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G = D245order 490 = 2·5·72

Dihedral group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: D245, C49⋊D5, C5⋊D49, C7.D35, C2451C2, C35.1D7, sometimes denoted D490 or Dih245 or Dih490, SmallGroup(490,3)

Series: Derived Chief Lower central Upper central

C1C245 — D245
C1C7C49C245 — D245
C245 — D245
C1

Generators and relations for D245
 G = < a,b | a245=b2=1, bab=a-1 >

245C2
49D5
35D7
7D35
5D49

Smallest permutation representation of D245
On 245 points
Generators in S245
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245)
(2 245)(3 244)(4 243)(5 242)(6 241)(7 240)(8 239)(9 238)(10 237)(11 236)(12 235)(13 234)(14 233)(15 232)(16 231)(17 230)(18 229)(19 228)(20 227)(21 226)(22 225)(23 224)(24 223)(25 222)(26 221)(27 220)(28 219)(29 218)(30 217)(31 216)(32 215)(33 214)(34 213)(35 212)(36 211)(37 210)(38 209)(39 208)(40 207)(41 206)(42 205)(43 204)(44 203)(45 202)(46 201)(47 200)(48 199)(49 198)(50 197)(51 196)(52 195)(53 194)(54 193)(55 192)(56 191)(57 190)(58 189)(59 188)(60 187)(61 186)(62 185)(63 184)(64 183)(65 182)(66 181)(67 180)(68 179)(69 178)(70 177)(71 176)(72 175)(73 174)(74 173)(75 172)(76 171)(77 170)(78 169)(79 168)(80 167)(81 166)(82 165)(83 164)(84 163)(85 162)(86 161)(87 160)(88 159)(89 158)(90 157)(91 156)(92 155)(93 154)(94 153)(95 152)(96 151)(97 150)(98 149)(99 148)(100 147)(101 146)(102 145)(103 144)(104 143)(105 142)(106 141)(107 140)(108 139)(109 138)(110 137)(111 136)(112 135)(113 134)(114 133)(115 132)(116 131)(117 130)(118 129)(119 128)(120 127)(121 126)(122 125)(123 124)

G:=sub<Sym(245)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245), (2,245)(3,244)(4,243)(5,242)(6,241)(7,240)(8,239)(9,238)(10,237)(11,236)(12,235)(13,234)(14,233)(15,232)(16,231)(17,230)(18,229)(19,228)(20,227)(21,226)(22,225)(23,224)(24,223)(25,222)(26,221)(27,220)(28,219)(29,218)(30,217)(31,216)(32,215)(33,214)(34,213)(35,212)(36,211)(37,210)(38,209)(39,208)(40,207)(41,206)(42,205)(43,204)(44,203)(45,202)(46,201)(47,200)(48,199)(49,198)(50,197)(51,196)(52,195)(53,194)(54,193)(55,192)(56,191)(57,190)(58,189)(59,188)(60,187)(61,186)(62,185)(63,184)(64,183)(65,182)(66,181)(67,180)(68,179)(69,178)(70,177)(71,176)(72,175)(73,174)(74,173)(75,172)(76,171)(77,170)(78,169)(79,168)(80,167)(81,166)(82,165)(83,164)(84,163)(85,162)(86,161)(87,160)(88,159)(89,158)(90,157)(91,156)(92,155)(93,154)(94,153)(95,152)(96,151)(97,150)(98,149)(99,148)(100,147)(101,146)(102,145)(103,144)(104,143)(105,142)(106,141)(107,140)(108,139)(109,138)(110,137)(111,136)(112,135)(113,134)(114,133)(115,132)(116,131)(117,130)(118,129)(119,128)(120,127)(121,126)(122,125)(123,124)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245), (2,245)(3,244)(4,243)(5,242)(6,241)(7,240)(8,239)(9,238)(10,237)(11,236)(12,235)(13,234)(14,233)(15,232)(16,231)(17,230)(18,229)(19,228)(20,227)(21,226)(22,225)(23,224)(24,223)(25,222)(26,221)(27,220)(28,219)(29,218)(30,217)(31,216)(32,215)(33,214)(34,213)(35,212)(36,211)(37,210)(38,209)(39,208)(40,207)(41,206)(42,205)(43,204)(44,203)(45,202)(46,201)(47,200)(48,199)(49,198)(50,197)(51,196)(52,195)(53,194)(54,193)(55,192)(56,191)(57,190)(58,189)(59,188)(60,187)(61,186)(62,185)(63,184)(64,183)(65,182)(66,181)(67,180)(68,179)(69,178)(70,177)(71,176)(72,175)(73,174)(74,173)(75,172)(76,171)(77,170)(78,169)(79,168)(80,167)(81,166)(82,165)(83,164)(84,163)(85,162)(86,161)(87,160)(88,159)(89,158)(90,157)(91,156)(92,155)(93,154)(94,153)(95,152)(96,151)(97,150)(98,149)(99,148)(100,147)(101,146)(102,145)(103,144)(104,143)(105,142)(106,141)(107,140)(108,139)(109,138)(110,137)(111,136)(112,135)(113,134)(114,133)(115,132)(116,131)(117,130)(118,129)(119,128)(120,127)(121,126)(122,125)(123,124) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245)], [(2,245),(3,244),(4,243),(5,242),(6,241),(7,240),(8,239),(9,238),(10,237),(11,236),(12,235),(13,234),(14,233),(15,232),(16,231),(17,230),(18,229),(19,228),(20,227),(21,226),(22,225),(23,224),(24,223),(25,222),(26,221),(27,220),(28,219),(29,218),(30,217),(31,216),(32,215),(33,214),(34,213),(35,212),(36,211),(37,210),(38,209),(39,208),(40,207),(41,206),(42,205),(43,204),(44,203),(45,202),(46,201),(47,200),(48,199),(49,198),(50,197),(51,196),(52,195),(53,194),(54,193),(55,192),(56,191),(57,190),(58,189),(59,188),(60,187),(61,186),(62,185),(63,184),(64,183),(65,182),(66,181),(67,180),(68,179),(69,178),(70,177),(71,176),(72,175),(73,174),(74,173),(75,172),(76,171),(77,170),(78,169),(79,168),(80,167),(81,166),(82,165),(83,164),(84,163),(85,162),(86,161),(87,160),(88,159),(89,158),(90,157),(91,156),(92,155),(93,154),(94,153),(95,152),(96,151),(97,150),(98,149),(99,148),(100,147),(101,146),(102,145),(103,144),(104,143),(105,142),(106,141),(107,140),(108,139),(109,138),(110,137),(111,136),(112,135),(113,134),(114,133),(115,132),(116,131),(117,130),(118,129),(119,128),(120,127),(121,126),(122,125),(123,124)])

124 conjugacy classes

class 1  2 5A5B7A7B7C35A···35L49A···49U245A···245CF
order125577735···3549···49245···245
size1245222222···22···22···2

124 irreducible representations

dim1122222
type+++++++
imageC1C2D5D7D35D49D245
kernelD245C245C49C35C7C5C1
# reps1123122184

Matrix representation of D245 in GL2(𝔽491) generated by

12994
397354
,
10
227490
G:=sub<GL(2,GF(491))| [129,397,94,354],[1,227,0,490] >;

D245 in GAP, Magma, Sage, TeX

D_{245}
% in TeX

G:=Group("D245");
// GroupNames label

G:=SmallGroup(490,3);
// by ID

G=gap.SmallGroup(490,3);
# by ID

G:=PCGroup([4,-2,-5,-7,-7,65,1562,514,6723]);
// Polycyclic

G:=Group<a,b|a^245=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D245 in TeX

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