Copied to
clipboard

G = C200⋊C2order 400 = 24·52

2nd semidirect product of C200 and C2 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C82D25, C2002C2, C50.2D4, C40.4D5, C4.9D50, C251SD16, C10.2D20, C2.4D100, Dic501C2, D100.1C2, C20.41D10, C100.9C22, C5.(C40⋊C2), SmallGroup(400,7)

Series: Derived Chief Lower central Upper central

C1C100 — C200⋊C2
C1C5C25C50C100D100 — C200⋊C2
C25C50C100 — C200⋊C2
C1C2C4C8

Generators and relations for C200⋊C2
 G = < a,b | a200=b2=1, bab=a99 >

100C2
50C4
50C22
20D5
25Q8
25D4
10Dic5
10D10
4D25
25SD16
5Dic10
5D20
2Dic25
2D50
5C40⋊C2

Smallest permutation representation of C200⋊C2
On 200 points
Generators in S200
(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)
(2 100)(3 199)(4 98)(5 197)(6 96)(7 195)(8 94)(9 193)(10 92)(11 191)(12 90)(13 189)(14 88)(15 187)(16 86)(17 185)(18 84)(19 183)(20 82)(21 181)(22 80)(23 179)(24 78)(25 177)(26 76)(27 175)(28 74)(29 173)(30 72)(31 171)(32 70)(33 169)(34 68)(35 167)(36 66)(37 165)(38 64)(39 163)(40 62)(41 161)(42 60)(43 159)(44 58)(45 157)(46 56)(47 155)(48 54)(49 153)(50 52)(51 151)(53 149)(55 147)(57 145)(59 143)(61 141)(63 139)(65 137)(67 135)(69 133)(71 131)(73 129)(75 127)(77 125)(79 123)(81 121)(83 119)(85 117)(87 115)(89 113)(91 111)(93 109)(95 107)(97 105)(99 103)(102 200)(104 198)(106 196)(108 194)(110 192)(112 190)(114 188)(116 186)(118 184)(120 182)(122 180)(124 178)(126 176)(128 174)(130 172)(132 170)(134 168)(136 166)(138 164)(140 162)(142 160)(144 158)(146 156)(148 154)(150 152)

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

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

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

103 conjugacy classes

class 1 2A2B4A4B5A5B8A8B10A10B20A20B20C20D25A···25J40A···40H50A···50J100A···100T200A···200AN
order12244558810102020202025···2540···4050···50100···100200···200
size11100210022222222222···22···22···22···22···2

103 irreducible representations

dim11112222222222
type+++++++++++
imageC1C2C2C2D4D5SD16D10D20D25C40⋊C2D50D100C200⋊C2
kernelC200⋊C2C200Dic50D100C50C40C25C20C10C8C5C4C2C1
# reps111112224108102040

Matrix representation of C200⋊C2 in GL4(𝔽401) generated by

22913100
2162900
00549
00392251
,
1000
7540000
0010
00111400
G:=sub<GL(4,GF(401))| [229,216,0,0,131,29,0,0,0,0,54,392,0,0,9,251],[1,75,0,0,0,400,0,0,0,0,1,111,0,0,0,400] >;

C200⋊C2 in GAP, Magma, Sage, TeX

C_{200}\rtimes C_2
% in TeX

G:=Group("C200:C2");
// GroupNames label

G:=SmallGroup(400,7);
// by ID

G=gap.SmallGroup(400,7);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,73,31,218,50,4324,628,11525]);
// Polycyclic

G:=Group<a,b|a^200=b^2=1,b*a*b=a^99>;
// generators/relations

Export

Subgroup lattice of C200⋊C2 in TeX

׿
×
𝔽