metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C61⋊2D4, C22⋊D61, Dic61⋊C2, D122⋊2C2, C2.5D122, C122.5C22, (C2×C122)⋊2C2, SmallGroup(488,8)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C61⋊D4
G = < a,b,c | a61=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C61⋊D4
►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)
(1 147 97 195)(2 146 98 194)(3 145 99 193)(4 144 100 192)(5 143 101 191)(6 142 102 190)(7 141 103 189)(8 140 104 188)(9 139 105 187)(10 138 106 186)(11 137 107 185)(12 136 108 184)(13 135 109 244)(14 134 110 243)(15 133 111 242)(16 132 112 241)(17 131 113 240)(18 130 114 239)(19 129 115 238)(20 128 116 237)(21 127 117 236)(22 126 118 235)(23 125 119 234)(24 124 120 233)(25 123 121 232)(26 183 122 231)(27 182 62 230)(28 181 63 229)(29 180 64 228)(30 179 65 227)(31 178 66 226)(32 177 67 225)(33 176 68 224)(34 175 69 223)(35 174 70 222)(36 173 71 221)(37 172 72 220)(38 171 73 219)(39 170 74 218)(40 169 75 217)(41 168 76 216)(42 167 77 215)(43 166 78 214)(44 165 79 213)(45 164 80 212)(46 163 81 211)(47 162 82 210)(48 161 83 209)(49 160 84 208)(50 159 85 207)(51 158 86 206)(52 157 87 205)(53 156 88 204)(54 155 89 203)(55 154 90 202)(56 153 91 201)(57 152 92 200)(58 151 93 199)(59 150 94 198)(60 149 95 197)(61 148 96 196)
(2 61)(3 60)(4 59)(5 58)(6 57)(7 56)(8 55)(9 54)(10 53)(11 52)(12 51)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 42)(22 41)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)(31 32)(62 71)(63 70)(64 69)(65 68)(66 67)(72 122)(73 121)(74 120)(75 119)(76 118)(77 117)(78 116)(79 115)(80 114)(81 113)(82 112)(83 111)(84 110)(85 109)(86 108)(87 107)(88 106)(89 105)(90 104)(91 103)(92 102)(93 101)(94 100)(95 99)(96 98)(123 219)(124 218)(125 217)(126 216)(127 215)(128 214)(129 213)(130 212)(131 211)(132 210)(133 209)(134 208)(135 207)(136 206)(137 205)(138 204)(139 203)(140 202)(141 201)(142 200)(143 199)(144 198)(145 197)(146 196)(147 195)(148 194)(149 193)(150 192)(151 191)(152 190)(153 189)(154 188)(155 187)(156 186)(157 185)(158 184)(159 244)(160 243)(161 242)(162 241)(163 240)(164 239)(165 238)(166 237)(167 236)(168 235)(169 234)(170 233)(171 232)(172 231)(173 230)(174 229)(175 228)(176 227)(177 226)(178 225)(179 224)(180 223)(181 222)(182 221)(183 220)
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), (1,147,97,195)(2,146,98,194)(3,145,99,193)(4,144,100,192)(5,143,101,191)(6,142,102,190)(7,141,103,189)(8,140,104,188)(9,139,105,187)(10,138,106,186)(11,137,107,185)(12,136,108,184)(13,135,109,244)(14,134,110,243)(15,133,111,242)(16,132,112,241)(17,131,113,240)(18,130,114,239)(19,129,115,238)(20,128,116,237)(21,127,117,236)(22,126,118,235)(23,125,119,234)(24,124,120,233)(25,123,121,232)(26,183,122,231)(27,182,62,230)(28,181,63,229)(29,180,64,228)(30,179,65,227)(31,178,66,226)(32,177,67,225)(33,176,68,224)(34,175,69,223)(35,174,70,222)(36,173,71,221)(37,172,72,220)(38,171,73,219)(39,170,74,218)(40,169,75,217)(41,168,76,216)(42,167,77,215)(43,166,78,214)(44,165,79,213)(45,164,80,212)(46,163,81,211)(47,162,82,210)(48,161,83,209)(49,160,84,208)(50,159,85,207)(51,158,86,206)(52,157,87,205)(53,156,88,204)(54,155,89,203)(55,154,90,202)(56,153,91,201)(57,152,92,200)(58,151,93,199)(59,150,94,198)(60,149,95,197)(61,148,96,196), (2,61)(3,60)(4,59)(5,58)(6,57)(7,56)(8,55)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)(62,71)(63,70)(64,69)(65,68)(66,67)(72,122)(73,121)(74,120)(75,119)(76,118)(77,117)(78,116)(79,115)(80,114)(81,113)(82,112)(83,111)(84,110)(85,109)(86,108)(87,107)(88,106)(89,105)(90,104)(91,103)(92,102)(93,101)(94,100)(95,99)(96,98)(123,219)(124,218)(125,217)(126,216)(127,215)(128,214)(129,213)(130,212)(131,211)(132,210)(133,209)(134,208)(135,207)(136,206)(137,205)(138,204)(139,203)(140,202)(141,201)(142,200)(143,199)(144,198)(145,197)(146,196)(147,195)(148,194)(149,193)(150,192)(151,191)(152,190)(153,189)(154,188)(155,187)(156,186)(157,185)(158,184)(159,244)(160,243)(161,242)(162,241)(163,240)(164,239)(165,238)(166,237)(167,236)(168,235)(169,234)(170,233)(171,232)(172,231)(173,230)(174,229)(175,228)(176,227)(177,226)(178,225)(179,224)(180,223)(181,222)(182,221)(183,220)>;
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), (1,147,97,195)(2,146,98,194)(3,145,99,193)(4,144,100,192)(5,143,101,191)(6,142,102,190)(7,141,103,189)(8,140,104,188)(9,139,105,187)(10,138,106,186)(11,137,107,185)(12,136,108,184)(13,135,109,244)(14,134,110,243)(15,133,111,242)(16,132,112,241)(17,131,113,240)(18,130,114,239)(19,129,115,238)(20,128,116,237)(21,127,117,236)(22,126,118,235)(23,125,119,234)(24,124,120,233)(25,123,121,232)(26,183,122,231)(27,182,62,230)(28,181,63,229)(29,180,64,228)(30,179,65,227)(31,178,66,226)(32,177,67,225)(33,176,68,224)(34,175,69,223)(35,174,70,222)(36,173,71,221)(37,172,72,220)(38,171,73,219)(39,170,74,218)(40,169,75,217)(41,168,76,216)(42,167,77,215)(43,166,78,214)(44,165,79,213)(45,164,80,212)(46,163,81,211)(47,162,82,210)(48,161,83,209)(49,160,84,208)(50,159,85,207)(51,158,86,206)(52,157,87,205)(53,156,88,204)(54,155,89,203)(55,154,90,202)(56,153,91,201)(57,152,92,200)(58,151,93,199)(59,150,94,198)(60,149,95,197)(61,148,96,196), (2,61)(3,60)(4,59)(5,58)(6,57)(7,56)(8,55)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)(62,71)(63,70)(64,69)(65,68)(66,67)(72,122)(73,121)(74,120)(75,119)(76,118)(77,117)(78,116)(79,115)(80,114)(81,113)(82,112)(83,111)(84,110)(85,109)(86,108)(87,107)(88,106)(89,105)(90,104)(91,103)(92,102)(93,101)(94,100)(95,99)(96,98)(123,219)(124,218)(125,217)(126,216)(127,215)(128,214)(129,213)(130,212)(131,211)(132,210)(133,209)(134,208)(135,207)(136,206)(137,205)(138,204)(139,203)(140,202)(141,201)(142,200)(143,199)(144,198)(145,197)(146,196)(147,195)(148,194)(149,193)(150,192)(151,191)(152,190)(153,189)(154,188)(155,187)(156,186)(157,185)(158,184)(159,244)(160,243)(161,242)(162,241)(163,240)(164,239)(165,238)(166,237)(167,236)(168,235)(169,234)(170,233)(171,232)(172,231)(173,230)(174,229)(175,228)(176,227)(177,226)(178,225)(179,224)(180,223)(181,222)(182,221)(183,220) );
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)], [(1,147,97,195),(2,146,98,194),(3,145,99,193),(4,144,100,192),(5,143,101,191),(6,142,102,190),(7,141,103,189),(8,140,104,188),(9,139,105,187),(10,138,106,186),(11,137,107,185),(12,136,108,184),(13,135,109,244),(14,134,110,243),(15,133,111,242),(16,132,112,241),(17,131,113,240),(18,130,114,239),(19,129,115,238),(20,128,116,237),(21,127,117,236),(22,126,118,235),(23,125,119,234),(24,124,120,233),(25,123,121,232),(26,183,122,231),(27,182,62,230),(28,181,63,229),(29,180,64,228),(30,179,65,227),(31,178,66,226),(32,177,67,225),(33,176,68,224),(34,175,69,223),(35,174,70,222),(36,173,71,221),(37,172,72,220),(38,171,73,219),(39,170,74,218),(40,169,75,217),(41,168,76,216),(42,167,77,215),(43,166,78,214),(44,165,79,213),(45,164,80,212),(46,163,81,211),(47,162,82,210),(48,161,83,209),(49,160,84,208),(50,159,85,207),(51,158,86,206),(52,157,87,205),(53,156,88,204),(54,155,89,203),(55,154,90,202),(56,153,91,201),(57,152,92,200),(58,151,93,199),(59,150,94,198),(60,149,95,197),(61,148,96,196)], [(2,61),(3,60),(4,59),(5,58),(6,57),(7,56),(8,55),(9,54),(10,53),(11,52),(12,51),(13,50),(14,49),(15,48),(16,47),(17,46),(18,45),(19,44),(20,43),(21,42),(22,41),(23,40),(24,39),(25,38),(26,37),(27,36),(28,35),(29,34),(30,33),(31,32),(62,71),(63,70),(64,69),(65,68),(66,67),(72,122),(73,121),(74,120),(75,119),(76,118),(77,117),(78,116),(79,115),(80,114),(81,113),(82,112),(83,111),(84,110),(85,109),(86,108),(87,107),(88,106),(89,105),(90,104),(91,103),(92,102),(93,101),(94,100),(95,99),(96,98),(123,219),(124,218),(125,217),(126,216),(127,215),(128,214),(129,213),(130,212),(131,211),(132,210),(133,209),(134,208),(135,207),(136,206),(137,205),(138,204),(139,203),(140,202),(141,201),(142,200),(143,199),(144,198),(145,197),(146,196),(147,195),(148,194),(149,193),(150,192),(151,191),(152,190),(153,189),(154,188),(155,187),(156,186),(157,185),(158,184),(159,244),(160,243),(161,242),(162,241),(163,240),(164,239),(165,238),(166,237),(167,236),(168,235),(169,234),(170,233),(171,232),(172,231),(173,230),(174,229),(175,228),(176,227),(177,226),(178,225),(179,224),(180,223),(181,222),(182,221),(183,220)]])
125 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 61A | ··· | 61AD | 122A | ··· | 122CL |
| order | 1 | 2 | 2 | 2 | 4 | 61 | ··· | 61 | 122 | ··· | 122 |
| size | 1 | 1 | 2 | 122 | 122 | 2 | ··· | 2 | 2 | ··· | 2 |
125 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | D61 | D122 | C61⋊D4 |
| kernel | C61⋊D4 | Dic61 | D122 | C2×C122 | C61 | C22 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 30 | 30 | 60 |
Matrix representation of C61⋊D4 ►in GL2(𝔽733) generated by
| 0 | 1 |
| 732 | 709 |
| 197 | 444 |
| 114 | 536 |
| 1 | 0 |
| 709 | 732 |
G:=sub<GL(2,GF(733))| [0,732,1,709],[197,114,444,536],[1,709,0,732] >;
C61⋊D4 in GAP, Magma, Sage, TeX
C_{61}\rtimes D_4 % in TeX
G:=Group("C61:D4"); // GroupNames label
G:=SmallGroup(488,8);
// by ID
G=gap.SmallGroup(488,8);
# by ID
G:=PCGroup([4,-2,-2,-2,-61,49,7683]);
// Polycyclic
G:=Group<a,b,c|a^61=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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