×C25⋊C10D5D10C5×D5D5×C10
kernelC2×C25⋊C10C25⋊C10C2×5- 1+2D50D25C50C2C1C5×C10C52C10C5
# reps121484222288

Matrix representation of C2×C25⋊C10 ►in GL10(𝔽101)

100000000000
010000000000
001000000000
000100000000
000010000000
000001000000
000000100000
000000010000
000000001000
000000000100
,
000000001001
2310000009978
2210000001000
000000001000
2317810000001000
001000001000
2310078100001000
000010001000
2310000781001000
000000101000
,
7910000000000
792200000000
78100000000231
1220000007878
78100000012300
000000010000
000010000000
78100002310000
1227878000000
78100123000000

G:=sub<GL(10,GF(101))| [100,0,0,0,0,0,0,0,0,0,0,100,0,0,0,0,0,0,0,0,0,0,100,0,0,0,0,0,0,0,0,0,0,100,0,0,0,0,0,0,0,0,0,0,100,0,0,0,0,0,0,0,0,0,0,100,0,0,0,0,0,0,0,0,0,0,100,0,0,0,0,0,0,0,0,0,0,100,0,0,0,0,0,0,0,0,0,0,100,0,0,0,0,0,0,0,0,0,0,100],[0,23,22,0,23,0,23,0,23,0,0,1,1,0,1,0,1,0,1,0,0,0,0,0,78,1,0,0,0,0,0,0,0,0,100,0,0,0,0,0,0,0,0,0,0,0,78,1,0,0,0,0,0,0,0,0,100,0,0,0,0,0,0,0,0,0,0,0,78,1,0,0,0,0,0,0,0,0,100,0,100,99,100,100,100,100,100,100,100,100,1,78,0,0,0,0,0,0,0,0],[79,79,78,1,78,0,0,78,1,78,100,22,100,22,100,0,0,100,22,100,0,0,0,0,0,0,0,0,78,1,0,0,0,0,0,0,0,0,78,23,0,0,0,0,0,0,100,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,23,100,0,0,0,0,0,0,23,78,0,0,0,0,0,0,0,0,1,78,0,0,0,0,0,0] >;
 

C2×C25⋊C10 in GAP, Magma, Sage, TeX 

C_2\times C_{25}\rtimes C_{10}
 
% in TeX
 
G:=Group("C2xC25:C10");
 
// GroupNames label
 
G:=SmallGroup(500,31);
 
// by ID
 
G=gap.SmallGroup(500,31);
 
# by ID
 
G:=PCGroup([5,-2,-2,-5,-5,-5,3603,613,418,10004]);
 
// Polycyclic
 
G:=Group<a,b,c|a^2=b^25=c^10=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^9>;
 
// generators/relations
 

Export

Subgroup lattice of C2×C25⋊C10 in TeX

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