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G = C22×C50order 200 = 23·52

Abelian group of type [2,2,50]

direct product, abelian, monomial, 2-elementary

Aliases: C22×C50, SmallGroup(200,14)

Series: Derived Chief Lower central Upper central

C1 — C22×C50
C1C5C25C50C2×C50 — C22×C50
C1 — C22×C50
C1 — C22×C50

Generators and relations for C22×C50
 G = < a,b,c | a2=b2=c50=1, ab=ba, ac=ca, bc=cb >


Smallest permutation representation of C22×C50
Regular action on 200 points
Generators in S200
(1 144)(2 145)(3 146)(4 147)(5 148)(6 149)(7 150)(8 101)(9 102)(10 103)(11 104)(12 105)(13 106)(14 107)(15 108)(16 109)(17 110)(18 111)(19 112)(20 113)(21 114)(22 115)(23 116)(24 117)(25 118)(26 119)(27 120)(28 121)(29 122)(30 123)(31 124)(32 125)(33 126)(34 127)(35 128)(36 129)(37 130)(38 131)(39 132)(40 133)(41 134)(42 135)(43 136)(44 137)(45 138)(46 139)(47 140)(48 141)(49 142)(50 143)(51 158)(52 159)(53 160)(54 161)(55 162)(56 163)(57 164)(58 165)(59 166)(60 167)(61 168)(62 169)(63 170)(64 171)(65 172)(66 173)(67 174)(68 175)(69 176)(70 177)(71 178)(72 179)(73 180)(74 181)(75 182)(76 183)(77 184)(78 185)(79 186)(80 187)(81 188)(82 189)(83 190)(84 191)(85 192)(86 193)(87 194)(88 195)(89 196)(90 197)(91 198)(92 199)(93 200)(94 151)(95 152)(96 153)(97 154)(98 155)(99 156)(100 157)
(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 92)(9 93)(10 94)(11 95)(12 96)(13 97)(14 98)(15 99)(16 100)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 57)(24 58)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 65)(32 66)(33 67)(34 68)(35 69)(36 70)(37 71)(38 72)(39 73)(40 74)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 81)(48 82)(49 83)(50 84)(101 199)(102 200)(103 151)(104 152)(105 153)(106 154)(107 155)(108 156)(109 157)(110 158)(111 159)(112 160)(113 161)(114 162)(115 163)(116 164)(117 165)(118 166)(119 167)(120 168)(121 169)(122 170)(123 171)(124 172)(125 173)(126 174)(127 175)(128 176)(129 177)(130 178)(131 179)(132 180)(133 181)(134 182)(135 183)(136 184)(137 185)(138 186)(139 187)(140 188)(141 189)(142 190)(143 191)(144 192)(145 193)(146 194)(147 195)(148 196)(149 197)(150 198)
(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)(101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200)

G:=sub<Sym(200)| (1,144)(2,145)(3,146)(4,147)(5,148)(6,149)(7,150)(8,101)(9,102)(10,103)(11,104)(12,105)(13,106)(14,107)(15,108)(16,109)(17,110)(18,111)(19,112)(20,113)(21,114)(22,115)(23,116)(24,117)(25,118)(26,119)(27,120)(28,121)(29,122)(30,123)(31,124)(32,125)(33,126)(34,127)(35,128)(36,129)(37,130)(38,131)(39,132)(40,133)(41,134)(42,135)(43,136)(44,137)(45,138)(46,139)(47,140)(48,141)(49,142)(50,143)(51,158)(52,159)(53,160)(54,161)(55,162)(56,163)(57,164)(58,165)(59,166)(60,167)(61,168)(62,169)(63,170)(64,171)(65,172)(66,173)(67,174)(68,175)(69,176)(70,177)(71,178)(72,179)(73,180)(74,181)(75,182)(76,183)(77,184)(78,185)(79,186)(80,187)(81,188)(82,189)(83,190)(84,191)(85,192)(86,193)(87,194)(88,195)(89,196)(90,197)(91,198)(92,199)(93,200)(94,151)(95,152)(96,153)(97,154)(98,155)(99,156)(100,157), (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,97)(14,98)(15,99)(16,100)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,57)(24,58)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,65)(32,66)(33,67)(34,68)(35,69)(36,70)(37,71)(38,72)(39,73)(40,74)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,81)(48,82)(49,83)(50,84)(101,199)(102,200)(103,151)(104,152)(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,159)(112,160)(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)(120,168)(121,169)(122,170)(123,171)(124,172)(125,173)(126,174)(127,175)(128,176)(129,177)(130,178)(131,179)(132,180)(133,181)(134,182)(135,183)(136,184)(137,185)(138,186)(139,187)(140,188)(141,189)(142,190)(143,191)(144,192)(145,193)(146,194)(147,195)(148,196)(149,197)(150,198), (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)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200)>;

G:=Group( (1,144)(2,145)(3,146)(4,147)(5,148)(6,149)(7,150)(8,101)(9,102)(10,103)(11,104)(12,105)(13,106)(14,107)(15,108)(16,109)(17,110)(18,111)(19,112)(20,113)(21,114)(22,115)(23,116)(24,117)(25,118)(26,119)(27,120)(28,121)(29,122)(30,123)(31,124)(32,125)(33,126)(34,127)(35,128)(36,129)(37,130)(38,131)(39,132)(40,133)(41,134)(42,135)(43,136)(44,137)(45,138)(46,139)(47,140)(48,141)(49,142)(50,143)(51,158)(52,159)(53,160)(54,161)(55,162)(56,163)(57,164)(58,165)(59,166)(60,167)(61,168)(62,169)(63,170)(64,171)(65,172)(66,173)(67,174)(68,175)(69,176)(70,177)(71,178)(72,179)(73,180)(74,181)(75,182)(76,183)(77,184)(78,185)(79,186)(80,187)(81,188)(82,189)(83,190)(84,191)(85,192)(86,193)(87,194)(88,195)(89,196)(90,197)(91,198)(92,199)(93,200)(94,151)(95,152)(96,153)(97,154)(98,155)(99,156)(100,157), (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,97)(14,98)(15,99)(16,100)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,57)(24,58)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,65)(32,66)(33,67)(34,68)(35,69)(36,70)(37,71)(38,72)(39,73)(40,74)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,81)(48,82)(49,83)(50,84)(101,199)(102,200)(103,151)(104,152)(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,159)(112,160)(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)(120,168)(121,169)(122,170)(123,171)(124,172)(125,173)(126,174)(127,175)(128,176)(129,177)(130,178)(131,179)(132,180)(133,181)(134,182)(135,183)(136,184)(137,185)(138,186)(139,187)(140,188)(141,189)(142,190)(143,191)(144,192)(145,193)(146,194)(147,195)(148,196)(149,197)(150,198), (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)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200) );

G=PermutationGroup([(1,144),(2,145),(3,146),(4,147),(5,148),(6,149),(7,150),(8,101),(9,102),(10,103),(11,104),(12,105),(13,106),(14,107),(15,108),(16,109),(17,110),(18,111),(19,112),(20,113),(21,114),(22,115),(23,116),(24,117),(25,118),(26,119),(27,120),(28,121),(29,122),(30,123),(31,124),(32,125),(33,126),(34,127),(35,128),(36,129),(37,130),(38,131),(39,132),(40,133),(41,134),(42,135),(43,136),(44,137),(45,138),(46,139),(47,140),(48,141),(49,142),(50,143),(51,158),(52,159),(53,160),(54,161),(55,162),(56,163),(57,164),(58,165),(59,166),(60,167),(61,168),(62,169),(63,170),(64,171),(65,172),(66,173),(67,174),(68,175),(69,176),(70,177),(71,178),(72,179),(73,180),(74,181),(75,182),(76,183),(77,184),(78,185),(79,186),(80,187),(81,188),(82,189),(83,190),(84,191),(85,192),(86,193),(87,194),(88,195),(89,196),(90,197),(91,198),(92,199),(93,200),(94,151),(95,152),(96,153),(97,154),(98,155),(99,156),(100,157)], [(1,85),(2,86),(3,87),(4,88),(5,89),(6,90),(7,91),(8,92),(9,93),(10,94),(11,95),(12,96),(13,97),(14,98),(15,99),(16,100),(17,51),(18,52),(19,53),(20,54),(21,55),(22,56),(23,57),(24,58),(25,59),(26,60),(27,61),(28,62),(29,63),(30,64),(31,65),(32,66),(33,67),(34,68),(35,69),(36,70),(37,71),(38,72),(39,73),(40,74),(41,75),(42,76),(43,77),(44,78),(45,79),(46,80),(47,81),(48,82),(49,83),(50,84),(101,199),(102,200),(103,151),(104,152),(105,153),(106,154),(107,155),(108,156),(109,157),(110,158),(111,159),(112,160),(113,161),(114,162),(115,163),(116,164),(117,165),(118,166),(119,167),(120,168),(121,169),(122,170),(123,171),(124,172),(125,173),(126,174),(127,175),(128,176),(129,177),(130,178),(131,179),(132,180),(133,181),(134,182),(135,183),(136,184),(137,185),(138,186),(139,187),(140,188),(141,189),(142,190),(143,191),(144,192),(145,193),(146,194),(147,195),(148,196),(149,197),(150,198)], [(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),(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150),(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200)])

C22×C50 is a maximal subgroup of   C23.D25

200 conjugacy classes

class 1 2A···2G5A5B5C5D10A···10AB25A···25T50A···50EJ
order12···2555510···1025···2550···50
size11···111111···11···11···1

200 irreducible representations

dim111111
type++
imageC1C2C5C10C25C50
kernelC22×C50C2×C50C22×C10C2×C10C23C22
# reps1742820140

Matrix representation of C22×C50 in GL3(𝔽101) generated by

100
01000
00100
,
10000
01000
001
,
1400
0160
0056
G:=sub<GL(3,GF(101))| [1,0,0,0,100,0,0,0,100],[100,0,0,0,100,0,0,0,1],[14,0,0,0,16,0,0,0,56] >;

C22×C50 in GAP, Magma, Sage, TeX

C_2^2\times C_{50}
% in TeX

G:=Group("C2^2xC50");
// GroupNames label

G:=SmallGroup(200,14);
// by ID

G=gap.SmallGroup(200,14);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-5,118]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^50=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

Export

Subgroup lattice of C22×C50 in TeX

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