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## G = C2×D100order 400 = 24·52

### Direct product of C2 and D100

Series: Derived Chief Lower central Upper central

 Derived series C1 — C50 — C2×D100
 Chief series C1 — C5 — C25 — C50 — D50 — C22×D25 — C2×D100
 Lower central C25 — C50 — C2×D100
 Upper central C1 — C22 — C2×C4

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

Smallest permutation representation of 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

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