direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C22⋊C4×C30, C23⋊3C60, C24.2C30, C2.1(D4×C30), C22⋊3(C2×C60), (C22×C6)⋊3C20, (C22×C20)⋊7C6, (C22×C60)⋊7C2, (C22×C4)⋊3C30, C6.64(D4×C10), C10.64(C6×D4), (C22×C30)⋊11C4, (C22×C12)⋊3C10, (C22×C10)⋊9C12, (C2×C60)⋊43C22, (C2×C30).193D4, C30.447(C2×D4), (C23×C6).1C10, (C23×C30).1C2, C2.1(C22×C60), (C23×C10).3C6, C23.15(C2×C30), C6.29(C22×C20), C22.12(D4×C15), C30.236(C22×C4), (C2×C30).450C23, C10.42(C22×C12), C22.4(C22×C30), (C22×C30).129C22, (C2×C4)⋊3(C2×C30), (C2×C6)⋊7(C2×C20), (C2×C20)⋊11(C2×C6), (C2×C30)⋊39(C2×C4), (C2×C6).50(C5×D4), (C2×C10)⋊14(C2×C12), (C2×C12)⋊11(C2×C10), (C2×C10).50(C3×D4), (C2×C6).70(C22×C10), (C2×C10).70(C22×C6), (C22×C6).24(C2×C10), (C22×C10).32(C2×C6), SmallGroup(480,920)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊C4×C30
G = < a,b,c,d | a30=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Subgroups: 376 in 264 conjugacy classes, 152 normal (24 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, C23, C23, C23, C10, C10, C10, C12, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C22×C4, C24, C20, C2×C10, C2×C10, C2×C10, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C30, C30, C30, C2×C22⋊C4, C2×C20, C2×C20, C22×C10, C22×C10, C22×C10, C3×C22⋊C4, C22×C12, C23×C6, C60, C2×C30, C2×C30, C2×C30, C5×C22⋊C4, C22×C20, C23×C10, C6×C22⋊C4, C2×C60, C2×C60, C22×C30, C22×C30, C22×C30, C10×C22⋊C4, C15×C22⋊C4, C22×C60, C23×C30, C22⋊C4×C30
Quotients: C1, C2, C3, C4, C22, C5, C6, C2×C4, D4, C23, C10, C12, C2×C6, C15, C22⋊C4, C22×C4, C2×D4, C20, C2×C10, C2×C12, C3×D4, C22×C6, C30, C2×C22⋊C4, C2×C20, C5×D4, C22×C10, C3×C22⋊C4, C22×C12, C6×D4, C60, C2×C30, C5×C22⋊C4, C22×C20, D4×C10, C6×C22⋊C4, C2×C60, D4×C15, C22×C30, C10×C22⋊C4, C15×C22⋊C4, C22×C60, D4×C30, C22⋊C4×C30
►On 240 points
Generators in S240
(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)
(1 161)(2 162)(3 163)(4 164)(5 165)(6 166)(7 167)(8 168)(9 169)(10 170)(11 171)(12 172)(13 173)(14 174)(15 175)(16 176)(17 177)(18 178)(19 179)(20 180)(21 151)(22 152)(23 153)(24 154)(25 155)(26 156)(27 157)(28 158)(29 159)(30 160)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)(61 76)(62 77)(63 78)(64 79)(65 80)(66 81)(67 82)(68 83)(69 84)(70 85)(71 86)(72 87)(73 88)(74 89)(75 90)(91 106)(92 107)(93 108)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)(105 120)(121 187)(122 188)(123 189)(124 190)(125 191)(126 192)(127 193)(128 194)(129 195)(130 196)(131 197)(132 198)(133 199)(134 200)(135 201)(136 202)(137 203)(138 204)(139 205)(140 206)(141 207)(142 208)(143 209)(144 210)(145 181)(146 182)(147 183)(148 184)(149 185)(150 186)(211 226)(212 227)(213 228)(214 229)(215 230)(216 231)(217 232)(218 233)(219 234)(220 235)(221 236)(222 237)(223 238)(224 239)(225 240)
(1 176)(2 177)(3 178)(4 179)(5 180)(6 151)(7 152)(8 153)(9 154)(10 155)(11 156)(12 157)(13 158)(14 159)(15 160)(16 161)(17 162)(18 163)(19 164)(20 165)(21 166)(22 167)(23 168)(24 169)(25 170)(26 171)(27 172)(28 173)(29 174)(30 175)(31 227)(32 228)(33 229)(34 230)(35 231)(36 232)(37 233)(38 234)(39 235)(40 236)(41 237)(42 238)(43 239)(44 240)(45 211)(46 212)(47 213)(48 214)(49 215)(50 216)(51 217)(52 218)(53 219)(54 220)(55 221)(56 222)(57 223)(58 224)(59 225)(60 226)(61 105)(62 106)(63 107)(64 108)(65 109)(66 110)(67 111)(68 112)(69 113)(70 114)(71 115)(72 116)(73 117)(74 118)(75 119)(76 120)(77 91)(78 92)(79 93)(80 94)(81 95)(82 96)(83 97)(84 98)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(121 202)(122 203)(123 204)(124 205)(125 206)(126 207)(127 208)(128 209)(129 210)(130 181)(131 182)(132 183)(133 184)(134 185)(135 186)(136 187)(137 188)(138 189)(139 190)(140 191)(141 192)(142 193)(143 194)(144 195)(145 196)(146 197)(147 198)(148 199)(149 200)(150 201)
(1 234 193 95)(2 235 194 96)(3 236 195 97)(4 237 196 98)(5 238 197 99)(6 239 198 100)(7 240 199 101)(8 211 200 102)(9 212 201 103)(10 213 202 104)(11 214 203 105)(12 215 204 106)(13 216 205 107)(14 217 206 108)(15 218 207 109)(16 219 208 110)(17 220 209 111)(18 221 210 112)(19 222 181 113)(20 223 182 114)(21 224 183 115)(22 225 184 116)(23 226 185 117)(24 227 186 118)(25 228 187 119)(26 229 188 120)(27 230 189 91)(28 231 190 92)(29 232 191 93)(30 233 192 94)(31 135 74 169)(32 136 75 170)(33 137 76 171)(34 138 77 172)(35 139 78 173)(36 140 79 174)(37 141 80 175)(38 142 81 176)(39 143 82 177)(40 144 83 178)(41 145 84 179)(42 146 85 180)(43 147 86 151)(44 148 87 152)(45 149 88 153)(46 150 89 154)(47 121 90 155)(48 122 61 156)(49 123 62 157)(50 124 63 158)(51 125 64 159)(52 126 65 160)(53 127 66 161)(54 128 67 162)(55 129 68 163)(56 130 69 164)(57 131 70 165)(58 132 71 166)(59 133 72 167)(60 134 73 168)
G:=sub<Sym(240)| (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), (1,161)(2,162)(3,163)(4,164)(5,165)(6,166)(7,167)(8,168)(9,169)(10,170)(11,171)(12,172)(13,173)(14,174)(15,175)(16,176)(17,177)(18,178)(19,179)(20,180)(21,151)(22,152)(23,153)(24,154)(25,155)(26,156)(27,157)(28,158)(29,159)(30,160)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,56)(42,57)(43,58)(44,59)(45,60)(61,76)(62,77)(63,78)(64,79)(65,80)(66,81)(67,82)(68,83)(69,84)(70,85)(71,86)(72,87)(73,88)(74,89)(75,90)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120)(121,187)(122,188)(123,189)(124,190)(125,191)(126,192)(127,193)(128,194)(129,195)(130,196)(131,197)(132,198)(133,199)(134,200)(135,201)(136,202)(137,203)(138,204)(139,205)(140,206)(141,207)(142,208)(143,209)(144,210)(145,181)(146,182)(147,183)(148,184)(149,185)(150,186)(211,226)(212,227)(213,228)(214,229)(215,230)(216,231)(217,232)(218,233)(219,234)(220,235)(221,236)(222,237)(223,238)(224,239)(225,240), (1,176)(2,177)(3,178)(4,179)(5,180)(6,151)(7,152)(8,153)(9,154)(10,155)(11,156)(12,157)(13,158)(14,159)(15,160)(16,161)(17,162)(18,163)(19,164)(20,165)(21,166)(22,167)(23,168)(24,169)(25,170)(26,171)(27,172)(28,173)(29,174)(30,175)(31,227)(32,228)(33,229)(34,230)(35,231)(36,232)(37,233)(38,234)(39,235)(40,236)(41,237)(42,238)(43,239)(44,240)(45,211)(46,212)(47,213)(48,214)(49,215)(50,216)(51,217)(52,218)(53,219)(54,220)(55,221)(56,222)(57,223)(58,224)(59,225)(60,226)(61,105)(62,106)(63,107)(64,108)(65,109)(66,110)(67,111)(68,112)(69,113)(70,114)(71,115)(72,116)(73,117)(74,118)(75,119)(76,120)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(121,202)(122,203)(123,204)(124,205)(125,206)(126,207)(127,208)(128,209)(129,210)(130,181)(131,182)(132,183)(133,184)(134,185)(135,186)(136,187)(137,188)(138,189)(139,190)(140,191)(141,192)(142,193)(143,194)(144,195)(145,196)(146,197)(147,198)(148,199)(149,200)(150,201), (1,234,193,95)(2,235,194,96)(3,236,195,97)(4,237,196,98)(5,238,197,99)(6,239,198,100)(7,240,199,101)(8,211,200,102)(9,212,201,103)(10,213,202,104)(11,214,203,105)(12,215,204,106)(13,216,205,107)(14,217,206,108)(15,218,207,109)(16,219,208,110)(17,220,209,111)(18,221,210,112)(19,222,181,113)(20,223,182,114)(21,224,183,115)(22,225,184,116)(23,226,185,117)(24,227,186,118)(25,228,187,119)(26,229,188,120)(27,230,189,91)(28,231,190,92)(29,232,191,93)(30,233,192,94)(31,135,74,169)(32,136,75,170)(33,137,76,171)(34,138,77,172)(35,139,78,173)(36,140,79,174)(37,141,80,175)(38,142,81,176)(39,143,82,177)(40,144,83,178)(41,145,84,179)(42,146,85,180)(43,147,86,151)(44,148,87,152)(45,149,88,153)(46,150,89,154)(47,121,90,155)(48,122,61,156)(49,123,62,157)(50,124,63,158)(51,125,64,159)(52,126,65,160)(53,127,66,161)(54,128,67,162)(55,129,68,163)(56,130,69,164)(57,131,70,165)(58,132,71,166)(59,133,72,167)(60,134,73,168)>;
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), (1,161)(2,162)(3,163)(4,164)(5,165)(6,166)(7,167)(8,168)(9,169)(10,170)(11,171)(12,172)(13,173)(14,174)(15,175)(16,176)(17,177)(18,178)(19,179)(20,180)(21,151)(22,152)(23,153)(24,154)(25,155)(26,156)(27,157)(28,158)(29,159)(30,160)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,56)(42,57)(43,58)(44,59)(45,60)(61,76)(62,77)(63,78)(64,79)(65,80)(66,81)(67,82)(68,83)(69,84)(70,85)(71,86)(72,87)(73,88)(74,89)(75,90)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120)(121,187)(122,188)(123,189)(124,190)(125,191)(126,192)(127,193)(128,194)(129,195)(130,196)(131,197)(132,198)(133,199)(134,200)(135,201)(136,202)(137,203)(138,204)(139,205)(140,206)(141,207)(142,208)(143,209)(144,210)(145,181)(146,182)(147,183)(148,184)(149,185)(150,186)(211,226)(212,227)(213,228)(214,229)(215,230)(216,231)(217,232)(218,233)(219,234)(220,235)(221,236)(222,237)(223,238)(224,239)(225,240), (1,176)(2,177)(3,178)(4,179)(5,180)(6,151)(7,152)(8,153)(9,154)(10,155)(11,156)(12,157)(13,158)(14,159)(15,160)(16,161)(17,162)(18,163)(19,164)(20,165)(21,166)(22,167)(23,168)(24,169)(25,170)(26,171)(27,172)(28,173)(29,174)(30,175)(31,227)(32,228)(33,229)(34,230)(35,231)(36,232)(37,233)(38,234)(39,235)(40,236)(41,237)(42,238)(43,239)(44,240)(45,211)(46,212)(47,213)(48,214)(49,215)(50,216)(51,217)(52,218)(53,219)(54,220)(55,221)(56,222)(57,223)(58,224)(59,225)(60,226)(61,105)(62,106)(63,107)(64,108)(65,109)(66,110)(67,111)(68,112)(69,113)(70,114)(71,115)(72,116)(73,117)(74,118)(75,119)(76,120)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(121,202)(122,203)(123,204)(124,205)(125,206)(126,207)(127,208)(128,209)(129,210)(130,181)(131,182)(132,183)(133,184)(134,185)(135,186)(136,187)(137,188)(138,189)(139,190)(140,191)(141,192)(142,193)(143,194)(144,195)(145,196)(146,197)(147,198)(148,199)(149,200)(150,201), (1,234,193,95)(2,235,194,96)(3,236,195,97)(4,237,196,98)(5,238,197,99)(6,239,198,100)(7,240,199,101)(8,211,200,102)(9,212,201,103)(10,213,202,104)(11,214,203,105)(12,215,204,106)(13,216,205,107)(14,217,206,108)(15,218,207,109)(16,219,208,110)(17,220,209,111)(18,221,210,112)(19,222,181,113)(20,223,182,114)(21,224,183,115)(22,225,184,116)(23,226,185,117)(24,227,186,118)(25,228,187,119)(26,229,188,120)(27,230,189,91)(28,231,190,92)(29,232,191,93)(30,233,192,94)(31,135,74,169)(32,136,75,170)(33,137,76,171)(34,138,77,172)(35,139,78,173)(36,140,79,174)(37,141,80,175)(38,142,81,176)(39,143,82,177)(40,144,83,178)(41,145,84,179)(42,146,85,180)(43,147,86,151)(44,148,87,152)(45,149,88,153)(46,150,89,154)(47,121,90,155)(48,122,61,156)(49,123,62,157)(50,124,63,158)(51,125,64,159)(52,126,65,160)(53,127,66,161)(54,128,67,162)(55,129,68,163)(56,130,69,164)(57,131,70,165)(58,132,71,166)(59,133,72,167)(60,134,73,168) );
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)], [(1,161),(2,162),(3,163),(4,164),(5,165),(6,166),(7,167),(8,168),(9,169),(10,170),(11,171),(12,172),(13,173),(14,174),(15,175),(16,176),(17,177),(18,178),(19,179),(20,180),(21,151),(22,152),(23,153),(24,154),(25,155),(26,156),(27,157),(28,158),(29,159),(30,160),(31,46),(32,47),(33,48),(34,49),(35,50),(36,51),(37,52),(38,53),(39,54),(40,55),(41,56),(42,57),(43,58),(44,59),(45,60),(61,76),(62,77),(63,78),(64,79),(65,80),(66,81),(67,82),(68,83),(69,84),(70,85),(71,86),(72,87),(73,88),(74,89),(75,90),(91,106),(92,107),(93,108),(94,109),(95,110),(96,111),(97,112),(98,113),(99,114),(100,115),(101,116),(102,117),(103,118),(104,119),(105,120),(121,187),(122,188),(123,189),(124,190),(125,191),(126,192),(127,193),(128,194),(129,195),(130,196),(131,197),(132,198),(133,199),(134,200),(135,201),(136,202),(137,203),(138,204),(139,205),(140,206),(141,207),(142,208),(143,209),(144,210),(145,181),(146,182),(147,183),(148,184),(149,185),(150,186),(211,226),(212,227),(213,228),(214,229),(215,230),(216,231),(217,232),(218,233),(219,234),(220,235),(221,236),(222,237),(223,238),(224,239),(225,240)], [(1,176),(2,177),(3,178),(4,179),(5,180),(6,151),(7,152),(8,153),(9,154),(10,155),(11,156),(12,157),(13,158),(14,159),(15,160),(16,161),(17,162),(18,163),(19,164),(20,165),(21,166),(22,167),(23,168),(24,169),(25,170),(26,171),(27,172),(28,173),(29,174),(30,175),(31,227),(32,228),(33,229),(34,230),(35,231),(36,232),(37,233),(38,234),(39,235),(40,236),(41,237),(42,238),(43,239),(44,240),(45,211),(46,212),(47,213),(48,214),(49,215),(50,216),(51,217),(52,218),(53,219),(54,220),(55,221),(56,222),(57,223),(58,224),(59,225),(60,226),(61,105),(62,106),(63,107),(64,108),(65,109),(66,110),(67,111),(68,112),(69,113),(70,114),(71,115),(72,116),(73,117),(74,118),(75,119),(76,120),(77,91),(78,92),(79,93),(80,94),(81,95),(82,96),(83,97),(84,98),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(121,202),(122,203),(123,204),(124,205),(125,206),(126,207),(127,208),(128,209),(129,210),(130,181),(131,182),(132,183),(133,184),(134,185),(135,186),(136,187),(137,188),(138,189),(139,190),(140,191),(141,192),(142,193),(143,194),(144,195),(145,196),(146,197),(147,198),(148,199),(149,200),(150,201)], [(1,234,193,95),(2,235,194,96),(3,236,195,97),(4,237,196,98),(5,238,197,99),(6,239,198,100),(7,240,199,101),(8,211,200,102),(9,212,201,103),(10,213,202,104),(11,214,203,105),(12,215,204,106),(13,216,205,107),(14,217,206,108),(15,218,207,109),(16,219,208,110),(17,220,209,111),(18,221,210,112),(19,222,181,113),(20,223,182,114),(21,224,183,115),(22,225,184,116),(23,226,185,117),(24,227,186,118),(25,228,187,119),(26,229,188,120),(27,230,189,91),(28,231,190,92),(29,232,191,93),(30,233,192,94),(31,135,74,169),(32,136,75,170),(33,137,76,171),(34,138,77,172),(35,139,78,173),(36,140,79,174),(37,141,80,175),(38,142,81,176),(39,143,82,177),(40,144,83,178),(41,145,84,179),(42,146,85,180),(43,147,86,151),(44,148,87,152),(45,149,88,153),(46,150,89,154),(47,121,90,155),(48,122,61,156),(49,123,62,157),(50,124,63,158),(51,125,64,159),(52,126,65,160),(53,127,66,161),(54,128,67,162),(55,129,68,163),(56,130,69,164),(57,131,70,165),(58,132,71,166),(59,133,72,167),(60,134,73,168)]])
300 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3A | 3B | 4A | ··· | 4H | 5A | 5B | 5C | 5D | 6A | ··· | 6N | 6O | ··· | 6V | 10A | ··· | 10AB | 10AC | ··· | 10AR | 12A | ··· | 12P | 15A | ··· | 15H | 20A | ··· | 20AF | 30A | ··· | 30BD | 30BE | ··· | 30CJ | 60A | ··· | 60BL |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
300 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C5 | C6 | C6 | C6 | C10 | C10 | C10 | C12 | C15 | C20 | C30 | C30 | C30 | C60 | D4 | C3×D4 | C5×D4 | D4×C15 |
| kernel | C22⋊C4×C30 | C15×C22⋊C4 | C22×C60 | C23×C30 | C10×C22⋊C4 | C22×C30 | C6×C22⋊C4 | C5×C22⋊C4 | C22×C20 | C23×C10 | C3×C22⋊C4 | C22×C12 | C23×C6 | C22×C10 | C2×C22⋊C4 | C22×C6 | C22⋊C4 | C22×C4 | C24 | C23 | C2×C30 | C2×C10 | C2×C6 | C22 |
| # reps | 1 | 4 | 2 | 1 | 2 | 8 | 4 | 8 | 4 | 2 | 16 | 8 | 4 | 16 | 8 | 32 | 32 | 16 | 8 | 64 | 4 | 8 | 16 | 32 |
Matrix representation of C22⋊C4×C30 ►in GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 25 | 0 |
| 0 | 0 | 0 | 25 |
| 1 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 50 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 60 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,16,0,0,0,0,25,0,0,0,0,25],[1,0,0,0,0,60,0,0,0,0,60,1,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[50,0,0,0,0,11,0,0,0,0,1,0,0,0,2,60] >;
C22⋊C4×C30 in GAP, Magma, Sage, TeX
C_2^2\rtimes C_4\times C_{30} % in TeX
G:=Group("C2^2:C4xC30"); // GroupNames label
G:=SmallGroup(480,920);
// by ID
G=gap.SmallGroup(480,920);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-5,-2,-2,1680,1709]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations