direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D30.C2, D15⋊2C42, C12⋊8(C4×D5), C5⋊2(S3×C42), C20⋊12(C4×S3), C3⋊1(D5×C42), C60⋊25(C2×C4), C15⋊7(C2×C42), (C4×D15)⋊11C4, Dic3⋊7(C4×D5), (C4×Dic5)⋊17S3, (C4×Dic3)⋊17D5, Dic5⋊11(C4×S3), D30.30(C2×C4), (C2×C20).338D6, (C12×Dic5)⋊11C2, (Dic3×C20)⋊11C2, Dic15⋊21(C2×C4), (C2×C12).342D10, (C2×C30).91C23, (Dic3×Dic5)⋊42C2, (C2×C60).240C22, C30.123(C22×C4), (C2×Dic5).214D6, (C2×Dic3).179D10, (C6×Dic5).197C22, (C10×Dic3).178C22, (C2×Dic15).203C22, (C22×D15).102C22, C2.4(C4×S3×D5), C6.47(C2×C4×D5), C10.53(S3×C2×C4), (C2×C4×D15).23C2, C22.46(C2×S3×D5), (C2×C4).243(S3×D5), C2.2(C2×D30.C2), (C3×Dic5)⋊14(C2×C4), (C5×Dic3)⋊15(C2×C4), (C2×D30.C2).14C2, (C2×C6).103(C22×D5), (C2×C10).103(C22×S3), SmallGroup(480,477)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C4×D30.C2 |
Generators and relations for C4×D30.C2
G = < a,b,c,d | a4=b30=c2=1, d2=b15, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b19, dcd-1=b18c >
Subgroups: 940 in 216 conjugacy classes, 92 normal (24 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, C23, D5, C10, C10, Dic3, Dic3, C12, C12, D6, C2×C6, C15, C42, C22×C4, Dic5, Dic5, C20, C20, D10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, D15, C30, C30, C2×C42, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×Dic3, C4×Dic3, C4×C12, S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, D30, C2×C30, C4×Dic5, C4×Dic5, C4×C20, C2×C4×D5, S3×C42, D30.C2, C6×Dic5, C10×Dic3, C4×D15, C2×Dic15, C2×C60, C22×D15, D5×C42, Dic3×Dic5, C12×Dic5, Dic3×C20, C2×D30.C2, C2×C4×D15, C4×D30.C2
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, C42, C22×C4, D10, C4×S3, C22×S3, C2×C42, C4×D5, C22×D5, S3×C2×C4, S3×D5, C2×C4×D5, S3×C42, D30.C2, C2×S3×D5, D5×C42, C4×S3×D5, C2×D30.C2, C4×D30.C2
►On 240 points
Generators in S240
(1 225 195 113)(2 226 196 114)(3 227 197 115)(4 228 198 116)(5 229 199 117)(6 230 200 118)(7 231 201 119)(8 232 202 120)(9 233 203 91)(10 234 204 92)(11 235 205 93)(12 236 206 94)(13 237 207 95)(14 238 208 96)(15 239 209 97)(16 240 210 98)(17 211 181 99)(18 212 182 100)(19 213 183 101)(20 214 184 102)(21 215 185 103)(22 216 186 104)(23 217 187 105)(24 218 188 106)(25 219 189 107)(26 220 190 108)(27 221 191 109)(28 222 192 110)(29 223 193 111)(30 224 194 112)(31 82 136 159)(32 83 137 160)(33 84 138 161)(34 85 139 162)(35 86 140 163)(36 87 141 164)(37 88 142 165)(38 89 143 166)(39 90 144 167)(40 61 145 168)(41 62 146 169)(42 63 147 170)(43 64 148 171)(44 65 149 172)(45 66 150 173)(46 67 121 174)(47 68 122 175)(48 69 123 176)(49 70 124 177)(50 71 125 178)(51 72 126 179)(52 73 127 180)(53 74 128 151)(54 75 129 152)(55 76 130 153)(56 77 131 154)(57 78 132 155)(58 79 133 156)(59 80 134 157)(60 81 135 158)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(31 55)(32 54)(33 53)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)(41 45)(42 44)(56 60)(57 59)(61 67)(62 66)(63 65)(68 90)(69 89)(70 88)(71 87)(72 86)(73 85)(74 84)(75 83)(76 82)(77 81)(78 80)(91 119)(92 118)(93 117)(94 116)(95 115)(96 114)(97 113)(98 112)(99 111)(100 110)(101 109)(102 108)(103 107)(104 106)(121 145)(122 144)(123 143)(124 142)(125 141)(126 140)(127 139)(128 138)(129 137)(130 136)(131 135)(132 134)(146 150)(147 149)(151 161)(152 160)(153 159)(154 158)(155 157)(162 180)(163 179)(164 178)(165 177)(166 176)(167 175)(168 174)(169 173)(170 172)(181 193)(182 192)(183 191)(184 190)(185 189)(186 188)(194 210)(195 209)(196 208)(197 207)(198 206)(199 205)(200 204)(201 203)(211 223)(212 222)(213 221)(214 220)(215 219)(216 218)(224 240)(225 239)(226 238)(227 237)(228 236)(229 235)(230 234)(231 233)
(1 126 16 141)(2 145 17 130)(3 134 18 149)(4 123 19 138)(5 142 20 127)(6 131 21 146)(7 150 22 135)(8 139 23 124)(9 128 24 143)(10 147 25 132)(11 136 26 121)(12 125 27 140)(13 144 28 129)(14 133 29 148)(15 122 30 137)(31 190 46 205)(32 209 47 194)(33 198 48 183)(34 187 49 202)(35 206 50 191)(36 195 51 210)(37 184 52 199)(38 203 53 188)(39 192 54 207)(40 181 55 196)(41 200 56 185)(42 189 57 204)(43 208 58 193)(44 197 59 182)(45 186 60 201)(61 99 76 114)(62 118 77 103)(63 107 78 92)(64 96 79 111)(65 115 80 100)(66 104 81 119)(67 93 82 108)(68 112 83 97)(69 101 84 116)(70 120 85 105)(71 109 86 94)(72 98 87 113)(73 117 88 102)(74 106 89 91)(75 95 90 110)(151 218 166 233)(152 237 167 222)(153 226 168 211)(154 215 169 230)(155 234 170 219)(156 223 171 238)(157 212 172 227)(158 231 173 216)(159 220 174 235)(160 239 175 224)(161 228 176 213)(162 217 177 232)(163 236 178 221)(164 225 179 240)(165 214 180 229)
G:=sub<Sym(240)| (1,225,195,113)(2,226,196,114)(3,227,197,115)(4,228,198,116)(5,229,199,117)(6,230,200,118)(7,231,201,119)(8,232,202,120)(9,233,203,91)(10,234,204,92)(11,235,205,93)(12,236,206,94)(13,237,207,95)(14,238,208,96)(15,239,209,97)(16,240,210,98)(17,211,181,99)(18,212,182,100)(19,213,183,101)(20,214,184,102)(21,215,185,103)(22,216,186,104)(23,217,187,105)(24,218,188,106)(25,219,189,107)(26,220,190,108)(27,221,191,109)(28,222,192,110)(29,223,193,111)(30,224,194,112)(31,82,136,159)(32,83,137,160)(33,84,138,161)(34,85,139,162)(35,86,140,163)(36,87,141,164)(37,88,142,165)(38,89,143,166)(39,90,144,167)(40,61,145,168)(41,62,146,169)(42,63,147,170)(43,64,148,171)(44,65,149,172)(45,66,150,173)(46,67,121,174)(47,68,122,175)(48,69,123,176)(49,70,124,177)(50,71,125,178)(51,72,126,179)(52,73,127,180)(53,74,128,151)(54,75,129,152)(55,76,130,153)(56,77,131,154)(57,78,132,155)(58,79,133,156)(59,80,134,157)(60,81,135,158), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(56,60)(57,59)(61,67)(62,66)(63,65)(68,90)(69,89)(70,88)(71,87)(72,86)(73,85)(74,84)(75,83)(76,82)(77,81)(78,80)(91,119)(92,118)(93,117)(94,116)(95,115)(96,114)(97,113)(98,112)(99,111)(100,110)(101,109)(102,108)(103,107)(104,106)(121,145)(122,144)(123,143)(124,142)(125,141)(126,140)(127,139)(128,138)(129,137)(130,136)(131,135)(132,134)(146,150)(147,149)(151,161)(152,160)(153,159)(154,158)(155,157)(162,180)(163,179)(164,178)(165,177)(166,176)(167,175)(168,174)(169,173)(170,172)(181,193)(182,192)(183,191)(184,190)(185,189)(186,188)(194,210)(195,209)(196,208)(197,207)(198,206)(199,205)(200,204)(201,203)(211,223)(212,222)(213,221)(214,220)(215,219)(216,218)(224,240)(225,239)(226,238)(227,237)(228,236)(229,235)(230,234)(231,233), (1,126,16,141)(2,145,17,130)(3,134,18,149)(4,123,19,138)(5,142,20,127)(6,131,21,146)(7,150,22,135)(8,139,23,124)(9,128,24,143)(10,147,25,132)(11,136,26,121)(12,125,27,140)(13,144,28,129)(14,133,29,148)(15,122,30,137)(31,190,46,205)(32,209,47,194)(33,198,48,183)(34,187,49,202)(35,206,50,191)(36,195,51,210)(37,184,52,199)(38,203,53,188)(39,192,54,207)(40,181,55,196)(41,200,56,185)(42,189,57,204)(43,208,58,193)(44,197,59,182)(45,186,60,201)(61,99,76,114)(62,118,77,103)(63,107,78,92)(64,96,79,111)(65,115,80,100)(66,104,81,119)(67,93,82,108)(68,112,83,97)(69,101,84,116)(70,120,85,105)(71,109,86,94)(72,98,87,113)(73,117,88,102)(74,106,89,91)(75,95,90,110)(151,218,166,233)(152,237,167,222)(153,226,168,211)(154,215,169,230)(155,234,170,219)(156,223,171,238)(157,212,172,227)(158,231,173,216)(159,220,174,235)(160,239,175,224)(161,228,176,213)(162,217,177,232)(163,236,178,221)(164,225,179,240)(165,214,180,229)>;
G:=Group( (1,225,195,113)(2,226,196,114)(3,227,197,115)(4,228,198,116)(5,229,199,117)(6,230,200,118)(7,231,201,119)(8,232,202,120)(9,233,203,91)(10,234,204,92)(11,235,205,93)(12,236,206,94)(13,237,207,95)(14,238,208,96)(15,239,209,97)(16,240,210,98)(17,211,181,99)(18,212,182,100)(19,213,183,101)(20,214,184,102)(21,215,185,103)(22,216,186,104)(23,217,187,105)(24,218,188,106)(25,219,189,107)(26,220,190,108)(27,221,191,109)(28,222,192,110)(29,223,193,111)(30,224,194,112)(31,82,136,159)(32,83,137,160)(33,84,138,161)(34,85,139,162)(35,86,140,163)(36,87,141,164)(37,88,142,165)(38,89,143,166)(39,90,144,167)(40,61,145,168)(41,62,146,169)(42,63,147,170)(43,64,148,171)(44,65,149,172)(45,66,150,173)(46,67,121,174)(47,68,122,175)(48,69,123,176)(49,70,124,177)(50,71,125,178)(51,72,126,179)(52,73,127,180)(53,74,128,151)(54,75,129,152)(55,76,130,153)(56,77,131,154)(57,78,132,155)(58,79,133,156)(59,80,134,157)(60,81,135,158), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(56,60)(57,59)(61,67)(62,66)(63,65)(68,90)(69,89)(70,88)(71,87)(72,86)(73,85)(74,84)(75,83)(76,82)(77,81)(78,80)(91,119)(92,118)(93,117)(94,116)(95,115)(96,114)(97,113)(98,112)(99,111)(100,110)(101,109)(102,108)(103,107)(104,106)(121,145)(122,144)(123,143)(124,142)(125,141)(126,140)(127,139)(128,138)(129,137)(130,136)(131,135)(132,134)(146,150)(147,149)(151,161)(152,160)(153,159)(154,158)(155,157)(162,180)(163,179)(164,178)(165,177)(166,176)(167,175)(168,174)(169,173)(170,172)(181,193)(182,192)(183,191)(184,190)(185,189)(186,188)(194,210)(195,209)(196,208)(197,207)(198,206)(199,205)(200,204)(201,203)(211,223)(212,222)(213,221)(214,220)(215,219)(216,218)(224,240)(225,239)(226,238)(227,237)(228,236)(229,235)(230,234)(231,233), (1,126,16,141)(2,145,17,130)(3,134,18,149)(4,123,19,138)(5,142,20,127)(6,131,21,146)(7,150,22,135)(8,139,23,124)(9,128,24,143)(10,147,25,132)(11,136,26,121)(12,125,27,140)(13,144,28,129)(14,133,29,148)(15,122,30,137)(31,190,46,205)(32,209,47,194)(33,198,48,183)(34,187,49,202)(35,206,50,191)(36,195,51,210)(37,184,52,199)(38,203,53,188)(39,192,54,207)(40,181,55,196)(41,200,56,185)(42,189,57,204)(43,208,58,193)(44,197,59,182)(45,186,60,201)(61,99,76,114)(62,118,77,103)(63,107,78,92)(64,96,79,111)(65,115,80,100)(66,104,81,119)(67,93,82,108)(68,112,83,97)(69,101,84,116)(70,120,85,105)(71,109,86,94)(72,98,87,113)(73,117,88,102)(74,106,89,91)(75,95,90,110)(151,218,166,233)(152,237,167,222)(153,226,168,211)(154,215,169,230)(155,234,170,219)(156,223,171,238)(157,212,172,227)(158,231,173,216)(159,220,174,235)(160,239,175,224)(161,228,176,213)(162,217,177,232)(163,236,178,221)(164,225,179,240)(165,214,180,229) );
G=PermutationGroup([[(1,225,195,113),(2,226,196,114),(3,227,197,115),(4,228,198,116),(5,229,199,117),(6,230,200,118),(7,231,201,119),(8,232,202,120),(9,233,203,91),(10,234,204,92),(11,235,205,93),(12,236,206,94),(13,237,207,95),(14,238,208,96),(15,239,209,97),(16,240,210,98),(17,211,181,99),(18,212,182,100),(19,213,183,101),(20,214,184,102),(21,215,185,103),(22,216,186,104),(23,217,187,105),(24,218,188,106),(25,219,189,107),(26,220,190,108),(27,221,191,109),(28,222,192,110),(29,223,193,111),(30,224,194,112),(31,82,136,159),(32,83,137,160),(33,84,138,161),(34,85,139,162),(35,86,140,163),(36,87,141,164),(37,88,142,165),(38,89,143,166),(39,90,144,167),(40,61,145,168),(41,62,146,169),(42,63,147,170),(43,64,148,171),(44,65,149,172),(45,66,150,173),(46,67,121,174),(47,68,122,175),(48,69,123,176),(49,70,124,177),(50,71,125,178),(51,72,126,179),(52,73,127,180),(53,74,128,151),(54,75,129,152),(55,76,130,153),(56,77,131,154),(57,78,132,155),(58,79,133,156),(59,80,134,157),(60,81,135,158)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,30),(17,29),(18,28),(19,27),(20,26),(21,25),(22,24),(31,55),(32,54),(33,53),(34,52),(35,51),(36,50),(37,49),(38,48),(39,47),(40,46),(41,45),(42,44),(56,60),(57,59),(61,67),(62,66),(63,65),(68,90),(69,89),(70,88),(71,87),(72,86),(73,85),(74,84),(75,83),(76,82),(77,81),(78,80),(91,119),(92,118),(93,117),(94,116),(95,115),(96,114),(97,113),(98,112),(99,111),(100,110),(101,109),(102,108),(103,107),(104,106),(121,145),(122,144),(123,143),(124,142),(125,141),(126,140),(127,139),(128,138),(129,137),(130,136),(131,135),(132,134),(146,150),(147,149),(151,161),(152,160),(153,159),(154,158),(155,157),(162,180),(163,179),(164,178),(165,177),(166,176),(167,175),(168,174),(169,173),(170,172),(181,193),(182,192),(183,191),(184,190),(185,189),(186,188),(194,210),(195,209),(196,208),(197,207),(198,206),(199,205),(200,204),(201,203),(211,223),(212,222),(213,221),(214,220),(215,219),(216,218),(224,240),(225,239),(226,238),(227,237),(228,236),(229,235),(230,234),(231,233)], [(1,126,16,141),(2,145,17,130),(3,134,18,149),(4,123,19,138),(5,142,20,127),(6,131,21,146),(7,150,22,135),(8,139,23,124),(9,128,24,143),(10,147,25,132),(11,136,26,121),(12,125,27,140),(13,144,28,129),(14,133,29,148),(15,122,30,137),(31,190,46,205),(32,209,47,194),(33,198,48,183),(34,187,49,202),(35,206,50,191),(36,195,51,210),(37,184,52,199),(38,203,53,188),(39,192,54,207),(40,181,55,196),(41,200,56,185),(42,189,57,204),(43,208,58,193),(44,197,59,182),(45,186,60,201),(61,99,76,114),(62,118,77,103),(63,107,78,92),(64,96,79,111),(65,115,80,100),(66,104,81,119),(67,93,82,108),(68,112,83,97),(69,101,84,116),(70,120,85,105),(71,109,86,94),(72,98,87,113),(73,117,88,102),(74,106,89,91),(75,95,90,110),(151,218,166,233),(152,237,167,222),(153,226,168,211),(154,215,169,230),(155,234,170,219),(156,223,171,238),(157,212,172,227),(158,231,173,216),(159,220,174,235),(160,239,175,224),(161,228,176,213),(162,217,177,232),(163,236,178,221),(164,225,179,240),(165,214,180,229)]])
96 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | ··· | 4T | 4U | 4V | 4W | 4X | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20X | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 15 | 15 | 15 | 15 | 2 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 5 | ··· | 5 | 15 | 15 | 15 | 15 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | S3 | D5 | D6 | D6 | D10 | D10 | C4×S3 | C4×S3 | C4×D5 | C4×D5 | S3×D5 | D30.C2 | C2×S3×D5 | C4×S3×D5 |
| kernel | C4×D30.C2 | Dic3×Dic5 | C12×Dic5 | Dic3×C20 | C2×D30.C2 | C2×C4×D15 | D30.C2 | C4×D15 | C4×Dic5 | C4×Dic3 | C2×Dic5 | C2×C20 | C2×Dic3 | C2×C12 | Dic5 | C20 | Dic3 | C12 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 16 | 8 | 1 | 2 | 2 | 1 | 4 | 2 | 8 | 4 | 16 | 8 | 2 | 4 | 2 | 8 |
Matrix representation of C4×D30.C2 ►in GL5(𝔽61)
| 50 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 43 | 60 | 0 | 0 |
| 0 | 19 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 1 | 60 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 60 |
| 0 | 0 | 0 | 0 | 60 |
| 11 | 0 | 0 | 0 | 0 |
| 0 | 15 | 4 | 0 | 0 |
| 0 | 35 | 46 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(61))| [50,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,0,43,19,0,0,0,60,1,0,0,0,0,0,0,1,0,0,0,60,60],[60,0,0,0,0,0,60,0,0,0,0,60,1,0,0,0,0,0,1,0,0,0,0,60,60],[11,0,0,0,0,0,15,35,0,0,0,4,46,0,0,0,0,0,1,0,0,0,0,0,1] >;
C4×D30.C2 in GAP, Magma, Sage, TeX
C_4\times D_{30}.C_2 % in TeX
G:=Group("C4xD30.C2"); // GroupNames label
G:=SmallGroup(480,477);
// by ID
G=gap.SmallGroup(480,477);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,64,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^30=c^2=1,d^2=b^15,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^19,d*c*d^-1=b^18*c>;
// generators/relations