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G = C2×C102order 200 = 23·52

Abelian group of type [2,10,10]

direct product, abelian, monomial

Aliases: C2×C102, SmallGroup(200,52)

Series: Derived Chief Lower central Upper central

C1 — C2×C102
C1C5C52C5×C10C102 — C2×C102
C1 — C2×C102
C1 — C2×C102

Generators and relations for C2×C102
 G = < a,b,c | a2=b10=c10=1, ab=ba, ac=ca, bc=cb >

Subgroups: 128, all normal (4 characteristic)
C1, C2 [×7], C22 [×7], C5 [×6], C23, C10 [×42], C2×C10 [×42], C52, C22×C10 [×6], C5×C10 [×7], C102 [×7], C2×C102
Quotients: C1, C2 [×7], C22 [×7], C5 [×6], C23, C10 [×42], C2×C10 [×42], C52, C22×C10 [×6], C5×C10 [×7], C102 [×7], C2×C102

Smallest permutation representation of C2×C102
Regular action on 200 points
Generators in S200
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 120)(12 111)(13 112)(14 113)(15 114)(16 115)(17 116)(18 117)(19 118)(20 119)(21 105)(22 106)(23 107)(24 108)(25 109)(26 110)(27 101)(28 102)(29 103)(30 104)(31 178)(32 179)(33 180)(34 171)(35 172)(36 173)(37 174)(38 175)(39 176)(40 177)(41 190)(42 181)(43 182)(44 183)(45 184)(46 185)(47 186)(48 187)(49 188)(50 189)(61 130)(62 121)(63 122)(64 123)(65 124)(66 125)(67 126)(68 127)(69 128)(70 129)(71 170)(72 161)(73 162)(74 163)(75 164)(76 165)(77 166)(78 167)(79 168)(80 169)(81 134)(82 135)(83 136)(84 137)(85 138)(86 139)(87 140)(88 131)(89 132)(90 133)(91 197)(92 198)(93 199)(94 200)(95 191)(96 192)(97 193)(98 194)(99 195)(100 196)(141 160)(142 151)(143 152)(144 153)(145 154)(146 155)(147 156)(148 157)(149 158)(150 159)
(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)
(1 93 116 110 83 167 46 123 38 151)(2 94 117 101 84 168 47 124 39 152)(3 95 118 102 85 169 48 125 40 153)(4 96 119 103 86 170 49 126 31 154)(5 97 120 104 87 161 50 127 32 155)(6 98 111 105 88 162 41 128 33 156)(7 99 112 106 89 163 42 129 34 157)(8 100 113 107 90 164 43 130 35 158)(9 91 114 108 81 165 44 121 36 159)(10 92 115 109 82 166 45 122 37 160)(11 30 140 72 189 68 179 146 55 193)(12 21 131 73 190 69 180 147 56 194)(13 22 132 74 181 70 171 148 57 195)(14 23 133 75 182 61 172 149 58 196)(15 24 134 76 183 62 173 150 59 197)(16 25 135 77 184 63 174 141 60 198)(17 26 136 78 185 64 175 142 51 199)(18 27 137 79 186 65 176 143 52 200)(19 28 138 80 187 66 177 144 53 191)(20 29 139 71 188 67 178 145 54 192)

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

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

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

C2×C102 is a maximal subgroup of   C10211C4

200 conjugacy classes

class 1 2A···2G5A···5X10A···10FL
order12···25···510···10
size11···11···11···1

200 irreducible representations

dim1111
type++
imageC1C2C5C10
kernelC2×C102C102C22×C10C2×C10
# reps1724168

Matrix representation of C2×C102 in GL3(𝔽11) generated by

100
0100
0010
,
600
050
009
,
400
070
009
G:=sub<GL(3,GF(11))| [1,0,0,0,10,0,0,0,10],[6,0,0,0,5,0,0,0,9],[4,0,0,0,7,0,0,0,9] >;

C2×C102 in GAP, Magma, Sage, TeX

C_2\times C_{10}^2
% in TeX

G:=Group("C2xC10^2");
// GroupNames label

G:=SmallGroup(200,52);
// by ID

G=gap.SmallGroup(200,52);
# by ID

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

G:=Group<a,b,c|a^2=b^10=c^10=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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