metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C30).67D4, C30.216(C2×D4), C23.27(S3×D5), C6.80(C4○D20), Dic15⋊5C4⋊30C2, (C2×Dic5).55D6, (C22×Dic3)⋊2D5, (C22×D5).28D6, (C22×C6).25D10, C30.137(C4○D4), C30.38D4⋊18C2, D10⋊Dic3⋊27C2, (C2×C30).178C23, (C22×C10).100D6, C5⋊4(C23.23D6), C10.52(D4⋊2S3), (C2×Dic3).161D10, C3⋊4(C23.23D10), C22.11(C15⋊D4), C15⋊16(C22.D4), (C22×C30).40C22, C2.25(Dic5.D6), (C6×Dic5).105C22, (C2×Dic15).126C22, (C10×Dic3).197C22, (Dic3×C2×C10)⋊2C2, (C6×C5⋊D4).3C2, (C2×C5⋊D4).3S3, C6.90(C2×C5⋊D4), C2.22(C2×C15⋊D4), C10.91(C2×C3⋊D4), (D5×C2×C6).46C22, C22.221(C2×S3×D5), (C2×C6).16(C5⋊D4), (C2×C10).57(C3⋊D4), (C2×C6).190(C22×D5), (C2×C10).190(C22×S3), SmallGroup(480,612)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C30).D4
G = < a,b,c,d | a2=b30=c4=d2=1, ab=ba, cac-1=dad=ab15, cbc-1=b-1, dbd=b19, dcd=b15c-1 >
Subgroups: 668 in 156 conjugacy classes, 52 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, C10, Dic3, C12, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C3×D4, C22×C6, C22×C6, C3×D5, C30, C30, C30, C22.D4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, Dic3⋊C4, C6.D4, C22×Dic3, C6×D4, C5×Dic3, C3×Dic5, Dic15, C6×D5, C2×C30, C2×C30, C2×C30, C10.D4, D10⋊C4, C23.D5, C2×C5⋊D4, C22×C20, C23.23D6, C6×Dic5, C3×C5⋊D4, C10×Dic3, C10×Dic3, C2×Dic15, D5×C2×C6, C22×C30, C23.23D10, D10⋊Dic3, Dic15⋊5C4, C30.38D4, C6×C5⋊D4, Dic3×C2×C10, (C2×C30).D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C3⋊D4, C22×S3, C22.D4, C5⋊D4, C22×D5, D4⋊2S3, C2×C3⋊D4, S3×D5, C4○D20, C2×C5⋊D4, C23.23D6, C15⋊D4, C2×S3×D5, C23.23D10, Dic5.D6, C2×C15⋊D4, (C2×C30).D4
►On 240 points
Generators in S240
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(61 98)(62 99)(63 100)(64 101)(65 102)(66 103)(67 104)(68 105)(69 106)(70 107)(71 108)(72 109)(73 110)(74 111)(75 112)(76 113)(77 114)(78 115)(79 116)(80 117)(81 118)(82 119)(83 120)(84 91)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(121 151)(122 152)(123 153)(124 154)(125 155)(126 156)(127 157)(128 158)(129 159)(130 160)(131 161)(132 162)(133 163)(134 164)(135 165)(136 166)(137 167)(138 168)(139 169)(140 170)(141 171)(142 172)(143 173)(144 174)(145 175)(146 176)(147 177)(148 178)(149 179)(150 180)(181 226)(182 227)(183 228)(184 229)(185 230)(186 231)(187 232)(188 233)(189 234)(190 235)(191 236)(192 237)(193 238)(194 239)(195 240)(196 211)(197 212)(198 213)(199 214)(200 215)(201 216)(202 217)(203 218)(204 219)(205 220)(206 221)(207 222)(208 223)(209 224)(210 225)
(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)
(1 221 36 206)(2 220 37 205)(3 219 38 204)(4 218 39 203)(5 217 40 202)(6 216 41 201)(7 215 42 200)(8 214 43 199)(9 213 44 198)(10 212 45 197)(11 211 46 196)(12 240 47 195)(13 239 48 194)(14 238 49 193)(15 237 50 192)(16 236 51 191)(17 235 52 190)(18 234 53 189)(19 233 54 188)(20 232 55 187)(21 231 56 186)(22 230 57 185)(23 229 58 184)(24 228 59 183)(25 227 60 182)(26 226 31 181)(27 225 32 210)(28 224 33 209)(29 223 34 208)(30 222 35 207)(61 160 113 130)(62 159 114 129)(63 158 115 128)(64 157 116 127)(65 156 117 126)(66 155 118 125)(67 154 119 124)(68 153 120 123)(69 152 91 122)(70 151 92 121)(71 180 93 150)(72 179 94 149)(73 178 95 148)(74 177 96 147)(75 176 97 146)(76 175 98 145)(77 174 99 144)(78 173 100 143)(79 172 101 142)(80 171 102 141)(81 170 103 140)(82 169 104 139)(83 168 105 138)(84 167 106 137)(85 166 107 136)(86 165 108 135)(87 164 109 134)(88 163 110 133)(89 162 111 132)(90 161 112 131)
(1 146)(2 135)(3 124)(4 143)(5 132)(6 121)(7 140)(8 129)(9 148)(10 137)(11 126)(12 145)(13 134)(14 123)(15 142)(16 131)(17 150)(18 139)(19 128)(20 147)(21 136)(22 125)(23 144)(24 133)(25 122)(26 141)(27 130)(28 149)(29 138)(30 127)(31 171)(32 160)(33 179)(34 168)(35 157)(36 176)(37 165)(38 154)(39 173)(40 162)(41 151)(42 170)(43 159)(44 178)(45 167)(46 156)(47 175)(48 164)(49 153)(50 172)(51 161)(52 180)(53 169)(54 158)(55 177)(56 166)(57 155)(58 174)(59 163)(60 152)(61 195)(62 184)(63 203)(64 192)(65 181)(66 200)(67 189)(68 208)(69 197)(70 186)(71 205)(72 194)(73 183)(74 202)(75 191)(76 210)(77 199)(78 188)(79 207)(80 196)(81 185)(82 204)(83 193)(84 182)(85 201)(86 190)(87 209)(88 198)(89 187)(90 206)(91 212)(92 231)(93 220)(94 239)(95 228)(96 217)(97 236)(98 225)(99 214)(100 233)(101 222)(102 211)(103 230)(104 219)(105 238)(106 227)(107 216)(108 235)(109 224)(110 213)(111 232)(112 221)(113 240)(114 229)(115 218)(116 237)(117 226)(118 215)(119 234)(120 223)
G:=sub<Sym(240)| (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(61,98)(62,99)(63,100)(64,101)(65,102)(66,103)(67,104)(68,105)(69,106)(70,107)(71,108)(72,109)(73,110)(74,111)(75,112)(76,113)(77,114)(78,115)(79,116)(80,117)(81,118)(82,119)(83,120)(84,91)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(121,151)(122,152)(123,153)(124,154)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(131,161)(132,162)(133,163)(134,164)(135,165)(136,166)(137,167)(138,168)(139,169)(140,170)(141,171)(142,172)(143,173)(144,174)(145,175)(146,176)(147,177)(148,178)(149,179)(150,180)(181,226)(182,227)(183,228)(184,229)(185,230)(186,231)(187,232)(188,233)(189,234)(190,235)(191,236)(192,237)(193,238)(194,239)(195,240)(196,211)(197,212)(198,213)(199,214)(200,215)(201,216)(202,217)(203,218)(204,219)(205,220)(206,221)(207,222)(208,223)(209,224)(210,225), (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), (1,221,36,206)(2,220,37,205)(3,219,38,204)(4,218,39,203)(5,217,40,202)(6,216,41,201)(7,215,42,200)(8,214,43,199)(9,213,44,198)(10,212,45,197)(11,211,46,196)(12,240,47,195)(13,239,48,194)(14,238,49,193)(15,237,50,192)(16,236,51,191)(17,235,52,190)(18,234,53,189)(19,233,54,188)(20,232,55,187)(21,231,56,186)(22,230,57,185)(23,229,58,184)(24,228,59,183)(25,227,60,182)(26,226,31,181)(27,225,32,210)(28,224,33,209)(29,223,34,208)(30,222,35,207)(61,160,113,130)(62,159,114,129)(63,158,115,128)(64,157,116,127)(65,156,117,126)(66,155,118,125)(67,154,119,124)(68,153,120,123)(69,152,91,122)(70,151,92,121)(71,180,93,150)(72,179,94,149)(73,178,95,148)(74,177,96,147)(75,176,97,146)(76,175,98,145)(77,174,99,144)(78,173,100,143)(79,172,101,142)(80,171,102,141)(81,170,103,140)(82,169,104,139)(83,168,105,138)(84,167,106,137)(85,166,107,136)(86,165,108,135)(87,164,109,134)(88,163,110,133)(89,162,111,132)(90,161,112,131), (1,146)(2,135)(3,124)(4,143)(5,132)(6,121)(7,140)(8,129)(9,148)(10,137)(11,126)(12,145)(13,134)(14,123)(15,142)(16,131)(17,150)(18,139)(19,128)(20,147)(21,136)(22,125)(23,144)(24,133)(25,122)(26,141)(27,130)(28,149)(29,138)(30,127)(31,171)(32,160)(33,179)(34,168)(35,157)(36,176)(37,165)(38,154)(39,173)(40,162)(41,151)(42,170)(43,159)(44,178)(45,167)(46,156)(47,175)(48,164)(49,153)(50,172)(51,161)(52,180)(53,169)(54,158)(55,177)(56,166)(57,155)(58,174)(59,163)(60,152)(61,195)(62,184)(63,203)(64,192)(65,181)(66,200)(67,189)(68,208)(69,197)(70,186)(71,205)(72,194)(73,183)(74,202)(75,191)(76,210)(77,199)(78,188)(79,207)(80,196)(81,185)(82,204)(83,193)(84,182)(85,201)(86,190)(87,209)(88,198)(89,187)(90,206)(91,212)(92,231)(93,220)(94,239)(95,228)(96,217)(97,236)(98,225)(99,214)(100,233)(101,222)(102,211)(103,230)(104,219)(105,238)(106,227)(107,216)(108,235)(109,224)(110,213)(111,232)(112,221)(113,240)(114,229)(115,218)(116,237)(117,226)(118,215)(119,234)(120,223)>;
G:=Group( (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(61,98)(62,99)(63,100)(64,101)(65,102)(66,103)(67,104)(68,105)(69,106)(70,107)(71,108)(72,109)(73,110)(74,111)(75,112)(76,113)(77,114)(78,115)(79,116)(80,117)(81,118)(82,119)(83,120)(84,91)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(121,151)(122,152)(123,153)(124,154)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(131,161)(132,162)(133,163)(134,164)(135,165)(136,166)(137,167)(138,168)(139,169)(140,170)(141,171)(142,172)(143,173)(144,174)(145,175)(146,176)(147,177)(148,178)(149,179)(150,180)(181,226)(182,227)(183,228)(184,229)(185,230)(186,231)(187,232)(188,233)(189,234)(190,235)(191,236)(192,237)(193,238)(194,239)(195,240)(196,211)(197,212)(198,213)(199,214)(200,215)(201,216)(202,217)(203,218)(204,219)(205,220)(206,221)(207,222)(208,223)(209,224)(210,225), (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), (1,221,36,206)(2,220,37,205)(3,219,38,204)(4,218,39,203)(5,217,40,202)(6,216,41,201)(7,215,42,200)(8,214,43,199)(9,213,44,198)(10,212,45,197)(11,211,46,196)(12,240,47,195)(13,239,48,194)(14,238,49,193)(15,237,50,192)(16,236,51,191)(17,235,52,190)(18,234,53,189)(19,233,54,188)(20,232,55,187)(21,231,56,186)(22,230,57,185)(23,229,58,184)(24,228,59,183)(25,227,60,182)(26,226,31,181)(27,225,32,210)(28,224,33,209)(29,223,34,208)(30,222,35,207)(61,160,113,130)(62,159,114,129)(63,158,115,128)(64,157,116,127)(65,156,117,126)(66,155,118,125)(67,154,119,124)(68,153,120,123)(69,152,91,122)(70,151,92,121)(71,180,93,150)(72,179,94,149)(73,178,95,148)(74,177,96,147)(75,176,97,146)(76,175,98,145)(77,174,99,144)(78,173,100,143)(79,172,101,142)(80,171,102,141)(81,170,103,140)(82,169,104,139)(83,168,105,138)(84,167,106,137)(85,166,107,136)(86,165,108,135)(87,164,109,134)(88,163,110,133)(89,162,111,132)(90,161,112,131), (1,146)(2,135)(3,124)(4,143)(5,132)(6,121)(7,140)(8,129)(9,148)(10,137)(11,126)(12,145)(13,134)(14,123)(15,142)(16,131)(17,150)(18,139)(19,128)(20,147)(21,136)(22,125)(23,144)(24,133)(25,122)(26,141)(27,130)(28,149)(29,138)(30,127)(31,171)(32,160)(33,179)(34,168)(35,157)(36,176)(37,165)(38,154)(39,173)(40,162)(41,151)(42,170)(43,159)(44,178)(45,167)(46,156)(47,175)(48,164)(49,153)(50,172)(51,161)(52,180)(53,169)(54,158)(55,177)(56,166)(57,155)(58,174)(59,163)(60,152)(61,195)(62,184)(63,203)(64,192)(65,181)(66,200)(67,189)(68,208)(69,197)(70,186)(71,205)(72,194)(73,183)(74,202)(75,191)(76,210)(77,199)(78,188)(79,207)(80,196)(81,185)(82,204)(83,193)(84,182)(85,201)(86,190)(87,209)(88,198)(89,187)(90,206)(91,212)(92,231)(93,220)(94,239)(95,228)(96,217)(97,236)(98,225)(99,214)(100,233)(101,222)(102,211)(103,230)(104,219)(105,238)(106,227)(107,216)(108,235)(109,224)(110,213)(111,232)(112,221)(113,240)(114,229)(115,218)(116,237)(117,226)(118,215)(119,234)(120,223) );
G=PermutationGroup([[(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,57),(8,58),(9,59),(10,60),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(21,41),(22,42),(23,43),(24,44),(25,45),(26,46),(27,47),(28,48),(29,49),(30,50),(61,98),(62,99),(63,100),(64,101),(65,102),(66,103),(67,104),(68,105),(69,106),(70,107),(71,108),(72,109),(73,110),(74,111),(75,112),(76,113),(77,114),(78,115),(79,116),(80,117),(81,118),(82,119),(83,120),(84,91),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(121,151),(122,152),(123,153),(124,154),(125,155),(126,156),(127,157),(128,158),(129,159),(130,160),(131,161),(132,162),(133,163),(134,164),(135,165),(136,166),(137,167),(138,168),(139,169),(140,170),(141,171),(142,172),(143,173),(144,174),(145,175),(146,176),(147,177),(148,178),(149,179),(150,180),(181,226),(182,227),(183,228),(184,229),(185,230),(186,231),(187,232),(188,233),(189,234),(190,235),(191,236),(192,237),(193,238),(194,239),(195,240),(196,211),(197,212),(198,213),(199,214),(200,215),(201,216),(202,217),(203,218),(204,219),(205,220),(206,221),(207,222),(208,223),(209,224),(210,225)], [(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)], [(1,221,36,206),(2,220,37,205),(3,219,38,204),(4,218,39,203),(5,217,40,202),(6,216,41,201),(7,215,42,200),(8,214,43,199),(9,213,44,198),(10,212,45,197),(11,211,46,196),(12,240,47,195),(13,239,48,194),(14,238,49,193),(15,237,50,192),(16,236,51,191),(17,235,52,190),(18,234,53,189),(19,233,54,188),(20,232,55,187),(21,231,56,186),(22,230,57,185),(23,229,58,184),(24,228,59,183),(25,227,60,182),(26,226,31,181),(27,225,32,210),(28,224,33,209),(29,223,34,208),(30,222,35,207),(61,160,113,130),(62,159,114,129),(63,158,115,128),(64,157,116,127),(65,156,117,126),(66,155,118,125),(67,154,119,124),(68,153,120,123),(69,152,91,122),(70,151,92,121),(71,180,93,150),(72,179,94,149),(73,178,95,148),(74,177,96,147),(75,176,97,146),(76,175,98,145),(77,174,99,144),(78,173,100,143),(79,172,101,142),(80,171,102,141),(81,170,103,140),(82,169,104,139),(83,168,105,138),(84,167,106,137),(85,166,107,136),(86,165,108,135),(87,164,109,134),(88,163,110,133),(89,162,111,132),(90,161,112,131)], [(1,146),(2,135),(3,124),(4,143),(5,132),(6,121),(7,140),(8,129),(9,148),(10,137),(11,126),(12,145),(13,134),(14,123),(15,142),(16,131),(17,150),(18,139),(19,128),(20,147),(21,136),(22,125),(23,144),(24,133),(25,122),(26,141),(27,130),(28,149),(29,138),(30,127),(31,171),(32,160),(33,179),(34,168),(35,157),(36,176),(37,165),(38,154),(39,173),(40,162),(41,151),(42,170),(43,159),(44,178),(45,167),(46,156),(47,175),(48,164),(49,153),(50,172),(51,161),(52,180),(53,169),(54,158),(55,177),(56,166),(57,155),(58,174),(59,163),(60,152),(61,195),(62,184),(63,203),(64,192),(65,181),(66,200),(67,189),(68,208),(69,197),(70,186),(71,205),(72,194),(73,183),(74,202),(75,191),(76,210),(77,199),(78,188),(79,207),(80,196),(81,185),(82,204),(83,193),(84,182),(85,201),(86,190),(87,209),(88,198),(89,187),(90,206),(91,212),(92,231),(93,220),(94,239),(95,228),(96,217),(97,236),(98,225),(99,214),(100,233),(101,222),(102,211),(103,230),(104,219),(105,238),(106,227),(107,216),(108,235),(109,224),(110,213),(111,232),(112,221),(113,240),(114,229),(115,218),(116,237),(117,226),(118,215),(119,234),(120,223)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10N | 12A | 12B | 15A | 15B | 20A | ··· | 20P | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 2 | 6 | 6 | 6 | 6 | 20 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 20 | 20 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | C3⋊D4 | C5⋊D4 | C4○D20 | D4⋊2S3 | S3×D5 | C15⋊D4 | C2×S3×D5 | Dic5.D6 |
| kernel | (C2×C30).D4 | D10⋊Dic3 | Dic15⋊5C4 | C30.38D4 | C6×C5⋊D4 | Dic3×C2×C10 | C2×C5⋊D4 | C2×C30 | C22×Dic3 | C2×Dic5 | C22×D5 | C22×C10 | C30 | C2×Dic3 | C22×C6 | C2×C10 | C2×C6 | C6 | C10 | C23 | C22 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 2 | 4 | 8 | 16 | 2 | 2 | 4 | 2 | 8 |
Matrix representation of (C2×C30).D4 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 0 | 0 | 0 |
| 0 | 0 | 13 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 19 |
| 0 | 0 | 0 | 0 | 48 | 59 |
| 0 | 50 | 0 | 0 | 0 | 0 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 14 | 0 | 0 |
| 0 | 0 | 22 | 39 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 54 |
| 0 | 0 | 0 | 0 | 41 | 46 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 39 | 47 | 0 | 0 |
| 0 | 0 | 4 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 37 |
| 0 | 0 | 0 | 0 | 10 | 34 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,34,13,0,0,0,0,0,9,0,0,0,0,0,0,1,48,0,0,0,0,19,59],[0,11,0,0,0,0,50,0,0,0,0,0,0,0,22,22,0,0,0,0,14,39,0,0,0,0,0,0,15,41,0,0,0,0,54,46],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,39,4,0,0,0,0,47,22,0,0,0,0,0,0,27,10,0,0,0,0,37,34] >;
(C2×C30).D4 in GAP, Magma, Sage, TeX
(C_2\times C_{30}).D_4 % in TeX
G:=Group("(C2xC30).D4"); // GroupNames label
G:=SmallGroup(480,612);
// by ID
G=gap.SmallGroup(480,612);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,120,422,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^30=c^4=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a*b^15,c*b*c^-1=b^-1,d*b*d=b^19,d*c*d=b^15*c^-1>;
// generators/relations