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G = D246order 492 = 22·3·41

Dihedral group

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D246, C2×D123, C82⋊S3, C6⋊D41, C32D82, C412D6, C2461C2, C1232C22, sometimes denoted D492 or Dih246 or Dih492, SmallGroup(492,11)

Series: Derived Chief Lower central Upper central

C1C123 — D246
C1C41C123D123 — D246
C123 — D246
C1C2

Generators and relations for D246
 G = < a,b | a246=b2=1, bab=a-1 >

123C2
123C2
123C22
41S3
41S3
3D41
3D41
41D6
3D82

Smallest permutation representation of D246
On 246 points
Generators in S246
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246)
(1 246)(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(246)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246), (1,246)(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( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246), (1,246)(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([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246)], [(1,246),(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)])

126 conjugacy classes

class 1 2A2B2C 3  6 41A···41T82A···82T123A···123AN246A···246AN
order12223641···4182···82123···123246···246
size11123123222···22···22···22···2

126 irreducible representations

dim111222222
type+++++++++
imageC1C2C2S3D6D41D82D123D246
kernelD246D123C246C82C41C6C3C2C1
# reps1211120204040

Matrix representation of D246 in GL3(𝔽739) generated by

73800
0221731
0890
,
100
0221731
0193518
G:=sub<GL(3,GF(739))| [738,0,0,0,221,8,0,731,90],[1,0,0,0,221,193,0,731,518] >;

D246 in GAP, Magma, Sage, TeX

D_{246}
% in TeX

G:=Group("D246");
// GroupNames label

G:=SmallGroup(492,11);
// by ID

G=gap.SmallGroup(492,11);
# by ID

G:=PCGroup([4,-2,-2,-3,-41,98,7683]);
// Polycyclic

G:=Group<a,b|a^246=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D246 in TeX

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