direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C12.28D10, Dic6⋊24D10, D60⋊37C22, C30.13C24, D30.4C23, C60.137C23, (C4×D5)⋊15D6, (C2×D60)⋊28C2, C30⋊7(C4○D4), C10⋊1(C4○D12), C6⋊1(Q8⋊2D5), (C10×Dic6)⋊8C2, (C2×Dic6)⋊15D5, (C2×C20).169D6, C3⋊D20⋊9C22, C6.13(C23×D5), (C2×C12).311D10, (D5×C12)⋊17C22, C10.13(S3×C23), D30.C2⋊6C22, (C6×D5).40C23, (C22×D5).98D6, C20.129(C22×S3), (C2×C60).155C22, (C2×C30).232C23, (C2×Dic5).220D6, (C5×Dic6)⋊21C22, D10.42(C22×S3), C12.161(C22×D5), (C5×Dic3).7C23, Dic3.7(C22×D5), (C2×Dic3).131D10, Dic5.56(C22×S3), (C3×Dic5).42C23, (C6×Dic5).229C22, (C22×D15).73C22, (C10×Dic3).130C22, (C2×C4×D5)⋊4S3, (D5×C2×C12)⋊5C2, C5⋊1(C2×C4○D12), C15⋊7(C2×C4○D4), C4.87(C2×S3×D5), C3⋊1(C2×Q8⋊2D5), (C2×C3⋊D20)⋊18C2, C2.17(C22×S3×D5), (C2×C4).168(S3×D5), C22.101(C2×S3×D5), (C2×D30.C2)⋊20C2, (D5×C2×C6).117C22, (C2×C6).242(C22×D5), (C2×C10).242(C22×S3), SmallGroup(480,1085)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1692 in 328 conjugacy classes, 116 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×2], C4 [×6], C22, C22 [×12], C5, S3 [×4], C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], D5 [×6], C10, C10 [×2], Dic3 [×4], C12 [×2], C12 [×2], D6 [×8], C2×C6, C2×C6 [×4], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×2], C20 [×2], C20 [×4], D10 [×2], D10 [×10], C2×C10, Dic6 [×4], C4×S3 [×8], D12 [×4], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12, C2×C12 [×5], C22×S3 [×2], C22×C6, C3×D5 [×2], D15 [×4], C30, C30 [×2], C2×C4○D4, C4×D5 [×4], C4×D5 [×8], D20 [×12], C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×4], C22×D5, C22×D5 [×2], C2×Dic6, S3×C2×C4 [×2], C2×D12, C4○D12 [×8], C2×C3⋊D4 [×2], C22×C12, C5×Dic3 [×4], C3×Dic5 [×2], C60 [×2], C6×D5 [×2], C6×D5 [×2], D30 [×4], D30 [×4], C2×C30, C2×C4×D5, C2×C4×D5 [×2], C2×D20 [×3], Q8⋊2D5 [×8], Q8×C10, C2×C4○D12, D30.C2 [×8], C3⋊D20 [×8], D5×C12 [×4], C6×Dic5, C5×Dic6 [×4], C10×Dic3 [×2], D60 [×4], C2×C60, D5×C2×C6, C22×D15 [×2], C2×Q8⋊2D5, C12.28D10 [×8], C2×D30.C2 [×2], C2×C3⋊D20 [×2], D5×C2×C12, C10×Dic6, C2×D60, C2×C12.28D10
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C4○D4 [×2], C24, D10 [×7], C22×S3 [×7], C2×C4○D4, C22×D5 [×7], C4○D12 [×2], S3×C23, S3×D5, Q8⋊2D5 [×2], C23×D5, C2×C4○D12, C2×S3×D5 [×3], C2×Q8⋊2D5, C12.28D10 [×2], C22×S3×D5, C2×C12.28D10
Generators and relations
G = < a,b,c,d | a2=b12=d2=1, c10=b6, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=b6c9 >
►On 240 points
Generators in S240
(1 116)(2 117)(3 118)(4 119)(5 120)(6 101)(7 102)(8 103)(9 104)(10 105)(11 106)(12 107)(13 108)(14 109)(15 110)(16 111)(17 112)(18 113)(19 114)(20 115)(21 133)(22 134)(23 135)(24 136)(25 137)(26 138)(27 139)(28 140)(29 121)(30 122)(31 123)(32 124)(33 125)(34 126)(35 127)(36 128)(37 129)(38 130)(39 131)(40 132)(41 150)(42 151)(43 152)(44 153)(45 154)(46 155)(47 156)(48 157)(49 158)(50 159)(51 160)(52 141)(53 142)(54 143)(55 144)(56 145)(57 146)(58 147)(59 148)(60 149)(61 227)(62 228)(63 229)(64 230)(65 231)(66 232)(67 233)(68 234)(69 235)(70 236)(71 237)(72 238)(73 239)(74 240)(75 221)(76 222)(77 223)(78 224)(79 225)(80 226)(81 219)(82 220)(83 201)(84 202)(85 203)(86 204)(87 205)(88 206)(89 207)(90 208)(91 209)(92 210)(93 211)(94 212)(95 213)(96 214)(97 215)(98 216)(99 217)(100 218)(161 195)(162 196)(163 197)(164 198)(165 199)(166 200)(167 181)(168 182)(169 183)(170 184)(171 185)(172 186)(173 187)(174 188)(175 189)(176 190)(177 191)(178 192)(179 193)(180 194)
(1 76 82 172 59 38 11 66 92 162 49 28)(2 29 50 163 93 67 12 39 60 173 83 77)(3 78 84 174 41 40 13 68 94 164 51 30)(4 31 52 165 95 69 14 21 42 175 85 79)(5 80 86 176 43 22 15 70 96 166 53 32)(6 33 54 167 97 71 16 23 44 177 87 61)(7 62 88 178 45 24 17 72 98 168 55 34)(8 35 56 169 99 73 18 25 46 179 89 63)(9 64 90 180 47 26 19 74 100 170 57 36)(10 37 58 171 81 75 20 27 48 161 91 65)(101 125 143 181 215 237 111 135 153 191 205 227)(102 228 206 192 154 136 112 238 216 182 144 126)(103 127 145 183 217 239 113 137 155 193 207 229)(104 230 208 194 156 138 114 240 218 184 146 128)(105 129 147 185 219 221 115 139 157 195 209 231)(106 232 210 196 158 140 116 222 220 186 148 130)(107 131 149 187 201 223 117 121 159 197 211 233)(108 234 212 198 160 122 118 224 202 188 150 132)(109 133 151 189 203 225 119 123 141 199 213 235)(110 236 214 200 142 124 120 226 204 190 152 134)
(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)
(1 10)(2 9)(3 8)(4 7)(5 6)(11 20)(12 19)(13 18)(14 17)(15 16)(21 24)(22 23)(25 40)(26 39)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(41 46)(42 45)(43 44)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)(93 100)(94 99)(95 98)(96 97)(101 120)(102 119)(103 118)(104 117)(105 116)(106 115)(107 114)(108 113)(109 112)(110 111)(121 128)(122 127)(123 126)(124 125)(129 140)(130 139)(131 138)(132 137)(133 136)(134 135)(141 144)(142 143)(145 160)(146 159)(147 158)(148 157)(149 156)(150 155)(151 154)(152 153)(161 172)(162 171)(163 170)(164 169)(165 168)(166 167)(173 180)(174 179)(175 178)(176 177)(181 200)(182 199)(183 198)(184 197)(185 196)(186 195)(187 194)(188 193)(189 192)(190 191)(201 208)(202 207)(203 206)(204 205)(209 220)(210 219)(211 218)(212 217)(213 216)(214 215)(221 232)(222 231)(223 230)(224 229)(225 228)(226 227)(233 240)(234 239)(235 238)(236 237)
G:=sub<Sym(240)| (1,116)(2,117)(3,118)(4,119)(5,120)(6,101)(7,102)(8,103)(9,104)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,112)(18,113)(19,114)(20,115)(21,133)(22,134)(23,135)(24,136)(25,137)(26,138)(27,139)(28,140)(29,121)(30,122)(31,123)(32,124)(33,125)(34,126)(35,127)(36,128)(37,129)(38,130)(39,131)(40,132)(41,150)(42,151)(43,152)(44,153)(45,154)(46,155)(47,156)(48,157)(49,158)(50,159)(51,160)(52,141)(53,142)(54,143)(55,144)(56,145)(57,146)(58,147)(59,148)(60,149)(61,227)(62,228)(63,229)(64,230)(65,231)(66,232)(67,233)(68,234)(69,235)(70,236)(71,237)(72,238)(73,239)(74,240)(75,221)(76,222)(77,223)(78,224)(79,225)(80,226)(81,219)(82,220)(83,201)(84,202)(85,203)(86,204)(87,205)(88,206)(89,207)(90,208)(91,209)(92,210)(93,211)(94,212)(95,213)(96,214)(97,215)(98,216)(99,217)(100,218)(161,195)(162,196)(163,197)(164,198)(165,199)(166,200)(167,181)(168,182)(169,183)(170,184)(171,185)(172,186)(173,187)(174,188)(175,189)(176,190)(177,191)(178,192)(179,193)(180,194), (1,76,82,172,59,38,11,66,92,162,49,28)(2,29,50,163,93,67,12,39,60,173,83,77)(3,78,84,174,41,40,13,68,94,164,51,30)(4,31,52,165,95,69,14,21,42,175,85,79)(5,80,86,176,43,22,15,70,96,166,53,32)(6,33,54,167,97,71,16,23,44,177,87,61)(7,62,88,178,45,24,17,72,98,168,55,34)(8,35,56,169,99,73,18,25,46,179,89,63)(9,64,90,180,47,26,19,74,100,170,57,36)(10,37,58,171,81,75,20,27,48,161,91,65)(101,125,143,181,215,237,111,135,153,191,205,227)(102,228,206,192,154,136,112,238,216,182,144,126)(103,127,145,183,217,239,113,137,155,193,207,229)(104,230,208,194,156,138,114,240,218,184,146,128)(105,129,147,185,219,221,115,139,157,195,209,231)(106,232,210,196,158,140,116,222,220,186,148,130)(107,131,149,187,201,223,117,121,159,197,211,233)(108,234,212,198,160,122,118,224,202,188,150,132)(109,133,151,189,203,225,119,123,141,199,213,235)(110,236,214,200,142,124,120,226,204,190,152,134), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(21,24)(22,23)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(41,46)(42,45)(43,44)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(93,100)(94,99)(95,98)(96,97)(101,120)(102,119)(103,118)(104,117)(105,116)(106,115)(107,114)(108,113)(109,112)(110,111)(121,128)(122,127)(123,126)(124,125)(129,140)(130,139)(131,138)(132,137)(133,136)(134,135)(141,144)(142,143)(145,160)(146,159)(147,158)(148,157)(149,156)(150,155)(151,154)(152,153)(161,172)(162,171)(163,170)(164,169)(165,168)(166,167)(173,180)(174,179)(175,178)(176,177)(181,200)(182,199)(183,198)(184,197)(185,196)(186,195)(187,194)(188,193)(189,192)(190,191)(201,208)(202,207)(203,206)(204,205)(209,220)(210,219)(211,218)(212,217)(213,216)(214,215)(221,232)(222,231)(223,230)(224,229)(225,228)(226,227)(233,240)(234,239)(235,238)(236,237)>;
G:=Group( (1,116)(2,117)(3,118)(4,119)(5,120)(6,101)(7,102)(8,103)(9,104)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,112)(18,113)(19,114)(20,115)(21,133)(22,134)(23,135)(24,136)(25,137)(26,138)(27,139)(28,140)(29,121)(30,122)(31,123)(32,124)(33,125)(34,126)(35,127)(36,128)(37,129)(38,130)(39,131)(40,132)(41,150)(42,151)(43,152)(44,153)(45,154)(46,155)(47,156)(48,157)(49,158)(50,159)(51,160)(52,141)(53,142)(54,143)(55,144)(56,145)(57,146)(58,147)(59,148)(60,149)(61,227)(62,228)(63,229)(64,230)(65,231)(66,232)(67,233)(68,234)(69,235)(70,236)(71,237)(72,238)(73,239)(74,240)(75,221)(76,222)(77,223)(78,224)(79,225)(80,226)(81,219)(82,220)(83,201)(84,202)(85,203)(86,204)(87,205)(88,206)(89,207)(90,208)(91,209)(92,210)(93,211)(94,212)(95,213)(96,214)(97,215)(98,216)(99,217)(100,218)(161,195)(162,196)(163,197)(164,198)(165,199)(166,200)(167,181)(168,182)(169,183)(170,184)(171,185)(172,186)(173,187)(174,188)(175,189)(176,190)(177,191)(178,192)(179,193)(180,194), (1,76,82,172,59,38,11,66,92,162,49,28)(2,29,50,163,93,67,12,39,60,173,83,77)(3,78,84,174,41,40,13,68,94,164,51,30)(4,31,52,165,95,69,14,21,42,175,85,79)(5,80,86,176,43,22,15,70,96,166,53,32)(6,33,54,167,97,71,16,23,44,177,87,61)(7,62,88,178,45,24,17,72,98,168,55,34)(8,35,56,169,99,73,18,25,46,179,89,63)(9,64,90,180,47,26,19,74,100,170,57,36)(10,37,58,171,81,75,20,27,48,161,91,65)(101,125,143,181,215,237,111,135,153,191,205,227)(102,228,206,192,154,136,112,238,216,182,144,126)(103,127,145,183,217,239,113,137,155,193,207,229)(104,230,208,194,156,138,114,240,218,184,146,128)(105,129,147,185,219,221,115,139,157,195,209,231)(106,232,210,196,158,140,116,222,220,186,148,130)(107,131,149,187,201,223,117,121,159,197,211,233)(108,234,212,198,160,122,118,224,202,188,150,132)(109,133,151,189,203,225,119,123,141,199,213,235)(110,236,214,200,142,124,120,226,204,190,152,134), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(21,24)(22,23)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(41,46)(42,45)(43,44)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(93,100)(94,99)(95,98)(96,97)(101,120)(102,119)(103,118)(104,117)(105,116)(106,115)(107,114)(108,113)(109,112)(110,111)(121,128)(122,127)(123,126)(124,125)(129,140)(130,139)(131,138)(132,137)(133,136)(134,135)(141,144)(142,143)(145,160)(146,159)(147,158)(148,157)(149,156)(150,155)(151,154)(152,153)(161,172)(162,171)(163,170)(164,169)(165,168)(166,167)(173,180)(174,179)(175,178)(176,177)(181,200)(182,199)(183,198)(184,197)(185,196)(186,195)(187,194)(188,193)(189,192)(190,191)(201,208)(202,207)(203,206)(204,205)(209,220)(210,219)(211,218)(212,217)(213,216)(214,215)(221,232)(222,231)(223,230)(224,229)(225,228)(226,227)(233,240)(234,239)(235,238)(236,237) );
G=PermutationGroup([(1,116),(2,117),(3,118),(4,119),(5,120),(6,101),(7,102),(8,103),(9,104),(10,105),(11,106),(12,107),(13,108),(14,109),(15,110),(16,111),(17,112),(18,113),(19,114),(20,115),(21,133),(22,134),(23,135),(24,136),(25,137),(26,138),(27,139),(28,140),(29,121),(30,122),(31,123),(32,124),(33,125),(34,126),(35,127),(36,128),(37,129),(38,130),(39,131),(40,132),(41,150),(42,151),(43,152),(44,153),(45,154),(46,155),(47,156),(48,157),(49,158),(50,159),(51,160),(52,141),(53,142),(54,143),(55,144),(56,145),(57,146),(58,147),(59,148),(60,149),(61,227),(62,228),(63,229),(64,230),(65,231),(66,232),(67,233),(68,234),(69,235),(70,236),(71,237),(72,238),(73,239),(74,240),(75,221),(76,222),(77,223),(78,224),(79,225),(80,226),(81,219),(82,220),(83,201),(84,202),(85,203),(86,204),(87,205),(88,206),(89,207),(90,208),(91,209),(92,210),(93,211),(94,212),(95,213),(96,214),(97,215),(98,216),(99,217),(100,218),(161,195),(162,196),(163,197),(164,198),(165,199),(166,200),(167,181),(168,182),(169,183),(170,184),(171,185),(172,186),(173,187),(174,188),(175,189),(176,190),(177,191),(178,192),(179,193),(180,194)], [(1,76,82,172,59,38,11,66,92,162,49,28),(2,29,50,163,93,67,12,39,60,173,83,77),(3,78,84,174,41,40,13,68,94,164,51,30),(4,31,52,165,95,69,14,21,42,175,85,79),(5,80,86,176,43,22,15,70,96,166,53,32),(6,33,54,167,97,71,16,23,44,177,87,61),(7,62,88,178,45,24,17,72,98,168,55,34),(8,35,56,169,99,73,18,25,46,179,89,63),(9,64,90,180,47,26,19,74,100,170,57,36),(10,37,58,171,81,75,20,27,48,161,91,65),(101,125,143,181,215,237,111,135,153,191,205,227),(102,228,206,192,154,136,112,238,216,182,144,126),(103,127,145,183,217,239,113,137,155,193,207,229),(104,230,208,194,156,138,114,240,218,184,146,128),(105,129,147,185,219,221,115,139,157,195,209,231),(106,232,210,196,158,140,116,222,220,186,148,130),(107,131,149,187,201,223,117,121,159,197,211,233),(108,234,212,198,160,122,118,224,202,188,150,132),(109,133,151,189,203,225,119,123,141,199,213,235),(110,236,214,200,142,124,120,226,204,190,152,134)], [(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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,20),(12,19),(13,18),(14,17),(15,16),(21,24),(22,23),(25,40),(26,39),(27,38),(28,37),(29,36),(30,35),(31,34),(32,33),(41,46),(42,45),(43,44),(47,60),(48,59),(49,58),(50,57),(51,56),(52,55),(53,54),(61,80),(62,79),(63,78),(64,77),(65,76),(66,75),(67,74),(68,73),(69,72),(70,71),(81,92),(82,91),(83,90),(84,89),(85,88),(86,87),(93,100),(94,99),(95,98),(96,97),(101,120),(102,119),(103,118),(104,117),(105,116),(106,115),(107,114),(108,113),(109,112),(110,111),(121,128),(122,127),(123,126),(124,125),(129,140),(130,139),(131,138),(132,137),(133,136),(134,135),(141,144),(142,143),(145,160),(146,159),(147,158),(148,157),(149,156),(150,155),(151,154),(152,153),(161,172),(162,171),(163,170),(164,169),(165,168),(166,167),(173,180),(174,179),(175,178),(176,177),(181,200),(182,199),(183,198),(184,197),(185,196),(186,195),(187,194),(188,193),(189,192),(190,191),(201,208),(202,207),(203,206),(204,205),(209,220),(210,219),(211,218),(212,217),(213,216),(214,215),(221,232),(222,231),(223,230),(224,229),(225,228),(226,227),(233,240),(234,239),(235,238),(236,237)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 19 | 29 |
| 1 | 44 | 0 | 0 |
| 17 | 17 | 0 | 0 |
| 0 | 0 | 32 | 20 |
| 0 | 0 | 25 | 29 |
| 1 | 44 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 29 | 41 |
| 0 | 0 | 42 | 32 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,40,19,0,0,0,29],[1,17,0,0,44,17,0,0,0,0,32,25,0,0,20,29],[1,0,0,0,44,60,0,0,0,0,29,42,0,0,41,32] >;
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D10 | C4○D12 | S3×D5 | Q8⋊2D5 | C2×S3×D5 | C2×S3×D5 | C12.28D10 |
| kernel | C2×C12.28D10 | C12.28D10 | C2×D30.C2 | C2×C3⋊D20 | D5×C2×C12 | C10×Dic6 | C2×D60 | C2×C4×D5 | C2×Dic6 | C4×D5 | C2×Dic5 | C2×C20 | C22×D5 | C30 | Dic6 | C2×Dic3 | C2×C12 | C10 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 8 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 1 | 1 | 1 | 4 | 8 | 4 | 2 | 8 | 2 | 4 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_{12}._{28}D_{10} % in TeX
G:=Group("C2xC12.28D10"); // GroupNames label
G:=SmallGroup(480,1085);
// by ID
G=gap.SmallGroup(480,1085);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=d^2=1,c^10=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=b^6*c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀