metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22⋊2Dic30, C23.16D30, Dic15.33D4, (C2×C30)⋊1Q8, C60⋊5C4⋊8C2, (C2×C4).5D30, C2.6(D4×D15), C6.95(D4×D5), (C2×C10)⋊6Dic6, (C2×C6)⋊3Dic10, (C2×C20).31D6, C10.97(S3×D4), C30.68(C2×Q8), (C2×Dic30)⋊5C2, (C2×C12).31D10, C30.303(C2×D4), C22⋊C4.1D15, C15⋊25(C22⋊Q8), C30.4Q8⋊4C2, C2.6(C2×Dic30), (C2×C60).16C22, C6.35(C2×Dic10), C10.35(C2×Dic6), (C22×C6).53D10, (C22×C10).68D6, C30.214(C4○D4), C2.6(D4⋊2D15), C6.89(D4⋊2D5), (C2×C30).275C23, C30.38D4.2C2, C5⋊5(Dic3.D4), (C22×C30).9C22, C10.89(D4⋊2S3), C3⋊5(Dic5.14D4), (C22×Dic15).3C2, (C2×Dic15).5C22, C22.39(C22×D15), (C5×C22⋊C4).1S3, (C3×C22⋊C4).1D5, (C15×C22⋊C4).1C2, (C2×C6).271(C22×D5), (C2×C10).270(C22×S3), SmallGroup(480,843)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊2Dic30
G = < a,b,c,d | a2=b2=c60=1, d2=c30, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 740 in 148 conjugacy classes, 57 normal (47 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C2×C4, C2×C4, Q8, C23, C10, C10, Dic3, C12, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, C2×C10, C2×C10, C2×C10, Dic6, C2×Dic3, C2×C12, C22×C6, C30, C30, C22⋊Q8, Dic10, C2×Dic5, C2×C20, C22×C10, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C22⋊C4, C2×Dic6, C22×Dic3, Dic15, Dic15, C60, C2×C30, C2×C30, C2×C30, C10.D4, C4⋊Dic5, C23.D5, C5×C22⋊C4, C2×Dic10, C22×Dic5, Dic3.D4, Dic30, C2×Dic15, C2×Dic15, C2×C60, C22×C30, Dic5.14D4, C30.4Q8, C60⋊5C4, C30.38D4, C15×C22⋊C4, C2×Dic30, C22×Dic15, C22⋊2Dic30
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, C4○D4, D10, Dic6, C22×S3, D15, C22⋊Q8, Dic10, C22×D5, C2×Dic6, S3×D4, D4⋊2S3, D30, C2×Dic10, D4×D5, D4⋊2D5, Dic3.D4, Dic30, C22×D15, Dic5.14D4, C2×Dic30, D4×D15, D4⋊2D15, C22⋊2Dic30
►On 240 points
Generators in S240
(1 31)(2 206)(3 33)(4 208)(5 35)(6 210)(7 37)(8 212)(9 39)(10 214)(11 41)(12 216)(13 43)(14 218)(15 45)(16 220)(17 47)(18 222)(19 49)(20 224)(21 51)(22 226)(23 53)(24 228)(25 55)(26 230)(27 57)(28 232)(29 59)(30 234)(32 236)(34 238)(36 240)(38 182)(40 184)(42 186)(44 188)(46 190)(48 192)(50 194)(52 196)(54 198)(56 200)(58 202)(60 204)(61 161)(62 92)(63 163)(64 94)(65 165)(66 96)(67 167)(68 98)(69 169)(70 100)(71 171)(72 102)(73 173)(74 104)(75 175)(76 106)(77 177)(78 108)(79 179)(80 110)(81 121)(82 112)(83 123)(84 114)(85 125)(86 116)(87 127)(88 118)(89 129)(90 120)(91 131)(93 133)(95 135)(97 137)(99 139)(101 141)(103 143)(105 145)(107 147)(109 149)(111 151)(113 153)(115 155)(117 157)(119 159)(122 152)(124 154)(126 156)(128 158)(130 160)(132 162)(134 164)(136 166)(138 168)(140 170)(142 172)(144 174)(146 176)(148 178)(150 180)(181 211)(183 213)(185 215)(187 217)(189 219)(191 221)(193 223)(195 225)(197 227)(199 229)(201 231)(203 233)(205 235)(207 237)(209 239)
(1 235)(2 236)(3 237)(4 238)(5 239)(6 240)(7 181)(8 182)(9 183)(10 184)(11 185)(12 186)(13 187)(14 188)(15 189)(16 190)(17 191)(18 192)(19 193)(20 194)(21 195)(22 196)(23 197)(24 198)(25 199)(26 200)(27 201)(28 202)(29 203)(30 204)(31 205)(32 206)(33 207)(34 208)(35 209)(36 210)(37 211)(38 212)(39 213)(40 214)(41 215)(42 216)(43 217)(44 218)(45 219)(46 220)(47 221)(48 222)(49 223)(50 224)(51 225)(52 226)(53 227)(54 228)(55 229)(56 230)(57 231)(58 232)(59 233)(60 234)(61 131)(62 132)(63 133)(64 134)(65 135)(66 136)(67 137)(68 138)(69 139)(70 140)(71 141)(72 142)(73 143)(74 144)(75 145)(76 146)(77 147)(78 148)(79 149)(80 150)(81 151)(82 152)(83 153)(84 154)(85 155)(86 156)(87 157)(88 158)(89 159)(90 160)(91 161)(92 162)(93 163)(94 164)(95 165)(96 166)(97 167)(98 168)(99 169)(100 170)(101 171)(102 172)(103 173)(104 174)(105 175)(106 176)(107 177)(108 178)(109 179)(110 180)(111 121)(112 122)(113 123)(114 124)(115 125)(116 126)(117 127)(118 128)(119 129)(120 130)
(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)
(1 162 31 132)(2 161 32 131)(3 160 33 130)(4 159 34 129)(5 158 35 128)(6 157 36 127)(7 156 37 126)(8 155 38 125)(9 154 39 124)(10 153 40 123)(11 152 41 122)(12 151 42 121)(13 150 43 180)(14 149 44 179)(15 148 45 178)(16 147 46 177)(17 146 47 176)(18 145 48 175)(19 144 49 174)(20 143 50 173)(21 142 51 172)(22 141 52 171)(23 140 53 170)(24 139 54 169)(25 138 55 168)(26 137 56 167)(27 136 57 166)(28 135 58 165)(29 134 59 164)(30 133 60 163)(61 236 91 206)(62 235 92 205)(63 234 93 204)(64 233 94 203)(65 232 95 202)(66 231 96 201)(67 230 97 200)(68 229 98 199)(69 228 99 198)(70 227 100 197)(71 226 101 196)(72 225 102 195)(73 224 103 194)(74 223 104 193)(75 222 105 192)(76 221 106 191)(77 220 107 190)(78 219 108 189)(79 218 109 188)(80 217 110 187)(81 216 111 186)(82 215 112 185)(83 214 113 184)(84 213 114 183)(85 212 115 182)(86 211 116 181)(87 210 117 240)(88 209 118 239)(89 208 119 238)(90 207 120 237)
G:=sub<Sym(240)| (1,31)(2,206)(3,33)(4,208)(5,35)(6,210)(7,37)(8,212)(9,39)(10,214)(11,41)(12,216)(13,43)(14,218)(15,45)(16,220)(17,47)(18,222)(19,49)(20,224)(21,51)(22,226)(23,53)(24,228)(25,55)(26,230)(27,57)(28,232)(29,59)(30,234)(32,236)(34,238)(36,240)(38,182)(40,184)(42,186)(44,188)(46,190)(48,192)(50,194)(52,196)(54,198)(56,200)(58,202)(60,204)(61,161)(62,92)(63,163)(64,94)(65,165)(66,96)(67,167)(68,98)(69,169)(70,100)(71,171)(72,102)(73,173)(74,104)(75,175)(76,106)(77,177)(78,108)(79,179)(80,110)(81,121)(82,112)(83,123)(84,114)(85,125)(86,116)(87,127)(88,118)(89,129)(90,120)(91,131)(93,133)(95,135)(97,137)(99,139)(101,141)(103,143)(105,145)(107,147)(109,149)(111,151)(113,153)(115,155)(117,157)(119,159)(122,152)(124,154)(126,156)(128,158)(130,160)(132,162)(134,164)(136,166)(138,168)(140,170)(142,172)(144,174)(146,176)(148,178)(150,180)(181,211)(183,213)(185,215)(187,217)(189,219)(191,221)(193,223)(195,225)(197,227)(199,229)(201,231)(203,233)(205,235)(207,237)(209,239), (1,235)(2,236)(3,237)(4,238)(5,239)(6,240)(7,181)(8,182)(9,183)(10,184)(11,185)(12,186)(13,187)(14,188)(15,189)(16,190)(17,191)(18,192)(19,193)(20,194)(21,195)(22,196)(23,197)(24,198)(25,199)(26,200)(27,201)(28,202)(29,203)(30,204)(31,205)(32,206)(33,207)(34,208)(35,209)(36,210)(37,211)(38,212)(39,213)(40,214)(41,215)(42,216)(43,217)(44,218)(45,219)(46,220)(47,221)(48,222)(49,223)(50,224)(51,225)(52,226)(53,227)(54,228)(55,229)(56,230)(57,231)(58,232)(59,233)(60,234)(61,131)(62,132)(63,133)(64,134)(65,135)(66,136)(67,137)(68,138)(69,139)(70,140)(71,141)(72,142)(73,143)(74,144)(75,145)(76,146)(77,147)(78,148)(79,149)(80,150)(81,151)(82,152)(83,153)(84,154)(85,155)(86,156)(87,157)(88,158)(89,159)(90,160)(91,161)(92,162)(93,163)(94,164)(95,165)(96,166)(97,167)(98,168)(99,169)(100,170)(101,171)(102,172)(103,173)(104,174)(105,175)(106,176)(107,177)(108,178)(109,179)(110,180)(111,121)(112,122)(113,123)(114,124)(115,125)(116,126)(117,127)(118,128)(119,129)(120,130), (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), (1,162,31,132)(2,161,32,131)(3,160,33,130)(4,159,34,129)(5,158,35,128)(6,157,36,127)(7,156,37,126)(8,155,38,125)(9,154,39,124)(10,153,40,123)(11,152,41,122)(12,151,42,121)(13,150,43,180)(14,149,44,179)(15,148,45,178)(16,147,46,177)(17,146,47,176)(18,145,48,175)(19,144,49,174)(20,143,50,173)(21,142,51,172)(22,141,52,171)(23,140,53,170)(24,139,54,169)(25,138,55,168)(26,137,56,167)(27,136,57,166)(28,135,58,165)(29,134,59,164)(30,133,60,163)(61,236,91,206)(62,235,92,205)(63,234,93,204)(64,233,94,203)(65,232,95,202)(66,231,96,201)(67,230,97,200)(68,229,98,199)(69,228,99,198)(70,227,100,197)(71,226,101,196)(72,225,102,195)(73,224,103,194)(74,223,104,193)(75,222,105,192)(76,221,106,191)(77,220,107,190)(78,219,108,189)(79,218,109,188)(80,217,110,187)(81,216,111,186)(82,215,112,185)(83,214,113,184)(84,213,114,183)(85,212,115,182)(86,211,116,181)(87,210,117,240)(88,209,118,239)(89,208,119,238)(90,207,120,237)>;
G:=Group( (1,31)(2,206)(3,33)(4,208)(5,35)(6,210)(7,37)(8,212)(9,39)(10,214)(11,41)(12,216)(13,43)(14,218)(15,45)(16,220)(17,47)(18,222)(19,49)(20,224)(21,51)(22,226)(23,53)(24,228)(25,55)(26,230)(27,57)(28,232)(29,59)(30,234)(32,236)(34,238)(36,240)(38,182)(40,184)(42,186)(44,188)(46,190)(48,192)(50,194)(52,196)(54,198)(56,200)(58,202)(60,204)(61,161)(62,92)(63,163)(64,94)(65,165)(66,96)(67,167)(68,98)(69,169)(70,100)(71,171)(72,102)(73,173)(74,104)(75,175)(76,106)(77,177)(78,108)(79,179)(80,110)(81,121)(82,112)(83,123)(84,114)(85,125)(86,116)(87,127)(88,118)(89,129)(90,120)(91,131)(93,133)(95,135)(97,137)(99,139)(101,141)(103,143)(105,145)(107,147)(109,149)(111,151)(113,153)(115,155)(117,157)(119,159)(122,152)(124,154)(126,156)(128,158)(130,160)(132,162)(134,164)(136,166)(138,168)(140,170)(142,172)(144,174)(146,176)(148,178)(150,180)(181,211)(183,213)(185,215)(187,217)(189,219)(191,221)(193,223)(195,225)(197,227)(199,229)(201,231)(203,233)(205,235)(207,237)(209,239), (1,235)(2,236)(3,237)(4,238)(5,239)(6,240)(7,181)(8,182)(9,183)(10,184)(11,185)(12,186)(13,187)(14,188)(15,189)(16,190)(17,191)(18,192)(19,193)(20,194)(21,195)(22,196)(23,197)(24,198)(25,199)(26,200)(27,201)(28,202)(29,203)(30,204)(31,205)(32,206)(33,207)(34,208)(35,209)(36,210)(37,211)(38,212)(39,213)(40,214)(41,215)(42,216)(43,217)(44,218)(45,219)(46,220)(47,221)(48,222)(49,223)(50,224)(51,225)(52,226)(53,227)(54,228)(55,229)(56,230)(57,231)(58,232)(59,233)(60,234)(61,131)(62,132)(63,133)(64,134)(65,135)(66,136)(67,137)(68,138)(69,139)(70,140)(71,141)(72,142)(73,143)(74,144)(75,145)(76,146)(77,147)(78,148)(79,149)(80,150)(81,151)(82,152)(83,153)(84,154)(85,155)(86,156)(87,157)(88,158)(89,159)(90,160)(91,161)(92,162)(93,163)(94,164)(95,165)(96,166)(97,167)(98,168)(99,169)(100,170)(101,171)(102,172)(103,173)(104,174)(105,175)(106,176)(107,177)(108,178)(109,179)(110,180)(111,121)(112,122)(113,123)(114,124)(115,125)(116,126)(117,127)(118,128)(119,129)(120,130), (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), (1,162,31,132)(2,161,32,131)(3,160,33,130)(4,159,34,129)(5,158,35,128)(6,157,36,127)(7,156,37,126)(8,155,38,125)(9,154,39,124)(10,153,40,123)(11,152,41,122)(12,151,42,121)(13,150,43,180)(14,149,44,179)(15,148,45,178)(16,147,46,177)(17,146,47,176)(18,145,48,175)(19,144,49,174)(20,143,50,173)(21,142,51,172)(22,141,52,171)(23,140,53,170)(24,139,54,169)(25,138,55,168)(26,137,56,167)(27,136,57,166)(28,135,58,165)(29,134,59,164)(30,133,60,163)(61,236,91,206)(62,235,92,205)(63,234,93,204)(64,233,94,203)(65,232,95,202)(66,231,96,201)(67,230,97,200)(68,229,98,199)(69,228,99,198)(70,227,100,197)(71,226,101,196)(72,225,102,195)(73,224,103,194)(74,223,104,193)(75,222,105,192)(76,221,106,191)(77,220,107,190)(78,219,108,189)(79,218,109,188)(80,217,110,187)(81,216,111,186)(82,215,112,185)(83,214,113,184)(84,213,114,183)(85,212,115,182)(86,211,116,181)(87,210,117,240)(88,209,118,239)(89,208,119,238)(90,207,120,237) );
G=PermutationGroup([[(1,31),(2,206),(3,33),(4,208),(5,35),(6,210),(7,37),(8,212),(9,39),(10,214),(11,41),(12,216),(13,43),(14,218),(15,45),(16,220),(17,47),(18,222),(19,49),(20,224),(21,51),(22,226),(23,53),(24,228),(25,55),(26,230),(27,57),(28,232),(29,59),(30,234),(32,236),(34,238),(36,240),(38,182),(40,184),(42,186),(44,188),(46,190),(48,192),(50,194),(52,196),(54,198),(56,200),(58,202),(60,204),(61,161),(62,92),(63,163),(64,94),(65,165),(66,96),(67,167),(68,98),(69,169),(70,100),(71,171),(72,102),(73,173),(74,104),(75,175),(76,106),(77,177),(78,108),(79,179),(80,110),(81,121),(82,112),(83,123),(84,114),(85,125),(86,116),(87,127),(88,118),(89,129),(90,120),(91,131),(93,133),(95,135),(97,137),(99,139),(101,141),(103,143),(105,145),(107,147),(109,149),(111,151),(113,153),(115,155),(117,157),(119,159),(122,152),(124,154),(126,156),(128,158),(130,160),(132,162),(134,164),(136,166),(138,168),(140,170),(142,172),(144,174),(146,176),(148,178),(150,180),(181,211),(183,213),(185,215),(187,217),(189,219),(191,221),(193,223),(195,225),(197,227),(199,229),(201,231),(203,233),(205,235),(207,237),(209,239)], [(1,235),(2,236),(3,237),(4,238),(5,239),(6,240),(7,181),(8,182),(9,183),(10,184),(11,185),(12,186),(13,187),(14,188),(15,189),(16,190),(17,191),(18,192),(19,193),(20,194),(21,195),(22,196),(23,197),(24,198),(25,199),(26,200),(27,201),(28,202),(29,203),(30,204),(31,205),(32,206),(33,207),(34,208),(35,209),(36,210),(37,211),(38,212),(39,213),(40,214),(41,215),(42,216),(43,217),(44,218),(45,219),(46,220),(47,221),(48,222),(49,223),(50,224),(51,225),(52,226),(53,227),(54,228),(55,229),(56,230),(57,231),(58,232),(59,233),(60,234),(61,131),(62,132),(63,133),(64,134),(65,135),(66,136),(67,137),(68,138),(69,139),(70,140),(71,141),(72,142),(73,143),(74,144),(75,145),(76,146),(77,147),(78,148),(79,149),(80,150),(81,151),(82,152),(83,153),(84,154),(85,155),(86,156),(87,157),(88,158),(89,159),(90,160),(91,161),(92,162),(93,163),(94,164),(95,165),(96,166),(97,167),(98,168),(99,169),(100,170),(101,171),(102,172),(103,173),(104,174),(105,175),(106,176),(107,177),(108,178),(109,179),(110,180),(111,121),(112,122),(113,123),(114,124),(115,125),(116,126),(117,127),(118,128),(119,129),(120,130)], [(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)], [(1,162,31,132),(2,161,32,131),(3,160,33,130),(4,159,34,129),(5,158,35,128),(6,157,36,127),(7,156,37,126),(8,155,38,125),(9,154,39,124),(10,153,40,123),(11,152,41,122),(12,151,42,121),(13,150,43,180),(14,149,44,179),(15,148,45,178),(16,147,46,177),(17,146,47,176),(18,145,48,175),(19,144,49,174),(20,143,50,173),(21,142,51,172),(22,141,52,171),(23,140,53,170),(24,139,54,169),(25,138,55,168),(26,137,56,167),(27,136,57,166),(28,135,58,165),(29,134,59,164),(30,133,60,163),(61,236,91,206),(62,235,92,205),(63,234,93,204),(64,233,94,203),(65,232,95,202),(66,231,96,201),(67,230,97,200),(68,229,98,199),(69,228,99,198),(70,227,100,197),(71,226,101,196),(72,225,102,195),(73,224,103,194),(74,223,104,193),(75,222,105,192),(76,221,106,191),(77,220,107,190),(78,219,108,189),(79,218,109,188),(80,217,110,187),(81,216,111,186),(82,215,112,185),(83,214,113,184),(84,213,114,183),(85,212,115,182),(86,211,116,181),(87,210,117,240),(88,209,118,239),(89,208,119,238),(90,207,120,237)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 30M | ··· | 30T | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 30 | 30 | 30 | 30 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | + | - | + | + | - | + | - | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D5 | D6 | D6 | C4○D4 | D10 | D10 | Dic6 | D15 | Dic10 | D30 | D30 | Dic30 | S3×D4 | D4⋊2S3 | D4×D5 | D4⋊2D5 | D4×D15 | D4⋊2D15 |
| kernel | C22⋊2Dic30 | C30.4Q8 | C60⋊5C4 | C30.38D4 | C15×C22⋊C4 | C2×Dic30 | C22×Dic15 | C5×C22⋊C4 | Dic15 | C2×C30 | C3×C22⋊C4 | C2×C20 | C22×C10 | C30 | C2×C12 | C22×C6 | C2×C10 | C22⋊C4 | C2×C6 | C2×C4 | C23 | C22 | C10 | C10 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 4 | 16 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of C22⋊2Dic30 ►in GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 33 | 41 | 0 | 0 |
| 14 | 47 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 0 |
| 0 | 57 | 0 | 0 |
| 46 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 50 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[33,14,0,0,41,47,0,0,0,0,0,60,0,0,1,0],[0,46,0,0,57,0,0,0,0,0,11,0,0,0,0,50] >;
C22⋊2Dic30 in GAP, Magma, Sage, TeX
C_2^2\rtimes_2{\rm Dic}_{30} % in TeX
G:=Group("C2^2:2Dic30"); // GroupNames label
G:=SmallGroup(480,843);
// by ID
G=gap.SmallGroup(480,843);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,254,219,58,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^60=1,d^2=c^30,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations