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G = C22×5- 1+2  order 500 = 22·53

Direct product of C22 and 5- 1+2

direct product, metacyclic, nilpotent (class 2), monomial

Aliases: C22×5- 1+2, C50⋊2C10, C102.C5, C5.2C102, (C2×C50)⋊C5, C25⋊2(C2×C10), C52.(C2×C10), C10.5(C5×C10), (C5×C10).4C10, (C2×C10).2C52, SmallGroup(500,36)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5 — C22×5- 1+2
Chief series
C1 — C5 — C52 — 5- 1+2 — C2×5- 1+2 — C22×5- 1+2
Lower central
C1 — C5 — C22×5- 1+2
Upper central
C1 — C2×C10 — C22×5- 1+2

Generators and relations for C22×5- 1+2
 G = < a,b,c,d | a2=b2=c25=d5=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c6 >

C1
C2
C2
C2
C5
5C5
C22
C10
C10
C10
5C10
5C10
5C10
C25
C25
C25
C25
C25
C52
C2×C10
5C2×C10
C50
C50
C50
C50
C50
C50
C50
C5×C10
C5×C10
C50
C50
C50
C50
C50
C5×C10
C50
C50
C50
5- 1+2
C2×C50
C2×C50
C2×C50
C2×C50
C102
C2×C50
C2×5- 1+2
C2×5- 1+2
C2×5- 1+2
C22×5- 1+2

Smallest permutation representation of C22×5- 1+2
►On 100 points
Generators in S100
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 73)(16 74)(17 75)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 57)(25 58)(26 86)(27 87)(28 88)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 76)(42 77)(43 78)(44 79)(45 80)(46 81)(47 82)(48 83)(49 84)(50 85)

(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 26)(25 27)(51 80)(52 81)(53 82)(54 83)(55 84)(56 85)(57 86)(58 87)(59 88)(60 89)(61 90)(62 91)(63 92)(64 93)(65 94)(66 95)(67 96)(68 97)(69 98)(70 99)(71 100)(72 76)(73 77)(74 78)(75 79)

(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)(51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75)(76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)

(1 6 11 16 21)(3 23 18 13 8)(4 19 9 24 14)(5 15 25 10 20)(26 41 31 46 36)(27 37 47 32 42)(28 33 38 43 48)(30 50 45 40 35)(51 71 66 61 56)(52 67 57 72 62)(53 63 73 58 68)(54 59 64 69 74)(76 91 81 96 86)(77 87 97 82 92)(78 83 88 93 98)(80 100 95 90 85)
 
G:=sub<Sym(100)| (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,75)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,57)(25,58)(26,86)(27,87)(28,88)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,76)(42,77)(43,78)(44,79)(45,80)(46,81)(47,82)(48,83)(49,84)(50,85), (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,26)(25,27)(51,80)(52,81)(53,82)(54,83)(55,84)(56,85)(57,86)(58,87)(59,88)(60,89)(61,90)(62,91)(63,92)(64,93)(65,94)(66,95)(67,96)(68,97)(69,98)(70,99)(71,100)(72,76)(73,77)(74,78)(75,79), (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)(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75)(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100), (1,6,11,16,21)(3,23,18,13,8)(4,19,9,24,14)(5,15,25,10,20)(26,41,31,46,36)(27,37,47,32,42)(28,33,38,43,48)(30,50,45,40,35)(51,71,66,61,56)(52,67,57,72,62)(53,63,73,58,68)(54,59,64,69,74)(76,91,81,96,86)(77,87,97,82,92)(78,83,88,93,98)(80,100,95,90,85)>;
 
G:=Group( (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,75)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,57)(25,58)(26,86)(27,87)(28,88)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,76)(42,77)(43,78)(44,79)(45,80)(46,81)(47,82)(48,83)(49,84)(50,85), (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,26)(25,27)(51,80)(52,81)(53,82)(54,83)(55,84)(56,85)(57,86)(58,87)(59,88)(60,89)(61,90)(62,91)(63,92)(64,93)(65,94)(66,95)(67,96)(68,97)(69,98)(70,99)(71,100)(72,76)(73,77)(74,78)(75,79), (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)(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75)(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100), (1,6,11,16,21)(3,23,18,13,8)(4,19,9,24,14)(5,15,25,10,20)(26,41,31,46,36)(27,37,47,32,42)(28,33,38,43,48)(30,50,45,40,35)(51,71,66,61,56)(52,67,57,72,62)(53,63,73,58,68)(54,59,64,69,74)(76,91,81,96,86)(77,87,97,82,92)(78,83,88,93,98)(80,100,95,90,85) );
 
G=PermutationGroup([[(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,65),(8,66),(9,67),(10,68),(11,69),(12,70),(13,71),(14,72),(15,73),(16,74),(17,75),(18,51),(19,52),(20,53),(21,54),(22,55),(23,56),(24,57),(25,58),(26,86),(27,87),(28,88),(29,89),(30,90),(31,91),(32,92),(33,93),(34,94),(35,95),(36,96),(37,97),(38,98),(39,99),(40,100),(41,76),(42,77),(43,78),(44,79),(45,80),(46,81),(47,82),(48,83),(49,84),(50,85)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,37),(11,38),(12,39),(13,40),(14,41),(15,42),(16,43),(17,44),(18,45),(19,46),(20,47),(21,48),(22,49),(23,50),(24,26),(25,27),(51,80),(52,81),(53,82),(54,83),(55,84),(56,85),(57,86),(58,87),(59,88),(60,89),(61,90),(62,91),(63,92),(64,93),(65,94),(66,95),(67,96),(68,97),(69,98),(70,99),(71,100),(72,76),(73,77),(74,78),(75,79)], [(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),(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75),(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)], [(1,6,11,16,21),(3,23,18,13,8),(4,19,9,24,14),(5,15,25,10,20),(26,41,31,46,36),(27,37,47,32,42),(28,33,38,43,48),(30,50,45,40,35),(51,71,66,61,56),(52,67,57,72,62),(53,63,73,58,68),(54,59,64,69,74),(76,91,81,96,86),(77,87,97,82,92),(78,83,88,93,98),(80,100,95,90,85)]])
 

116 conjugacy classes

class 1 2A2B2C5A5B5C5D5E5F5G5H10A···10L10M···10X25A···25T50A···50BH
order12225555555510···1010···1025···2550···50
size1111111155551···15···55···55···5

116 irreducible representations

dim11111155
type++
imageC1C2C5C5C10C105- 1+2C2×5- 1+2
kernelC22×5- 1+2C2×5- 1+2C2×C50C102C50C5×C10C22C2
# reps132046012412

Matrix representation of C22×5- 1+2 ►in GL6(𝔽101)

100000
01000000
00100000
00010000
00001000
00000100
,
10000000
010000
001000
000100
000010
000001
,
100000
0036000
000100
0000870
0000095
010000
,
8700000
0950000
0036000
0008700
0000840
000001

G:=sub<GL(6,GF(101))| [1,0,0,0,0,0,0,100,0,0,0,0,0,0,100,0,0,0,0,0,0,100,0,0,0,0,0,0,100,0,0,0,0,0,0,100],[100,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,87,0,0,0,0,0,0,95,0],[87,0,0,0,0,0,0,95,0,0,0,0,0,0,36,0,0,0,0,0,0,87,0,0,0,0,0,0,84,0,0,0,0,0,0,1] >;
 

C22×5- 1+2 in GAP, Magma, Sage, TeX 

C_2^2\times 5_-^{1+2}
 
% in TeX
 
G:=Group("C2^2xES-(5,1)");
 
// GroupNames label
 
G:=SmallGroup(500,36);
 
// by ID
 
G=gap.SmallGroup(500,36);
 
# by ID
 
G:=PCGroup([5,-2,-2,-5,-5,-5,387,2113]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^2=c^25=d^5=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^6>;
 
// generators/relations
 

Export

Subgroup lattice of C22×5- 1+2 in TeX

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