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G = Dic62order 248 = 23·31

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic62, C31⋊Q8, C4.D31, C2.3D62, C124.1C2, Dic31.C2, C62.1C22, SmallGroup(248,3)

Series: Derived Chief Lower central Upper central

C1C62 — Dic62
C1C31C62Dic31 — Dic62
C31C62 — Dic62
C1C2C4

Generators and relations for Dic62
 G = < a,b | a124=1, b2=a62, bab-1=a-1 >

31C4
31C4
31Q8

Smallest permutation representation of Dic62
Regular action on 248 points
Generators in S248
(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)(125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248)
(1 238 63 176)(2 237 64 175)(3 236 65 174)(4 235 66 173)(5 234 67 172)(6 233 68 171)(7 232 69 170)(8 231 70 169)(9 230 71 168)(10 229 72 167)(11 228 73 166)(12 227 74 165)(13 226 75 164)(14 225 76 163)(15 224 77 162)(16 223 78 161)(17 222 79 160)(18 221 80 159)(19 220 81 158)(20 219 82 157)(21 218 83 156)(22 217 84 155)(23 216 85 154)(24 215 86 153)(25 214 87 152)(26 213 88 151)(27 212 89 150)(28 211 90 149)(29 210 91 148)(30 209 92 147)(31 208 93 146)(32 207 94 145)(33 206 95 144)(34 205 96 143)(35 204 97 142)(36 203 98 141)(37 202 99 140)(38 201 100 139)(39 200 101 138)(40 199 102 137)(41 198 103 136)(42 197 104 135)(43 196 105 134)(44 195 106 133)(45 194 107 132)(46 193 108 131)(47 192 109 130)(48 191 110 129)(49 190 111 128)(50 189 112 127)(51 188 113 126)(52 187 114 125)(53 186 115 248)(54 185 116 247)(55 184 117 246)(56 183 118 245)(57 182 119 244)(58 181 120 243)(59 180 121 242)(60 179 122 241)(61 178 123 240)(62 177 124 239)

G:=sub<Sym(248)| (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)(125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248), (1,238,63,176)(2,237,64,175)(3,236,65,174)(4,235,66,173)(5,234,67,172)(6,233,68,171)(7,232,69,170)(8,231,70,169)(9,230,71,168)(10,229,72,167)(11,228,73,166)(12,227,74,165)(13,226,75,164)(14,225,76,163)(15,224,77,162)(16,223,78,161)(17,222,79,160)(18,221,80,159)(19,220,81,158)(20,219,82,157)(21,218,83,156)(22,217,84,155)(23,216,85,154)(24,215,86,153)(25,214,87,152)(26,213,88,151)(27,212,89,150)(28,211,90,149)(29,210,91,148)(30,209,92,147)(31,208,93,146)(32,207,94,145)(33,206,95,144)(34,205,96,143)(35,204,97,142)(36,203,98,141)(37,202,99,140)(38,201,100,139)(39,200,101,138)(40,199,102,137)(41,198,103,136)(42,197,104,135)(43,196,105,134)(44,195,106,133)(45,194,107,132)(46,193,108,131)(47,192,109,130)(48,191,110,129)(49,190,111,128)(50,189,112,127)(51,188,113,126)(52,187,114,125)(53,186,115,248)(54,185,116,247)(55,184,117,246)(56,183,118,245)(57,182,119,244)(58,181,120,243)(59,180,121,242)(60,179,122,241)(61,178,123,240)(62,177,124,239)>;

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)(125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248), (1,238,63,176)(2,237,64,175)(3,236,65,174)(4,235,66,173)(5,234,67,172)(6,233,68,171)(7,232,69,170)(8,231,70,169)(9,230,71,168)(10,229,72,167)(11,228,73,166)(12,227,74,165)(13,226,75,164)(14,225,76,163)(15,224,77,162)(16,223,78,161)(17,222,79,160)(18,221,80,159)(19,220,81,158)(20,219,82,157)(21,218,83,156)(22,217,84,155)(23,216,85,154)(24,215,86,153)(25,214,87,152)(26,213,88,151)(27,212,89,150)(28,211,90,149)(29,210,91,148)(30,209,92,147)(31,208,93,146)(32,207,94,145)(33,206,95,144)(34,205,96,143)(35,204,97,142)(36,203,98,141)(37,202,99,140)(38,201,100,139)(39,200,101,138)(40,199,102,137)(41,198,103,136)(42,197,104,135)(43,196,105,134)(44,195,106,133)(45,194,107,132)(46,193,108,131)(47,192,109,130)(48,191,110,129)(49,190,111,128)(50,189,112,127)(51,188,113,126)(52,187,114,125)(53,186,115,248)(54,185,116,247)(55,184,117,246)(56,183,118,245)(57,182,119,244)(58,181,120,243)(59,180,121,242)(60,179,122,241)(61,178,123,240)(62,177,124,239) );

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),(125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248)], [(1,238,63,176),(2,237,64,175),(3,236,65,174),(4,235,66,173),(5,234,67,172),(6,233,68,171),(7,232,69,170),(8,231,70,169),(9,230,71,168),(10,229,72,167),(11,228,73,166),(12,227,74,165),(13,226,75,164),(14,225,76,163),(15,224,77,162),(16,223,78,161),(17,222,79,160),(18,221,80,159),(19,220,81,158),(20,219,82,157),(21,218,83,156),(22,217,84,155),(23,216,85,154),(24,215,86,153),(25,214,87,152),(26,213,88,151),(27,212,89,150),(28,211,90,149),(29,210,91,148),(30,209,92,147),(31,208,93,146),(32,207,94,145),(33,206,95,144),(34,205,96,143),(35,204,97,142),(36,203,98,141),(37,202,99,140),(38,201,100,139),(39,200,101,138),(40,199,102,137),(41,198,103,136),(42,197,104,135),(43,196,105,134),(44,195,106,133),(45,194,107,132),(46,193,108,131),(47,192,109,130),(48,191,110,129),(49,190,111,128),(50,189,112,127),(51,188,113,126),(52,187,114,125),(53,186,115,248),(54,185,116,247),(55,184,117,246),(56,183,118,245),(57,182,119,244),(58,181,120,243),(59,180,121,242),(60,179,122,241),(61,178,123,240),(62,177,124,239)])

Dic62 is a maximal subgroup of   C248⋊C2  Dic124  D4.D31  C31⋊Q16  D1245C2  D42D31  Q8×D31
Dic62 is a maximal quotient of   Dic31⋊C4  C4⋊Dic31

65 conjugacy classes

class 1  2 4A4B4C31A···31O62A···62O124A···124AD
order1244431···3162···62124···124
size11262622···22···22···2

65 irreducible representations

dim1112222
type+++-++-
imageC1C2C2Q8D31D62Dic62
kernelDic62Dic31C124C31C4C2C1
# reps1211151530

Matrix representation of Dic62 in GL2(𝔽373) generated by

111284
31027
,
368115
1365
G:=sub<GL(2,GF(373))| [111,310,284,27],[368,136,115,5] >;

Dic62 in GAP, Magma, Sage, TeX

{\rm Dic}_{62}
% in TeX

G:=Group("Dic62");
// GroupNames label

G:=SmallGroup(248,3);
// by ID

G=gap.SmallGroup(248,3);
# by ID

G:=PCGroup([4,-2,-2,-2,-31,16,49,21,3843]);
// Polycyclic

G:=Group<a,b|a^124=1,b^2=a^62,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic62 in TeX

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