metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic61, C61⋊2C4, C2.D61, C122.C2, SmallGroup(244,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C61 — Dic61 |
Generators and relations for Dic61
G = < a,b | a122=1, b2=a61, bab-1=a-1 >
Smallest permutation representation of Dic61
►Regular action on 244 points
Generators in S244
(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)
(1 236 62 175)(2 235 63 174)(3 234 64 173)(4 233 65 172)(5 232 66 171)(6 231 67 170)(7 230 68 169)(8 229 69 168)(9 228 70 167)(10 227 71 166)(11 226 72 165)(12 225 73 164)(13 224 74 163)(14 223 75 162)(15 222 76 161)(16 221 77 160)(17 220 78 159)(18 219 79 158)(19 218 80 157)(20 217 81 156)(21 216 82 155)(22 215 83 154)(23 214 84 153)(24 213 85 152)(25 212 86 151)(26 211 87 150)(27 210 88 149)(28 209 89 148)(29 208 90 147)(30 207 91 146)(31 206 92 145)(32 205 93 144)(33 204 94 143)(34 203 95 142)(35 202 96 141)(36 201 97 140)(37 200 98 139)(38 199 99 138)(39 198 100 137)(40 197 101 136)(41 196 102 135)(42 195 103 134)(43 194 104 133)(44 193 105 132)(45 192 106 131)(46 191 107 130)(47 190 108 129)(48 189 109 128)(49 188 110 127)(50 187 111 126)(51 186 112 125)(52 185 113 124)(53 184 114 123)(54 183 115 244)(55 182 116 243)(56 181 117 242)(57 180 118 241)(58 179 119 240)(59 178 120 239)(60 177 121 238)(61 176 122 237)
G:=sub<Sym(244)| (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), (1,236,62,175)(2,235,63,174)(3,234,64,173)(4,233,65,172)(5,232,66,171)(6,231,67,170)(7,230,68,169)(8,229,69,168)(9,228,70,167)(10,227,71,166)(11,226,72,165)(12,225,73,164)(13,224,74,163)(14,223,75,162)(15,222,76,161)(16,221,77,160)(17,220,78,159)(18,219,79,158)(19,218,80,157)(20,217,81,156)(21,216,82,155)(22,215,83,154)(23,214,84,153)(24,213,85,152)(25,212,86,151)(26,211,87,150)(27,210,88,149)(28,209,89,148)(29,208,90,147)(30,207,91,146)(31,206,92,145)(32,205,93,144)(33,204,94,143)(34,203,95,142)(35,202,96,141)(36,201,97,140)(37,200,98,139)(38,199,99,138)(39,198,100,137)(40,197,101,136)(41,196,102,135)(42,195,103,134)(43,194,104,133)(44,193,105,132)(45,192,106,131)(46,191,107,130)(47,190,108,129)(48,189,109,128)(49,188,110,127)(50,187,111,126)(51,186,112,125)(52,185,113,124)(53,184,114,123)(54,183,115,244)(55,182,116,243)(56,181,117,242)(57,180,118,241)(58,179,119,240)(59,178,120,239)(60,177,121,238)(61,176,122,237)>;
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), (1,236,62,175)(2,235,63,174)(3,234,64,173)(4,233,65,172)(5,232,66,171)(6,231,67,170)(7,230,68,169)(8,229,69,168)(9,228,70,167)(10,227,71,166)(11,226,72,165)(12,225,73,164)(13,224,74,163)(14,223,75,162)(15,222,76,161)(16,221,77,160)(17,220,78,159)(18,219,79,158)(19,218,80,157)(20,217,81,156)(21,216,82,155)(22,215,83,154)(23,214,84,153)(24,213,85,152)(25,212,86,151)(26,211,87,150)(27,210,88,149)(28,209,89,148)(29,208,90,147)(30,207,91,146)(31,206,92,145)(32,205,93,144)(33,204,94,143)(34,203,95,142)(35,202,96,141)(36,201,97,140)(37,200,98,139)(38,199,99,138)(39,198,100,137)(40,197,101,136)(41,196,102,135)(42,195,103,134)(43,194,104,133)(44,193,105,132)(45,192,106,131)(46,191,107,130)(47,190,108,129)(48,189,109,128)(49,188,110,127)(50,187,111,126)(51,186,112,125)(52,185,113,124)(53,184,114,123)(54,183,115,244)(55,182,116,243)(56,181,117,242)(57,180,118,241)(58,179,119,240)(59,178,120,239)(60,177,121,238)(61,176,122,237) );
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)], [(1,236,62,175),(2,235,63,174),(3,234,64,173),(4,233,65,172),(5,232,66,171),(6,231,67,170),(7,230,68,169),(8,229,69,168),(9,228,70,167),(10,227,71,166),(11,226,72,165),(12,225,73,164),(13,224,74,163),(14,223,75,162),(15,222,76,161),(16,221,77,160),(17,220,78,159),(18,219,79,158),(19,218,80,157),(20,217,81,156),(21,216,82,155),(22,215,83,154),(23,214,84,153),(24,213,85,152),(25,212,86,151),(26,211,87,150),(27,210,88,149),(28,209,89,148),(29,208,90,147),(30,207,91,146),(31,206,92,145),(32,205,93,144),(33,204,94,143),(34,203,95,142),(35,202,96,141),(36,201,97,140),(37,200,98,139),(38,199,99,138),(39,198,100,137),(40,197,101,136),(41,196,102,135),(42,195,103,134),(43,194,104,133),(44,193,105,132),(45,192,106,131),(46,191,107,130),(47,190,108,129),(48,189,109,128),(49,188,110,127),(50,187,111,126),(51,186,112,125),(52,185,113,124),(53,184,114,123),(54,183,115,244),(55,182,116,243),(56,181,117,242),(57,180,118,241),(58,179,119,240),(59,178,120,239),(60,177,121,238),(61,176,122,237)]])
Dic61 is a maximal subgroup of
C61⋊C8 Dic122 C4×D61 C61⋊D4
Dic61 is a maximal quotient of C61⋊2C8
64 conjugacy classes
| class | 1 | 2 | 4A | 4B | 61A | ··· | 61AD | 122A | ··· | 122AD |
| order | 1 | 2 | 4 | 4 | 61 | ··· | 61 | 122 | ··· | 122 |
| size | 1 | 1 | 61 | 61 | 2 | ··· | 2 | 2 | ··· | 2 |
64 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | - | |
| image | C1 | C2 | C4 | D61 | Dic61 |
| kernel | Dic61 | C122 | C61 | C2 | C1 |
| # reps | 1 | 1 | 2 | 30 | 30 |
Matrix representation of Dic61 ►in GL3(𝔽733) generated by
| 732 | 0 | 0 |
| 0 | 695 | 732 |
| 0 | 1 | 0 |
| 353 | 0 | 0 |
| 0 | 90 | 162 |
| 0 | 407 | 643 |
G:=sub<GL(3,GF(733))| [732,0,0,0,695,1,0,732,0],[353,0,0,0,90,407,0,162,643] >;
Dic61 in GAP, Magma, Sage, TeX
{\rm Dic}_{61} % in TeX
G:=Group("Dic61"); // GroupNames label
G:=SmallGroup(244,1);
// by ID
G=gap.SmallGroup(244,1);
# by ID
G:=PCGroup([3,-2,-2,-61,6,2162]);
// Polycyclic
G:=Group<a,b|a^122=1,b^2=a^61,b*a*b^-1=a^-1>;
// generators/relations
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