direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D100, C50⋊1D4, C4⋊2D50, C10.6D20, C100⋊2C22, C20.44D10, D50⋊1C22, C50.3C23, C22.10D50, C25⋊1(C2×D4), C5.(C2×D20), (C2×C4)⋊2D25, (C2×C100)⋊3C2, (C2×C20).7D5, (C2×C10).25D10, (C22×D25)⋊1C2, C2.4(C22×D25), (C2×C50).10C22, C10.21(C22×D5), SmallGroup(400,37)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D100
G = < a,b,c | a2=b100=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 861 in 81 conjugacy classes, 35 normal (13 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C2×C4, D4, C23, D5, C10, C10, C2×D4, C20, D10, C2×C10, C25, D20, C2×C20, C22×D5, D25, C50, C50, C2×D20, C100, D50, D50, C2×C50, D100, C2×C100, C22×D25, C2×D100
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, D20, C22×D5, D25, C2×D20, D50, D100, C22×D25, C2×D100
►On 200 points
Generators in S200
(1 169)(2 170)(3 171)(4 172)(5 173)(6 174)(7 175)(8 176)(9 177)(10 178)(11 179)(12 180)(13 181)(14 182)(15 183)(16 184)(17 185)(18 186)(19 187)(20 188)(21 189)(22 190)(23 191)(24 192)(25 193)(26 194)(27 195)(28 196)(29 197)(30 198)(31 199)(32 200)(33 101)(34 102)(35 103)(36 104)(37 105)(38 106)(39 107)(40 108)(41 109)(42 110)(43 111)(44 112)(45 113)(46 114)(47 115)(48 116)(49 117)(50 118)(51 119)(52 120)(53 121)(54 122)(55 123)(56 124)(57 125)(58 126)(59 127)(60 128)(61 129)(62 130)(63 131)(64 132)(65 133)(66 134)(67 135)(68 136)(69 137)(70 138)(71 139)(72 140)(73 141)(74 142)(75 143)(76 144)(77 145)(78 146)(79 147)(80 148)(81 149)(82 150)(83 151)(84 152)(85 153)(86 154)(87 155)(88 156)(89 157)(90 158)(91 159)(92 160)(93 161)(94 162)(95 163)(96 164)(97 165)(98 166)(99 167)(100 168)
(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)
(1 100)(2 99)(3 98)(4 97)(5 96)(6 95)(7 94)(8 93)(9 92)(10 91)(11 90)(12 89)(13 88)(14 87)(15 86)(16 85)(17 84)(18 83)(19 82)(20 81)(21 80)(22 79)(23 78)(24 77)(25 76)(26 75)(27 74)(28 73)(29 72)(30 71)(31 70)(32 69)(33 68)(34 67)(35 66)(36 65)(37 64)(38 63)(39 62)(40 61)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(101 136)(102 135)(103 134)(104 133)(105 132)(106 131)(107 130)(108 129)(109 128)(110 127)(111 126)(112 125)(113 124)(114 123)(115 122)(116 121)(117 120)(118 119)(137 200)(138 199)(139 198)(140 197)(141 196)(142 195)(143 194)(144 193)(145 192)(146 191)(147 190)(148 189)(149 188)(150 187)(151 186)(152 185)(153 184)(154 183)(155 182)(156 181)(157 180)(158 179)(159 178)(160 177)(161 176)(162 175)(163 174)(164 173)(165 172)(166 171)(167 170)(168 169)
G:=sub<Sym(200)| (1,169)(2,170)(3,171)(4,172)(5,173)(6,174)(7,175)(8,176)(9,177)(10,178)(11,179)(12,180)(13,181)(14,182)(15,183)(16,184)(17,185)(18,186)(19,187)(20,188)(21,189)(22,190)(23,191)(24,192)(25,193)(26,194)(27,195)(28,196)(29,197)(30,198)(31,199)(32,200)(33,101)(34,102)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(41,109)(42,110)(43,111)(44,112)(45,113)(46,114)(47,115)(48,116)(49,117)(50,118)(51,119)(52,120)(53,121)(54,122)(55,123)(56,124)(57,125)(58,126)(59,127)(60,128)(61,129)(62,130)(63,131)(64,132)(65,133)(66,134)(67,135)(68,136)(69,137)(70,138)(71,139)(72,140)(73,141)(74,142)(75,143)(76,144)(77,145)(78,146)(79,147)(80,148)(81,149)(82,150)(83,151)(84,152)(85,153)(86,154)(87,155)(88,156)(89,157)(90,158)(91,159)(92,160)(93,161)(94,162)(95,163)(96,164)(97,165)(98,166)(99,167)(100,168), (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), (1,100)(2,99)(3,98)(4,97)(5,96)(6,95)(7,94)(8,93)(9,92)(10,91)(11,90)(12,89)(13,88)(14,87)(15,86)(16,85)(17,84)(18,83)(19,82)(20,81)(21,80)(22,79)(23,78)(24,77)(25,76)(26,75)(27,74)(28,73)(29,72)(30,71)(31,70)(32,69)(33,68)(34,67)(35,66)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(101,136)(102,135)(103,134)(104,133)(105,132)(106,131)(107,130)(108,129)(109,128)(110,127)(111,126)(112,125)(113,124)(114,123)(115,122)(116,121)(117,120)(118,119)(137,200)(138,199)(139,198)(140,197)(141,196)(142,195)(143,194)(144,193)(145,192)(146,191)(147,190)(148,189)(149,188)(150,187)(151,186)(152,185)(153,184)(154,183)(155,182)(156,181)(157,180)(158,179)(159,178)(160,177)(161,176)(162,175)(163,174)(164,173)(165,172)(166,171)(167,170)(168,169)>;
G:=Group( (1,169)(2,170)(3,171)(4,172)(5,173)(6,174)(7,175)(8,176)(9,177)(10,178)(11,179)(12,180)(13,181)(14,182)(15,183)(16,184)(17,185)(18,186)(19,187)(20,188)(21,189)(22,190)(23,191)(24,192)(25,193)(26,194)(27,195)(28,196)(29,197)(30,198)(31,199)(32,200)(33,101)(34,102)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(41,109)(42,110)(43,111)(44,112)(45,113)(46,114)(47,115)(48,116)(49,117)(50,118)(51,119)(52,120)(53,121)(54,122)(55,123)(56,124)(57,125)(58,126)(59,127)(60,128)(61,129)(62,130)(63,131)(64,132)(65,133)(66,134)(67,135)(68,136)(69,137)(70,138)(71,139)(72,140)(73,141)(74,142)(75,143)(76,144)(77,145)(78,146)(79,147)(80,148)(81,149)(82,150)(83,151)(84,152)(85,153)(86,154)(87,155)(88,156)(89,157)(90,158)(91,159)(92,160)(93,161)(94,162)(95,163)(96,164)(97,165)(98,166)(99,167)(100,168), (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), (1,100)(2,99)(3,98)(4,97)(5,96)(6,95)(7,94)(8,93)(9,92)(10,91)(11,90)(12,89)(13,88)(14,87)(15,86)(16,85)(17,84)(18,83)(19,82)(20,81)(21,80)(22,79)(23,78)(24,77)(25,76)(26,75)(27,74)(28,73)(29,72)(30,71)(31,70)(32,69)(33,68)(34,67)(35,66)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(101,136)(102,135)(103,134)(104,133)(105,132)(106,131)(107,130)(108,129)(109,128)(110,127)(111,126)(112,125)(113,124)(114,123)(115,122)(116,121)(117,120)(118,119)(137,200)(138,199)(139,198)(140,197)(141,196)(142,195)(143,194)(144,193)(145,192)(146,191)(147,190)(148,189)(149,188)(150,187)(151,186)(152,185)(153,184)(154,183)(155,182)(156,181)(157,180)(158,179)(159,178)(160,177)(161,176)(162,175)(163,174)(164,173)(165,172)(166,171)(167,170)(168,169) );
G=PermutationGroup([[(1,169),(2,170),(3,171),(4,172),(5,173),(6,174),(7,175),(8,176),(9,177),(10,178),(11,179),(12,180),(13,181),(14,182),(15,183),(16,184),(17,185),(18,186),(19,187),(20,188),(21,189),(22,190),(23,191),(24,192),(25,193),(26,194),(27,195),(28,196),(29,197),(30,198),(31,199),(32,200),(33,101),(34,102),(35,103),(36,104),(37,105),(38,106),(39,107),(40,108),(41,109),(42,110),(43,111),(44,112),(45,113),(46,114),(47,115),(48,116),(49,117),(50,118),(51,119),(52,120),(53,121),(54,122),(55,123),(56,124),(57,125),(58,126),(59,127),(60,128),(61,129),(62,130),(63,131),(64,132),(65,133),(66,134),(67,135),(68,136),(69,137),(70,138),(71,139),(72,140),(73,141),(74,142),(75,143),(76,144),(77,145),(78,146),(79,147),(80,148),(81,149),(82,150),(83,151),(84,152),(85,153),(86,154),(87,155),(88,156),(89,157),(90,158),(91,159),(92,160),(93,161),(94,162),(95,163),(96,164),(97,165),(98,166),(99,167),(100,168)], [(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)], [(1,100),(2,99),(3,98),(4,97),(5,96),(6,95),(7,94),(8,93),(9,92),(10,91),(11,90),(12,89),(13,88),(14,87),(15,86),(16,85),(17,84),(18,83),(19,82),(20,81),(21,80),(22,79),(23,78),(24,77),(25,76),(26,75),(27,74),(28,73),(29,72),(30,71),(31,70),(32,69),(33,68),(34,67),(35,66),(36,65),(37,64),(38,63),(39,62),(40,61),(41,60),(42,59),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53),(49,52),(50,51),(101,136),(102,135),(103,134),(104,133),(105,132),(106,131),(107,130),(108,129),(109,128),(110,127),(111,126),(112,125),(113,124),(114,123),(115,122),(116,121),(117,120),(118,119),(137,200),(138,199),(139,198),(140,197),(141,196),(142,195),(143,194),(144,193),(145,192),(146,191),(147,190),(148,189),(149,188),(150,187),(151,186),(152,185),(153,184),(154,183),(155,182),(156,181),(157,180),(158,179),(159,178),(160,177),(161,176),(162,175),(163,174),(164,173),(165,172),(166,171),(167,170),(168,169)]])
106 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 25A | ··· | 25J | 50A | ··· | 50AD | 100A | ··· | 100AN |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 25 | ··· | 25 | 50 | ··· | 50 | 100 | ··· | 100 |
| size | 1 | 1 | 1 | 1 | 50 | 50 | 50 | 50 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
106 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | D4 | D5 | D10 | D10 | D20 | D25 | D50 | D50 | D100 |
| kernel | C2×D100 | D100 | C2×C100 | C22×D25 | C50 | C2×C20 | C20 | C2×C10 | C10 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 2 | 2 | 4 | 2 | 8 | 10 | 20 | 10 | 40 |
Matrix representation of C2×D100 ►in GL3(𝔽101) generated by
| 100 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 100 | 0 | 0 |
| 0 | 83 | 77 |
| 0 | 24 | 60 |
| 1 | 0 | 0 |
| 0 | 60 | 24 |
| 0 | 31 | 41 |
G:=sub<GL(3,GF(101))| [100,0,0,0,1,0,0,0,1],[100,0,0,0,83,24,0,77,60],[1,0,0,0,60,31,0,24,41] >;
C2×D100 in GAP, Magma, Sage, TeX
C_2\times D_{100} % in TeX
G:=Group("C2xD100"); // GroupNames label
G:=SmallGroup(400,37);
// by ID
G=gap.SmallGroup(400,37);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,218,50,4324,628,11525]);
// Polycyclic
G:=Group<a,b,c|a^2=b^100=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations