direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D61, C244⋊2C2, C2.1D122, Dic61⋊2C2, D122.2C2, C122.2C22, C61⋊2(C2×C4), SmallGroup(488,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C61 — C4×D61 |
Generators and relations for C4×D61
G = < a,b,c | a4=b61=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C4×D61
►On 244 points
Generators in S244
(1 230 84 132)(2 231 85 133)(3 232 86 134)(4 233 87 135)(5 234 88 136)(6 235 89 137)(7 236 90 138)(8 237 91 139)(9 238 92 140)(10 239 93 141)(11 240 94 142)(12 241 95 143)(13 242 96 144)(14 243 97 145)(15 244 98 146)(16 184 99 147)(17 185 100 148)(18 186 101 149)(19 187 102 150)(20 188 103 151)(21 189 104 152)(22 190 105 153)(23 191 106 154)(24 192 107 155)(25 193 108 156)(26 194 109 157)(27 195 110 158)(28 196 111 159)(29 197 112 160)(30 198 113 161)(31 199 114 162)(32 200 115 163)(33 201 116 164)(34 202 117 165)(35 203 118 166)(36 204 119 167)(37 205 120 168)(38 206 121 169)(39 207 122 170)(40 208 62 171)(41 209 63 172)(42 210 64 173)(43 211 65 174)(44 212 66 175)(45 213 67 176)(46 214 68 177)(47 215 69 178)(48 216 70 179)(49 217 71 180)(50 218 72 181)(51 219 73 182)(52 220 74 183)(53 221 75 123)(54 222 76 124)(55 223 77 125)(56 224 78 126)(57 225 79 127)(58 226 80 128)(59 227 81 129)(60 228 82 130)(61 229 83 131)
(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 61)(2 60)(3 59)(4 58)(5 57)(6 56)(7 55)(8 54)(9 53)(10 52)(11 51)(12 50)(13 49)(14 48)(15 47)(16 46)(17 45)(18 44)(19 43)(20 42)(21 41)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(62 105)(63 104)(64 103)(65 102)(66 101)(67 100)(68 99)(69 98)(70 97)(71 96)(72 95)(73 94)(74 93)(75 92)(76 91)(77 90)(78 89)(79 88)(80 87)(81 86)(82 85)(83 84)(106 122)(107 121)(108 120)(109 119)(110 118)(111 117)(112 116)(113 115)(123 140)(124 139)(125 138)(126 137)(127 136)(128 135)(129 134)(130 133)(131 132)(141 183)(142 182)(143 181)(144 180)(145 179)(146 178)(147 177)(148 176)(149 175)(150 174)(151 173)(152 172)(153 171)(154 170)(155 169)(156 168)(157 167)(158 166)(159 165)(160 164)(161 163)(184 214)(185 213)(186 212)(187 211)(188 210)(189 209)(190 208)(191 207)(192 206)(193 205)(194 204)(195 203)(196 202)(197 201)(198 200)(215 244)(216 243)(217 242)(218 241)(219 240)(220 239)(221 238)(222 237)(223 236)(224 235)(225 234)(226 233)(227 232)(228 231)(229 230)
G:=sub<Sym(244)| (1,230,84,132)(2,231,85,133)(3,232,86,134)(4,233,87,135)(5,234,88,136)(6,235,89,137)(7,236,90,138)(8,237,91,139)(9,238,92,140)(10,239,93,141)(11,240,94,142)(12,241,95,143)(13,242,96,144)(14,243,97,145)(15,244,98,146)(16,184,99,147)(17,185,100,148)(18,186,101,149)(19,187,102,150)(20,188,103,151)(21,189,104,152)(22,190,105,153)(23,191,106,154)(24,192,107,155)(25,193,108,156)(26,194,109,157)(27,195,110,158)(28,196,111,159)(29,197,112,160)(30,198,113,161)(31,199,114,162)(32,200,115,163)(33,201,116,164)(34,202,117,165)(35,203,118,166)(36,204,119,167)(37,205,120,168)(38,206,121,169)(39,207,122,170)(40,208,62,171)(41,209,63,172)(42,210,64,173)(43,211,65,174)(44,212,66,175)(45,213,67,176)(46,214,68,177)(47,215,69,178)(48,216,70,179)(49,217,71,180)(50,218,72,181)(51,219,73,182)(52,220,74,183)(53,221,75,123)(54,222,76,124)(55,223,77,125)(56,224,78,126)(57,225,79,127)(58,226,80,128)(59,227,81,129)(60,228,82,130)(61,229,83,131), (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), (1,61)(2,60)(3,59)(4,58)(5,57)(6,56)(7,55)(8,54)(9,53)(10,52)(11,51)(12,50)(13,49)(14,48)(15,47)(16,46)(17,45)(18,44)(19,43)(20,42)(21,41)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(62,105)(63,104)(64,103)(65,102)(66,101)(67,100)(68,99)(69,98)(70,97)(71,96)(72,95)(73,94)(74,93)(75,92)(76,91)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,84)(106,122)(107,121)(108,120)(109,119)(110,118)(111,117)(112,116)(113,115)(123,140)(124,139)(125,138)(126,137)(127,136)(128,135)(129,134)(130,133)(131,132)(141,183)(142,182)(143,181)(144,180)(145,179)(146,178)(147,177)(148,176)(149,175)(150,174)(151,173)(152,172)(153,171)(154,170)(155,169)(156,168)(157,167)(158,166)(159,165)(160,164)(161,163)(184,214)(185,213)(186,212)(187,211)(188,210)(189,209)(190,208)(191,207)(192,206)(193,205)(194,204)(195,203)(196,202)(197,201)(198,200)(215,244)(216,243)(217,242)(218,241)(219,240)(220,239)(221,238)(222,237)(223,236)(224,235)(225,234)(226,233)(227,232)(228,231)(229,230)>;
G:=Group( (1,230,84,132)(2,231,85,133)(3,232,86,134)(4,233,87,135)(5,234,88,136)(6,235,89,137)(7,236,90,138)(8,237,91,139)(9,238,92,140)(10,239,93,141)(11,240,94,142)(12,241,95,143)(13,242,96,144)(14,243,97,145)(15,244,98,146)(16,184,99,147)(17,185,100,148)(18,186,101,149)(19,187,102,150)(20,188,103,151)(21,189,104,152)(22,190,105,153)(23,191,106,154)(24,192,107,155)(25,193,108,156)(26,194,109,157)(27,195,110,158)(28,196,111,159)(29,197,112,160)(30,198,113,161)(31,199,114,162)(32,200,115,163)(33,201,116,164)(34,202,117,165)(35,203,118,166)(36,204,119,167)(37,205,120,168)(38,206,121,169)(39,207,122,170)(40,208,62,171)(41,209,63,172)(42,210,64,173)(43,211,65,174)(44,212,66,175)(45,213,67,176)(46,214,68,177)(47,215,69,178)(48,216,70,179)(49,217,71,180)(50,218,72,181)(51,219,73,182)(52,220,74,183)(53,221,75,123)(54,222,76,124)(55,223,77,125)(56,224,78,126)(57,225,79,127)(58,226,80,128)(59,227,81,129)(60,228,82,130)(61,229,83,131), (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), (1,61)(2,60)(3,59)(4,58)(5,57)(6,56)(7,55)(8,54)(9,53)(10,52)(11,51)(12,50)(13,49)(14,48)(15,47)(16,46)(17,45)(18,44)(19,43)(20,42)(21,41)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(62,105)(63,104)(64,103)(65,102)(66,101)(67,100)(68,99)(69,98)(70,97)(71,96)(72,95)(73,94)(74,93)(75,92)(76,91)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,84)(106,122)(107,121)(108,120)(109,119)(110,118)(111,117)(112,116)(113,115)(123,140)(124,139)(125,138)(126,137)(127,136)(128,135)(129,134)(130,133)(131,132)(141,183)(142,182)(143,181)(144,180)(145,179)(146,178)(147,177)(148,176)(149,175)(150,174)(151,173)(152,172)(153,171)(154,170)(155,169)(156,168)(157,167)(158,166)(159,165)(160,164)(161,163)(184,214)(185,213)(186,212)(187,211)(188,210)(189,209)(190,208)(191,207)(192,206)(193,205)(194,204)(195,203)(196,202)(197,201)(198,200)(215,244)(216,243)(217,242)(218,241)(219,240)(220,239)(221,238)(222,237)(223,236)(224,235)(225,234)(226,233)(227,232)(228,231)(229,230) );
G=PermutationGroup([[(1,230,84,132),(2,231,85,133),(3,232,86,134),(4,233,87,135),(5,234,88,136),(6,235,89,137),(7,236,90,138),(8,237,91,139),(9,238,92,140),(10,239,93,141),(11,240,94,142),(12,241,95,143),(13,242,96,144),(14,243,97,145),(15,244,98,146),(16,184,99,147),(17,185,100,148),(18,186,101,149),(19,187,102,150),(20,188,103,151),(21,189,104,152),(22,190,105,153),(23,191,106,154),(24,192,107,155),(25,193,108,156),(26,194,109,157),(27,195,110,158),(28,196,111,159),(29,197,112,160),(30,198,113,161),(31,199,114,162),(32,200,115,163),(33,201,116,164),(34,202,117,165),(35,203,118,166),(36,204,119,167),(37,205,120,168),(38,206,121,169),(39,207,122,170),(40,208,62,171),(41,209,63,172),(42,210,64,173),(43,211,65,174),(44,212,66,175),(45,213,67,176),(46,214,68,177),(47,215,69,178),(48,216,70,179),(49,217,71,180),(50,218,72,181),(51,219,73,182),(52,220,74,183),(53,221,75,123),(54,222,76,124),(55,223,77,125),(56,224,78,126),(57,225,79,127),(58,226,80,128),(59,227,81,129),(60,228,82,130),(61,229,83,131)], [(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)], [(1,61),(2,60),(3,59),(4,58),(5,57),(6,56),(7,55),(8,54),(9,53),(10,52),(11,51),(12,50),(13,49),(14,48),(15,47),(16,46),(17,45),(18,44),(19,43),(20,42),(21,41),(22,40),(23,39),(24,38),(25,37),(26,36),(27,35),(28,34),(29,33),(30,32),(62,105),(63,104),(64,103),(65,102),(66,101),(67,100),(68,99),(69,98),(70,97),(71,96),(72,95),(73,94),(74,93),(75,92),(76,91),(77,90),(78,89),(79,88),(80,87),(81,86),(82,85),(83,84),(106,122),(107,121),(108,120),(109,119),(110,118),(111,117),(112,116),(113,115),(123,140),(124,139),(125,138),(126,137),(127,136),(128,135),(129,134),(130,133),(131,132),(141,183),(142,182),(143,181),(144,180),(145,179),(146,178),(147,177),(148,176),(149,175),(150,174),(151,173),(152,172),(153,171),(154,170),(155,169),(156,168),(157,167),(158,166),(159,165),(160,164),(161,163),(184,214),(185,213),(186,212),(187,211),(188,210),(189,209),(190,208),(191,207),(192,206),(193,205),(194,204),(195,203),(196,202),(197,201),(198,200),(215,244),(216,243),(217,242),(218,241),(219,240),(220,239),(221,238),(222,237),(223,236),(224,235),(225,234),(226,233),(227,232),(228,231),(229,230)]])
128 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 61A | ··· | 61AD | 122A | ··· | 122AD | 244A | ··· | 244BH |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 61 | ··· | 61 | 122 | ··· | 122 | 244 | ··· | 244 |
| size | 1 | 1 | 61 | 61 | 1 | 1 | 61 | 61 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
128 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C4 | D61 | D122 | C4×D61 |
| kernel | C4×D61 | Dic61 | C244 | D122 | D61 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 30 | 30 | 60 |
Matrix representation of C4×D61 ►in GL2(𝔽733) generated by
| 380 | 0 |
| 0 | 380 |
| 652 | 27 |
| 732 | 308 |
| 440 | 340 |
| 515 | 293 |
G:=sub<GL(2,GF(733))| [380,0,0,380],[652,732,27,308],[440,515,340,293] >;
C4×D61 in GAP, Magma, Sage, TeX
C_4\times D_{61} % in TeX
G:=Group("C4xD61"); // GroupNames label
G:=SmallGroup(488,5);
// by ID
G=gap.SmallGroup(488,5);
# by ID
G:=PCGroup([4,-2,-2,-2,-61,21,7683]);
// Polycyclic
G:=Group<a,b,c|a^4=b^61=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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