direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C61, C4⋊C122, C244⋊3C2, C22⋊C122, C122.6C22, (C2×C122)⋊1C2, C2.1(C2×C122), SmallGroup(488,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C61
G = < a,b,c | a61=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D4×C61
►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)
(1 227 68 165)(2 228 69 166)(3 229 70 167)(4 230 71 168)(5 231 72 169)(6 232 73 170)(7 233 74 171)(8 234 75 172)(9 235 76 173)(10 236 77 174)(11 237 78 175)(12 238 79 176)(13 239 80 177)(14 240 81 178)(15 241 82 179)(16 242 83 180)(17 243 84 181)(18 244 85 182)(19 184 86 183)(20 185 87 123)(21 186 88 124)(22 187 89 125)(23 188 90 126)(24 189 91 127)(25 190 92 128)(26 191 93 129)(27 192 94 130)(28 193 95 131)(29 194 96 132)(30 195 97 133)(31 196 98 134)(32 197 99 135)(33 198 100 136)(34 199 101 137)(35 200 102 138)(36 201 103 139)(37 202 104 140)(38 203 105 141)(39 204 106 142)(40 205 107 143)(41 206 108 144)(42 207 109 145)(43 208 110 146)(44 209 111 147)(45 210 112 148)(46 211 113 149)(47 212 114 150)(48 213 115 151)(49 214 116 152)(50 215 117 153)(51 216 118 154)(52 217 119 155)(53 218 120 156)(54 219 121 157)(55 220 122 158)(56 221 62 159)(57 222 63 160)(58 223 64 161)(59 224 65 162)(60 225 66 163)(61 226 67 164)
(123 185)(124 186)(125 187)(126 188)(127 189)(128 190)(129 191)(130 192)(131 193)(132 194)(133 195)(134 196)(135 197)(136 198)(137 199)(138 200)(139 201)(140 202)(141 203)(142 204)(143 205)(144 206)(145 207)(146 208)(147 209)(148 210)(149 211)(150 212)(151 213)(152 214)(153 215)(154 216)(155 217)(156 218)(157 219)(158 220)(159 221)(160 222)(161 223)(162 224)(163 225)(164 226)(165 227)(166 228)(167 229)(168 230)(169 231)(170 232)(171 233)(172 234)(173 235)(174 236)(175 237)(176 238)(177 239)(178 240)(179 241)(180 242)(181 243)(182 244)(183 184)
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), (1,227,68,165)(2,228,69,166)(3,229,70,167)(4,230,71,168)(5,231,72,169)(6,232,73,170)(7,233,74,171)(8,234,75,172)(9,235,76,173)(10,236,77,174)(11,237,78,175)(12,238,79,176)(13,239,80,177)(14,240,81,178)(15,241,82,179)(16,242,83,180)(17,243,84,181)(18,244,85,182)(19,184,86,183)(20,185,87,123)(21,186,88,124)(22,187,89,125)(23,188,90,126)(24,189,91,127)(25,190,92,128)(26,191,93,129)(27,192,94,130)(28,193,95,131)(29,194,96,132)(30,195,97,133)(31,196,98,134)(32,197,99,135)(33,198,100,136)(34,199,101,137)(35,200,102,138)(36,201,103,139)(37,202,104,140)(38,203,105,141)(39,204,106,142)(40,205,107,143)(41,206,108,144)(42,207,109,145)(43,208,110,146)(44,209,111,147)(45,210,112,148)(46,211,113,149)(47,212,114,150)(48,213,115,151)(49,214,116,152)(50,215,117,153)(51,216,118,154)(52,217,119,155)(53,218,120,156)(54,219,121,157)(55,220,122,158)(56,221,62,159)(57,222,63,160)(58,223,64,161)(59,224,65,162)(60,225,66,163)(61,226,67,164), (123,185)(124,186)(125,187)(126,188)(127,189)(128,190)(129,191)(130,192)(131,193)(132,194)(133,195)(134,196)(135,197)(136,198)(137,199)(138,200)(139,201)(140,202)(141,203)(142,204)(143,205)(144,206)(145,207)(146,208)(147,209)(148,210)(149,211)(150,212)(151,213)(152,214)(153,215)(154,216)(155,217)(156,218)(157,219)(158,220)(159,221)(160,222)(161,223)(162,224)(163,225)(164,226)(165,227)(166,228)(167,229)(168,230)(169,231)(170,232)(171,233)(172,234)(173,235)(174,236)(175,237)(176,238)(177,239)(178,240)(179,241)(180,242)(181,243)(182,244)(183,184)>;
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), (1,227,68,165)(2,228,69,166)(3,229,70,167)(4,230,71,168)(5,231,72,169)(6,232,73,170)(7,233,74,171)(8,234,75,172)(9,235,76,173)(10,236,77,174)(11,237,78,175)(12,238,79,176)(13,239,80,177)(14,240,81,178)(15,241,82,179)(16,242,83,180)(17,243,84,181)(18,244,85,182)(19,184,86,183)(20,185,87,123)(21,186,88,124)(22,187,89,125)(23,188,90,126)(24,189,91,127)(25,190,92,128)(26,191,93,129)(27,192,94,130)(28,193,95,131)(29,194,96,132)(30,195,97,133)(31,196,98,134)(32,197,99,135)(33,198,100,136)(34,199,101,137)(35,200,102,138)(36,201,103,139)(37,202,104,140)(38,203,105,141)(39,204,106,142)(40,205,107,143)(41,206,108,144)(42,207,109,145)(43,208,110,146)(44,209,111,147)(45,210,112,148)(46,211,113,149)(47,212,114,150)(48,213,115,151)(49,214,116,152)(50,215,117,153)(51,216,118,154)(52,217,119,155)(53,218,120,156)(54,219,121,157)(55,220,122,158)(56,221,62,159)(57,222,63,160)(58,223,64,161)(59,224,65,162)(60,225,66,163)(61,226,67,164), (123,185)(124,186)(125,187)(126,188)(127,189)(128,190)(129,191)(130,192)(131,193)(132,194)(133,195)(134,196)(135,197)(136,198)(137,199)(138,200)(139,201)(140,202)(141,203)(142,204)(143,205)(144,206)(145,207)(146,208)(147,209)(148,210)(149,211)(150,212)(151,213)(152,214)(153,215)(154,216)(155,217)(156,218)(157,219)(158,220)(159,221)(160,222)(161,223)(162,224)(163,225)(164,226)(165,227)(166,228)(167,229)(168,230)(169,231)(170,232)(171,233)(172,234)(173,235)(174,236)(175,237)(176,238)(177,239)(178,240)(179,241)(180,242)(181,243)(182,244)(183,184) );
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)], [(1,227,68,165),(2,228,69,166),(3,229,70,167),(4,230,71,168),(5,231,72,169),(6,232,73,170),(7,233,74,171),(8,234,75,172),(9,235,76,173),(10,236,77,174),(11,237,78,175),(12,238,79,176),(13,239,80,177),(14,240,81,178),(15,241,82,179),(16,242,83,180),(17,243,84,181),(18,244,85,182),(19,184,86,183),(20,185,87,123),(21,186,88,124),(22,187,89,125),(23,188,90,126),(24,189,91,127),(25,190,92,128),(26,191,93,129),(27,192,94,130),(28,193,95,131),(29,194,96,132),(30,195,97,133),(31,196,98,134),(32,197,99,135),(33,198,100,136),(34,199,101,137),(35,200,102,138),(36,201,103,139),(37,202,104,140),(38,203,105,141),(39,204,106,142),(40,205,107,143),(41,206,108,144),(42,207,109,145),(43,208,110,146),(44,209,111,147),(45,210,112,148),(46,211,113,149),(47,212,114,150),(48,213,115,151),(49,214,116,152),(50,215,117,153),(51,216,118,154),(52,217,119,155),(53,218,120,156),(54,219,121,157),(55,220,122,158),(56,221,62,159),(57,222,63,160),(58,223,64,161),(59,224,65,162),(60,225,66,163),(61,226,67,164)], [(123,185),(124,186),(125,187),(126,188),(127,189),(128,190),(129,191),(130,192),(131,193),(132,194),(133,195),(134,196),(135,197),(136,198),(137,199),(138,200),(139,201),(140,202),(141,203),(142,204),(143,205),(144,206),(145,207),(146,208),(147,209),(148,210),(149,211),(150,212),(151,213),(152,214),(153,215),(154,216),(155,217),(156,218),(157,219),(158,220),(159,221),(160,222),(161,223),(162,224),(163,225),(164,226),(165,227),(166,228),(167,229),(168,230),(169,231),(170,232),(171,233),(172,234),(173,235),(174,236),(175,237),(176,238),(177,239),(178,240),(179,241),(180,242),(181,243),(182,244),(183,184)]])
305 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 61A | ··· | 61BH | 122A | ··· | 122BH | 122BI | ··· | 122FX | 244A | ··· | 244BH |
| order | 1 | 2 | 2 | 2 | 4 | 61 | ··· | 61 | 122 | ··· | 122 | 122 | ··· | 122 | 244 | ··· | 244 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
305 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C61 | C122 | C122 | D4 | D4×C61 |
| kernel | D4×C61 | C244 | C2×C122 | D4 | C4 | C22 | C61 | C1 |
| # reps | 1 | 1 | 2 | 60 | 60 | 120 | 1 | 60 |
Matrix representation of D4×C61 ►in GL2(𝔽733) generated by
| 612 | 0 |
| 0 | 612 |
| 0 | 732 |
| 1 | 0 |
| 1 | 0 |
| 0 | 732 |
G:=sub<GL(2,GF(733))| [612,0,0,612],[0,1,732,0],[1,0,0,732] >;
D4×C61 in GAP, Magma, Sage, TeX
D_4\times C_{61} % in TeX
G:=Group("D4xC61"); // GroupNames label
G:=SmallGroup(488,10);
// by ID
G=gap.SmallGroup(488,10);
# by ID
G:=PCGroup([4,-2,-2,-61,-2,1969]);
// Polycyclic
G:=Group<a,b,c|a^61=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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