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
Generators and relations for Dic62
G = < a,b | a124=1, b2=a62, bab-1=a-1 >
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)
(1 194 63 132)(2 193 64 131)(3 192 65 130)(4 191 66 129)(5 190 67 128)(6 189 68 127)(7 188 69 126)(8 187 70 125)(9 186 71 248)(10 185 72 247)(11 184 73 246)(12 183 74 245)(13 182 75 244)(14 181 76 243)(15 180 77 242)(16 179 78 241)(17 178 79 240)(18 177 80 239)(19 176 81 238)(20 175 82 237)(21 174 83 236)(22 173 84 235)(23 172 85 234)(24 171 86 233)(25 170 87 232)(26 169 88 231)(27 168 89 230)(28 167 90 229)(29 166 91 228)(30 165 92 227)(31 164 93 226)(32 163 94 225)(33 162 95 224)(34 161 96 223)(35 160 97 222)(36 159 98 221)(37 158 99 220)(38 157 100 219)(39 156 101 218)(40 155 102 217)(41 154 103 216)(42 153 104 215)(43 152 105 214)(44 151 106 213)(45 150 107 212)(46 149 108 211)(47 148 109 210)(48 147 110 209)(49 146 111 208)(50 145 112 207)(51 144 113 206)(52 143 114 205)(53 142 115 204)(54 141 116 203)(55 140 117 202)(56 139 118 201)(57 138 119 200)(58 137 120 199)(59 136 121 198)(60 135 122 197)(61 134 123 196)(62 133 124 195)
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), (1,194,63,132)(2,193,64,131)(3,192,65,130)(4,191,66,129)(5,190,67,128)(6,189,68,127)(7,188,69,126)(8,187,70,125)(9,186,71,248)(10,185,72,247)(11,184,73,246)(12,183,74,245)(13,182,75,244)(14,181,76,243)(15,180,77,242)(16,179,78,241)(17,178,79,240)(18,177,80,239)(19,176,81,238)(20,175,82,237)(21,174,83,236)(22,173,84,235)(23,172,85,234)(24,171,86,233)(25,170,87,232)(26,169,88,231)(27,168,89,230)(28,167,90,229)(29,166,91,228)(30,165,92,227)(31,164,93,226)(32,163,94,225)(33,162,95,224)(34,161,96,223)(35,160,97,222)(36,159,98,221)(37,158,99,220)(38,157,100,219)(39,156,101,218)(40,155,102,217)(41,154,103,216)(42,153,104,215)(43,152,105,214)(44,151,106,213)(45,150,107,212)(46,149,108,211)(47,148,109,210)(48,147,110,209)(49,146,111,208)(50,145,112,207)(51,144,113,206)(52,143,114,205)(53,142,115,204)(54,141,116,203)(55,140,117,202)(56,139,118,201)(57,138,119,200)(58,137,120,199)(59,136,121,198)(60,135,122,197)(61,134,123,196)(62,133,124,195)>;
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), (1,194,63,132)(2,193,64,131)(3,192,65,130)(4,191,66,129)(5,190,67,128)(6,189,68,127)(7,188,69,126)(8,187,70,125)(9,186,71,248)(10,185,72,247)(11,184,73,246)(12,183,74,245)(13,182,75,244)(14,181,76,243)(15,180,77,242)(16,179,78,241)(17,178,79,240)(18,177,80,239)(19,176,81,238)(20,175,82,237)(21,174,83,236)(22,173,84,235)(23,172,85,234)(24,171,86,233)(25,170,87,232)(26,169,88,231)(27,168,89,230)(28,167,90,229)(29,166,91,228)(30,165,92,227)(31,164,93,226)(32,163,94,225)(33,162,95,224)(34,161,96,223)(35,160,97,222)(36,159,98,221)(37,158,99,220)(38,157,100,219)(39,156,101,218)(40,155,102,217)(41,154,103,216)(42,153,104,215)(43,152,105,214)(44,151,106,213)(45,150,107,212)(46,149,108,211)(47,148,109,210)(48,147,110,209)(49,146,111,208)(50,145,112,207)(51,144,113,206)(52,143,114,205)(53,142,115,204)(54,141,116,203)(55,140,117,202)(56,139,118,201)(57,138,119,200)(58,137,120,199)(59,136,121,198)(60,135,122,197)(61,134,123,196)(62,133,124,195) );
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)], [(1,194,63,132),(2,193,64,131),(3,192,65,130),(4,191,66,129),(5,190,67,128),(6,189,68,127),(7,188,69,126),(8,187,70,125),(9,186,71,248),(10,185,72,247),(11,184,73,246),(12,183,74,245),(13,182,75,244),(14,181,76,243),(15,180,77,242),(16,179,78,241),(17,178,79,240),(18,177,80,239),(19,176,81,238),(20,175,82,237),(21,174,83,236),(22,173,84,235),(23,172,85,234),(24,171,86,233),(25,170,87,232),(26,169,88,231),(27,168,89,230),(28,167,90,229),(29,166,91,228),(30,165,92,227),(31,164,93,226),(32,163,94,225),(33,162,95,224),(34,161,96,223),(35,160,97,222),(36,159,98,221),(37,158,99,220),(38,157,100,219),(39,156,101,218),(40,155,102,217),(41,154,103,216),(42,153,104,215),(43,152,105,214),(44,151,106,213),(45,150,107,212),(46,149,108,211),(47,148,109,210),(48,147,110,209),(49,146,111,208),(50,145,112,207),(51,144,113,206),(52,143,114,205),(53,142,115,204),(54,141,116,203),(55,140,117,202),(56,139,118,201),(57,138,119,200),(58,137,120,199),(59,136,121,198),(60,135,122,197),(61,134,123,196),(62,133,124,195)]])
Dic62 is a maximal subgroup of
C248⋊C2 Dic124 D4.D31 C31⋊Q16 D124⋊5C2 D4⋊2D31 Q8×D31
Dic62 is a maximal quotient of Dic31⋊C4 C4⋊Dic31
65 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 31A | ··· | 31O | 62A | ··· | 62O | 124A | ··· | 124AD |
| order | 1 | 2 | 4 | 4 | 4 | 31 | ··· | 31 | 62 | ··· | 62 | 124 | ··· | 124 |
| size | 1 | 1 | 2 | 62 | 62 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
65 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | Q8 | D31 | D62 | Dic62 |
| kernel | Dic62 | Dic31 | C124 | C31 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 15 | 15 | 30 |
Matrix representation of Dic62 ►in GL2(𝔽373) generated by
| 111 | 284 |
| 310 | 27 |
| 368 | 115 |
| 136 | 5 |
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