direct product, abelian, monomial
Aliases: C2×C102, SmallGroup(200,52)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — 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, C22, C5, C23, C10, C2×C10, C52, C22×C10, C5×C10, C102, C2×C102
Quotients: C1, C2, C22, C5, C23, C10, C2×C10, C52, C22×C10, C5×C10, C102, C2×C102
►Regular action on 200 points
Generators in S200
(1 190)(2 181)(3 182)(4 183)(5 184)(6 185)(7 186)(8 187)(9 188)(10 189)(11 155)(12 156)(13 157)(14 158)(15 159)(16 160)(17 151)(18 152)(19 153)(20 154)(21 115)(22 116)(23 117)(24 118)(25 119)(26 120)(27 111)(28 112)(29 113)(30 114)(31 131)(32 132)(33 133)(34 134)(35 135)(36 136)(37 137)(38 138)(39 139)(40 140)(41 141)(42 142)(43 143)(44 144)(45 145)(46 146)(47 147)(48 148)(49 149)(50 150)(51 105)(52 106)(53 107)(54 108)(55 109)(56 110)(57 101)(58 102)(59 103)(60 104)(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)(81 176)(82 177)(83 178)(84 179)(85 180)(86 171)(87 172)(88 173)(89 174)(90 175)(91 191)(92 192)(93 193)(94 194)(95 195)(96 196)(97 197)(98 198)(99 199)(100 200)
(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)
(1 167 116 99 81 32 160 123 43 102)(2 168 117 100 82 33 151 124 44 103)(3 169 118 91 83 34 152 125 45 104)(4 170 119 92 84 35 153 126 46 105)(5 161 120 93 85 36 154 127 47 106)(6 162 111 94 86 37 155 128 48 107)(7 163 112 95 87 38 156 129 49 108)(8 164 113 96 88 39 157 130 50 109)(9 165 114 97 89 40 158 121 41 110)(10 166 115 98 90 31 159 122 42 101)(11 69 148 53 185 73 27 194 171 137)(12 70 149 54 186 74 28 195 172 138)(13 61 150 55 187 75 29 196 173 139)(14 62 141 56 188 76 30 197 174 140)(15 63 142 57 189 77 21 198 175 131)(16 64 143 58 190 78 22 199 176 132)(17 65 144 59 181 79 23 200 177 133)(18 66 145 60 182 80 24 191 178 134)(19 67 146 51 183 71 25 192 179 135)(20 68 147 52 184 72 26 193 180 136)
G:=sub<Sym(200)| (1,190)(2,181)(3,182)(4,183)(5,184)(6,185)(7,186)(8,187)(9,188)(10,189)(11,155)(12,156)(13,157)(14,158)(15,159)(16,160)(17,151)(18,152)(19,153)(20,154)(21,115)(22,116)(23,117)(24,118)(25,119)(26,120)(27,111)(28,112)(29,113)(30,114)(31,131)(32,132)(33,133)(34,134)(35,135)(36,136)(37,137)(38,138)(39,139)(40,140)(41,141)(42,142)(43,143)(44,144)(45,145)(46,146)(47,147)(48,148)(49,149)(50,150)(51,105)(52,106)(53,107)(54,108)(55,109)(56,110)(57,101)(58,102)(59,103)(60,104)(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)(81,176)(82,177)(83,178)(84,179)(85,180)(86,171)(87,172)(88,173)(89,174)(90,175)(91,191)(92,192)(93,193)(94,194)(95,195)(96,196)(97,197)(98,198)(99,199)(100,200), (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), (1,167,116,99,81,32,160,123,43,102)(2,168,117,100,82,33,151,124,44,103)(3,169,118,91,83,34,152,125,45,104)(4,170,119,92,84,35,153,126,46,105)(5,161,120,93,85,36,154,127,47,106)(6,162,111,94,86,37,155,128,48,107)(7,163,112,95,87,38,156,129,49,108)(8,164,113,96,88,39,157,130,50,109)(9,165,114,97,89,40,158,121,41,110)(10,166,115,98,90,31,159,122,42,101)(11,69,148,53,185,73,27,194,171,137)(12,70,149,54,186,74,28,195,172,138)(13,61,150,55,187,75,29,196,173,139)(14,62,141,56,188,76,30,197,174,140)(15,63,142,57,189,77,21,198,175,131)(16,64,143,58,190,78,22,199,176,132)(17,65,144,59,181,79,23,200,177,133)(18,66,145,60,182,80,24,191,178,134)(19,67,146,51,183,71,25,192,179,135)(20,68,147,52,184,72,26,193,180,136)>;
G:=Group( (1,190)(2,181)(3,182)(4,183)(5,184)(6,185)(7,186)(8,187)(9,188)(10,189)(11,155)(12,156)(13,157)(14,158)(15,159)(16,160)(17,151)(18,152)(19,153)(20,154)(21,115)(22,116)(23,117)(24,118)(25,119)(26,120)(27,111)(28,112)(29,113)(30,114)(31,131)(32,132)(33,133)(34,134)(35,135)(36,136)(37,137)(38,138)(39,139)(40,140)(41,141)(42,142)(43,143)(44,144)(45,145)(46,146)(47,147)(48,148)(49,149)(50,150)(51,105)(52,106)(53,107)(54,108)(55,109)(56,110)(57,101)(58,102)(59,103)(60,104)(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)(81,176)(82,177)(83,178)(84,179)(85,180)(86,171)(87,172)(88,173)(89,174)(90,175)(91,191)(92,192)(93,193)(94,194)(95,195)(96,196)(97,197)(98,198)(99,199)(100,200), (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), (1,167,116,99,81,32,160,123,43,102)(2,168,117,100,82,33,151,124,44,103)(3,169,118,91,83,34,152,125,45,104)(4,170,119,92,84,35,153,126,46,105)(5,161,120,93,85,36,154,127,47,106)(6,162,111,94,86,37,155,128,48,107)(7,163,112,95,87,38,156,129,49,108)(8,164,113,96,88,39,157,130,50,109)(9,165,114,97,89,40,158,121,41,110)(10,166,115,98,90,31,159,122,42,101)(11,69,148,53,185,73,27,194,171,137)(12,70,149,54,186,74,28,195,172,138)(13,61,150,55,187,75,29,196,173,139)(14,62,141,56,188,76,30,197,174,140)(15,63,142,57,189,77,21,198,175,131)(16,64,143,58,190,78,22,199,176,132)(17,65,144,59,181,79,23,200,177,133)(18,66,145,60,182,80,24,191,178,134)(19,67,146,51,183,71,25,192,179,135)(20,68,147,52,184,72,26,193,180,136) );
G=PermutationGroup([[(1,190),(2,181),(3,182),(4,183),(5,184),(6,185),(7,186),(8,187),(9,188),(10,189),(11,155),(12,156),(13,157),(14,158),(15,159),(16,160),(17,151),(18,152),(19,153),(20,154),(21,115),(22,116),(23,117),(24,118),(25,119),(26,120),(27,111),(28,112),(29,113),(30,114),(31,131),(32,132),(33,133),(34,134),(35,135),(36,136),(37,137),(38,138),(39,139),(40,140),(41,141),(42,142),(43,143),(44,144),(45,145),(46,146),(47,147),(48,148),(49,149),(50,150),(51,105),(52,106),(53,107),(54,108),(55,109),(56,110),(57,101),(58,102),(59,103),(60,104),(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),(81,176),(82,177),(83,178),(84,179),(85,180),(86,171),(87,172),(88,173),(89,174),(90,175),(91,191),(92,192),(93,193),(94,194),(95,195),(96,196),(97,197),(98,198),(99,199),(100,200)], [(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)], [(1,167,116,99,81,32,160,123,43,102),(2,168,117,100,82,33,151,124,44,103),(3,169,118,91,83,34,152,125,45,104),(4,170,119,92,84,35,153,126,46,105),(5,161,120,93,85,36,154,127,47,106),(6,162,111,94,86,37,155,128,48,107),(7,163,112,95,87,38,156,129,49,108),(8,164,113,96,88,39,157,130,50,109),(9,165,114,97,89,40,158,121,41,110),(10,166,115,98,90,31,159,122,42,101),(11,69,148,53,185,73,27,194,171,137),(12,70,149,54,186,74,28,195,172,138),(13,61,150,55,187,75,29,196,173,139),(14,62,141,56,188,76,30,197,174,140),(15,63,142,57,189,77,21,198,175,131),(16,64,143,58,190,78,22,199,176,132),(17,65,144,59,181,79,23,200,177,133),(18,66,145,60,182,80,24,191,178,134),(19,67,146,51,183,71,25,192,179,135),(20,68,147,52,184,72,26,193,180,136)]])
C2×C102 is a maximal subgroup of
C102⋊11C4
200 conjugacy classes
| class | 1 | 2A | ··· | 2G | 5A | ··· | 5X | 10A | ··· | 10FL |
| order | 1 | 2 | ··· | 2 | 5 | ··· | 5 | 10 | ··· | 10 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
200 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C5 | C10 |
| kernel | C2×C102 | C102 | C22×C10 | C2×C10 |
| # reps | 1 | 7 | 24 | 168 |
Matrix representation of C2×C102 ►in GL3(𝔽11) generated by
| 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 |
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