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## G = C22×5- 1+2order 500 = 22·53

### Direct product of C22 and 5- 1+2

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

Aliases: C22×5- 1+2, C502C10, C102.C5, C5.2C102, (C2×C50)⋊C5, C252(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 >

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 2A 2B 2C 5A 5B 5C 5D 5E 5F 5G 5H 10A ··· 10L 10M ··· 10X 25A ··· 25T 50A ··· 50BH order 1 2 2 2 5 5 5 5 5 5 5 5 10 ··· 10 10 ··· 10 25 ··· 25 50 ··· 50 size 1 1 1 1 1 1 1 1 5 5 5 5 1 ··· 1 5 ··· 5 5 ··· 5 5 ··· 5

116 irreducible representations

 dim 1 1 1 1 1 1 5 5 type + + image C1 C2 C5 C5 C10 C10 5- 1+2 C2×5- 1+2 kernel C22×5- 1+2 C2×5- 1+2 C2×C50 C102 C50 C5×C10 C22 C2 # reps 1 3 20 4 60 12 4 12

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

 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 36 0 0 0 0 0 0 1 0 0 0 0 0 0 87 0 0 0 0 0 0 95 0 1 0 0 0 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

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

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