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## G = C2×C25⋊C10order 500 = 22·53

### Direct product of C2 and C25⋊C10

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 >

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 2A 2B 2C 5A 5B 5C 5D 5E 5F 10A 10B 10C 10D 10E 10F 10G ··· 10N 25A ··· 25J 50A ··· 50J order 1 2 2 2 5 5 5 5 5 5 10 10 10 10 10 10 10 ··· 10 25 ··· 25 50 ··· 50 size 1 1 25 25 2 2 5 5 5 5 2 2 5 5 5 5 25 ··· 25 10 ··· 10 10 ··· 10

44 irreducible representations

 dim 1 1 1 1 1 1 10 10 2 2 2 2 type + + + + + + + image C1 C2 C2 C5 C10 C10 C25⋊C10 C2×C25⋊C10 D5 D10 C5×D5 D5×C10 kernel C2×C25⋊C10 C25⋊C10 C2×5- 1+2 D50 D25 C50 C2 C1 C5×C10 C52 C10 C5 # reps 1 2 1 4 8 4 2 2 2 2 8 8

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

 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 0 0 0 0 0 0 0 100 1 23 1 0 0 0 0 0 0 99 78 22 1 0 0 0 0 0 0 100 0 0 0 0 0 0 0 0 0 100 0 23 1 78 100 0 0 0 0 100 0 0 0 1 0 0 0 0 0 100 0 23 1 0 0 78 100 0 0 100 0 0 0 0 0 1 0 0 0 100 0 23 1 0 0 0 0 78 100 100 0 0 0 0 0 0 0 1 0 100 0
,
 79 100 0 0 0 0 0 0 0 0 79 22 0 0 0 0 0 0 0 0 78 100 0 0 0 0 0 0 23 1 1 22 0 0 0 0 0 0 78 78 78 100 0 0 0 0 1 23 0 0 0 0 0 0 0 0 0 100 0 0 0 0 0 0 100 0 0 0 0 0 78 100 0 0 23 1 0 0 0 0 1 22 78 78 0 0 0 0 0 0 78 100 1 23 0 0 0 0 0 0

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

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