metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊2D15, (C4×C12)⋊9D5, (C4×C60)⋊11C2, (C4×C20)⋊11S3, (C4×D15)⋊7C4, C20.92(C4×S3), C12.60(C4×D5), C4.22(C4×D15), (C2×C4).97D30, C60.195(C2×C4), C5⋊5(C42⋊2S3), D30.33(C2×C4), (C2×C20).379D6, C3⋊4(C42⋊D5), (C4×Dic15)⋊14C2, C6.91(C4○D20), (C2×C12).379D10, C30.4Q8⋊39C2, C15⋊21(C42⋊C2), D30⋊3C4.17C2, C10.91(C4○D12), C30.165(C4○D4), C30.154(C22×C4), (C2×C30).269C23, (C2×C60).497C22, Dic15.41(C2×C4), C2.2(D60⋊11C2), C22.10(C22×D15), (C22×D15).77C22, (C2×Dic15).156C22, C2.5(C2×C4×D15), C6.59(C2×C4×D5), C10.91(S3×C2×C4), (C2×C4×D15).9C2, (C2×C6).265(C22×D5), (C2×C10).264(C22×S3), SmallGroup(480,837)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊2D15
G = < a,b,c,d | a4=b4=c15=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 804 in 152 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, C2×C4, C23, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C42, C42, C22⋊C4, C4⋊C4, C22×C4, Dic5, C20, C20, D10, C2×C10, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, D15, C30, C30, C42⋊C2, C4×D5, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×Dic3, Dic3⋊C4, D6⋊C4, C4×C12, S3×C2×C4, Dic15, Dic15, C60, C60, D30, D30, C2×C30, C4×Dic5, C10.D4, D10⋊C4, C4×C20, C2×C4×D5, C42⋊2S3, C4×D15, C2×Dic15, C2×Dic15, C2×C60, C2×C60, C22×D15, C42⋊D5, C4×Dic15, C30.4Q8, D30⋊3C4, C4×C60, C2×C4×D15, C42⋊2D15
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, C22×C4, C4○D4, D10, C4×S3, C22×S3, D15, C42⋊C2, C4×D5, C22×D5, S3×C2×C4, C4○D12, D30, C2×C4×D5, C4○D20, C42⋊2S3, C4×D15, C22×D15, C42⋊D5, C2×C4×D15, D60⋊11C2, C42⋊2D15
►On 240 points
Generators in S240
(1 144 28 133)(2 145 29 134)(3 146 30 135)(4 147 16 121)(5 148 17 122)(6 149 18 123)(7 150 19 124)(8 136 20 125)(9 137 21 126)(10 138 22 127)(11 139 23 128)(12 140 24 129)(13 141 25 130)(14 142 26 131)(15 143 27 132)(31 167 54 152)(32 168 55 153)(33 169 56 154)(34 170 57 155)(35 171 58 156)(36 172 59 157)(37 173 60 158)(38 174 46 159)(39 175 47 160)(40 176 48 161)(41 177 49 162)(42 178 50 163)(43 179 51 164)(44 180 52 165)(45 166 53 151)(61 200 76 188)(62 201 77 189)(63 202 78 190)(64 203 79 191)(65 204 80 192)(66 205 81 193)(67 206 82 194)(68 207 83 195)(69 208 84 181)(70 209 85 182)(71 210 86 183)(72 196 87 184)(73 197 88 185)(74 198 89 186)(75 199 90 187)(91 234 114 222)(92 235 115 223)(93 236 116 224)(94 237 117 225)(95 238 118 211)(96 239 119 212)(97 240 120 213)(98 226 106 214)(99 227 107 215)(100 228 108 216)(101 229 109 217)(102 230 110 218)(103 231 111 219)(104 232 112 220)(105 233 113 221)
(1 99 31 73)(2 100 32 74)(3 101 33 75)(4 102 34 61)(5 103 35 62)(6 104 36 63)(7 105 37 64)(8 91 38 65)(9 92 39 66)(10 93 40 67)(11 94 41 68)(12 95 42 69)(13 96 43 70)(14 97 44 71)(15 98 45 72)(16 110 57 76)(17 111 58 77)(18 112 59 78)(19 113 60 79)(20 114 46 80)(21 115 47 81)(22 116 48 82)(23 117 49 83)(24 118 50 84)(25 119 51 85)(26 120 52 86)(27 106 53 87)(28 107 54 88)(29 108 55 89)(30 109 56 90)(121 218 155 188)(122 219 156 189)(123 220 157 190)(124 221 158 191)(125 222 159 192)(126 223 160 193)(127 224 161 194)(128 225 162 195)(129 211 163 181)(130 212 164 182)(131 213 165 183)(132 214 151 184)(133 215 152 185)(134 216 153 186)(135 217 154 187)(136 234 174 204)(137 235 175 205)(138 236 176 206)(139 237 177 207)(140 238 178 208)(141 239 179 209)(142 240 180 210)(143 226 166 196)(144 227 167 197)(145 228 168 198)(146 229 169 199)(147 230 170 200)(148 231 171 201)(149 232 172 202)(150 233 173 203)
(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 45)(2 44)(3 43)(4 42)(5 41)(6 40)(7 39)(8 38)(9 37)(10 36)(11 35)(12 34)(13 33)(14 32)(15 31)(16 50)(17 49)(18 48)(19 47)(20 46)(21 60)(22 59)(23 58)(24 57)(25 56)(26 55)(27 54)(28 53)(29 52)(30 51)(61 118)(62 117)(63 116)(64 115)(65 114)(66 113)(67 112)(68 111)(69 110)(70 109)(71 108)(72 107)(73 106)(74 120)(75 119)(76 95)(77 94)(78 93)(79 92)(80 91)(81 105)(82 104)(83 103)(84 102)(85 101)(86 100)(87 99)(88 98)(89 97)(90 96)(121 163)(122 162)(123 161)(124 160)(125 159)(126 158)(127 157)(128 156)(129 155)(130 154)(131 153)(132 152)(133 151)(134 165)(135 164)(136 174)(137 173)(138 172)(139 171)(140 170)(141 169)(142 168)(143 167)(144 166)(145 180)(146 179)(147 178)(148 177)(149 176)(150 175)(181 230)(182 229)(183 228)(184 227)(185 226)(186 240)(187 239)(188 238)(189 237)(190 236)(191 235)(192 234)(193 233)(194 232)(195 231)(196 215)(197 214)(198 213)(199 212)(200 211)(201 225)(202 224)(203 223)(204 222)(205 221)(206 220)(207 219)(208 218)(209 217)(210 216)
G:=sub<Sym(240)| (1,144,28,133)(2,145,29,134)(3,146,30,135)(4,147,16,121)(5,148,17,122)(6,149,18,123)(7,150,19,124)(8,136,20,125)(9,137,21,126)(10,138,22,127)(11,139,23,128)(12,140,24,129)(13,141,25,130)(14,142,26,131)(15,143,27,132)(31,167,54,152)(32,168,55,153)(33,169,56,154)(34,170,57,155)(35,171,58,156)(36,172,59,157)(37,173,60,158)(38,174,46,159)(39,175,47,160)(40,176,48,161)(41,177,49,162)(42,178,50,163)(43,179,51,164)(44,180,52,165)(45,166,53,151)(61,200,76,188)(62,201,77,189)(63,202,78,190)(64,203,79,191)(65,204,80,192)(66,205,81,193)(67,206,82,194)(68,207,83,195)(69,208,84,181)(70,209,85,182)(71,210,86,183)(72,196,87,184)(73,197,88,185)(74,198,89,186)(75,199,90,187)(91,234,114,222)(92,235,115,223)(93,236,116,224)(94,237,117,225)(95,238,118,211)(96,239,119,212)(97,240,120,213)(98,226,106,214)(99,227,107,215)(100,228,108,216)(101,229,109,217)(102,230,110,218)(103,231,111,219)(104,232,112,220)(105,233,113,221), (1,99,31,73)(2,100,32,74)(3,101,33,75)(4,102,34,61)(5,103,35,62)(6,104,36,63)(7,105,37,64)(8,91,38,65)(9,92,39,66)(10,93,40,67)(11,94,41,68)(12,95,42,69)(13,96,43,70)(14,97,44,71)(15,98,45,72)(16,110,57,76)(17,111,58,77)(18,112,59,78)(19,113,60,79)(20,114,46,80)(21,115,47,81)(22,116,48,82)(23,117,49,83)(24,118,50,84)(25,119,51,85)(26,120,52,86)(27,106,53,87)(28,107,54,88)(29,108,55,89)(30,109,56,90)(121,218,155,188)(122,219,156,189)(123,220,157,190)(124,221,158,191)(125,222,159,192)(126,223,160,193)(127,224,161,194)(128,225,162,195)(129,211,163,181)(130,212,164,182)(131,213,165,183)(132,214,151,184)(133,215,152,185)(134,216,153,186)(135,217,154,187)(136,234,174,204)(137,235,175,205)(138,236,176,206)(139,237,177,207)(140,238,178,208)(141,239,179,209)(142,240,180,210)(143,226,166,196)(144,227,167,197)(145,228,168,198)(146,229,169,199)(147,230,170,200)(148,231,171,201)(149,232,172,202)(150,233,173,203), (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), (1,45)(2,44)(3,43)(4,42)(5,41)(6,40)(7,39)(8,38)(9,37)(10,36)(11,35)(12,34)(13,33)(14,32)(15,31)(16,50)(17,49)(18,48)(19,47)(20,46)(21,60)(22,59)(23,58)(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(61,118)(62,117)(63,116)(64,115)(65,114)(66,113)(67,112)(68,111)(69,110)(70,109)(71,108)(72,107)(73,106)(74,120)(75,119)(76,95)(77,94)(78,93)(79,92)(80,91)(81,105)(82,104)(83,103)(84,102)(85,101)(86,100)(87,99)(88,98)(89,97)(90,96)(121,163)(122,162)(123,161)(124,160)(125,159)(126,158)(127,157)(128,156)(129,155)(130,154)(131,153)(132,152)(133,151)(134,165)(135,164)(136,174)(137,173)(138,172)(139,171)(140,170)(141,169)(142,168)(143,167)(144,166)(145,180)(146,179)(147,178)(148,177)(149,176)(150,175)(181,230)(182,229)(183,228)(184,227)(185,226)(186,240)(187,239)(188,238)(189,237)(190,236)(191,235)(192,234)(193,233)(194,232)(195,231)(196,215)(197,214)(198,213)(199,212)(200,211)(201,225)(202,224)(203,223)(204,222)(205,221)(206,220)(207,219)(208,218)(209,217)(210,216)>;
G:=Group( (1,144,28,133)(2,145,29,134)(3,146,30,135)(4,147,16,121)(5,148,17,122)(6,149,18,123)(7,150,19,124)(8,136,20,125)(9,137,21,126)(10,138,22,127)(11,139,23,128)(12,140,24,129)(13,141,25,130)(14,142,26,131)(15,143,27,132)(31,167,54,152)(32,168,55,153)(33,169,56,154)(34,170,57,155)(35,171,58,156)(36,172,59,157)(37,173,60,158)(38,174,46,159)(39,175,47,160)(40,176,48,161)(41,177,49,162)(42,178,50,163)(43,179,51,164)(44,180,52,165)(45,166,53,151)(61,200,76,188)(62,201,77,189)(63,202,78,190)(64,203,79,191)(65,204,80,192)(66,205,81,193)(67,206,82,194)(68,207,83,195)(69,208,84,181)(70,209,85,182)(71,210,86,183)(72,196,87,184)(73,197,88,185)(74,198,89,186)(75,199,90,187)(91,234,114,222)(92,235,115,223)(93,236,116,224)(94,237,117,225)(95,238,118,211)(96,239,119,212)(97,240,120,213)(98,226,106,214)(99,227,107,215)(100,228,108,216)(101,229,109,217)(102,230,110,218)(103,231,111,219)(104,232,112,220)(105,233,113,221), (1,99,31,73)(2,100,32,74)(3,101,33,75)(4,102,34,61)(5,103,35,62)(6,104,36,63)(7,105,37,64)(8,91,38,65)(9,92,39,66)(10,93,40,67)(11,94,41,68)(12,95,42,69)(13,96,43,70)(14,97,44,71)(15,98,45,72)(16,110,57,76)(17,111,58,77)(18,112,59,78)(19,113,60,79)(20,114,46,80)(21,115,47,81)(22,116,48,82)(23,117,49,83)(24,118,50,84)(25,119,51,85)(26,120,52,86)(27,106,53,87)(28,107,54,88)(29,108,55,89)(30,109,56,90)(121,218,155,188)(122,219,156,189)(123,220,157,190)(124,221,158,191)(125,222,159,192)(126,223,160,193)(127,224,161,194)(128,225,162,195)(129,211,163,181)(130,212,164,182)(131,213,165,183)(132,214,151,184)(133,215,152,185)(134,216,153,186)(135,217,154,187)(136,234,174,204)(137,235,175,205)(138,236,176,206)(139,237,177,207)(140,238,178,208)(141,239,179,209)(142,240,180,210)(143,226,166,196)(144,227,167,197)(145,228,168,198)(146,229,169,199)(147,230,170,200)(148,231,171,201)(149,232,172,202)(150,233,173,203), (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), (1,45)(2,44)(3,43)(4,42)(5,41)(6,40)(7,39)(8,38)(9,37)(10,36)(11,35)(12,34)(13,33)(14,32)(15,31)(16,50)(17,49)(18,48)(19,47)(20,46)(21,60)(22,59)(23,58)(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(61,118)(62,117)(63,116)(64,115)(65,114)(66,113)(67,112)(68,111)(69,110)(70,109)(71,108)(72,107)(73,106)(74,120)(75,119)(76,95)(77,94)(78,93)(79,92)(80,91)(81,105)(82,104)(83,103)(84,102)(85,101)(86,100)(87,99)(88,98)(89,97)(90,96)(121,163)(122,162)(123,161)(124,160)(125,159)(126,158)(127,157)(128,156)(129,155)(130,154)(131,153)(132,152)(133,151)(134,165)(135,164)(136,174)(137,173)(138,172)(139,171)(140,170)(141,169)(142,168)(143,167)(144,166)(145,180)(146,179)(147,178)(148,177)(149,176)(150,175)(181,230)(182,229)(183,228)(184,227)(185,226)(186,240)(187,239)(188,238)(189,237)(190,236)(191,235)(192,234)(193,233)(194,232)(195,231)(196,215)(197,214)(198,213)(199,212)(200,211)(201,225)(202,224)(203,223)(204,222)(205,221)(206,220)(207,219)(208,218)(209,217)(210,216) );
G=PermutationGroup([[(1,144,28,133),(2,145,29,134),(3,146,30,135),(4,147,16,121),(5,148,17,122),(6,149,18,123),(7,150,19,124),(8,136,20,125),(9,137,21,126),(10,138,22,127),(11,139,23,128),(12,140,24,129),(13,141,25,130),(14,142,26,131),(15,143,27,132),(31,167,54,152),(32,168,55,153),(33,169,56,154),(34,170,57,155),(35,171,58,156),(36,172,59,157),(37,173,60,158),(38,174,46,159),(39,175,47,160),(40,176,48,161),(41,177,49,162),(42,178,50,163),(43,179,51,164),(44,180,52,165),(45,166,53,151),(61,200,76,188),(62,201,77,189),(63,202,78,190),(64,203,79,191),(65,204,80,192),(66,205,81,193),(67,206,82,194),(68,207,83,195),(69,208,84,181),(70,209,85,182),(71,210,86,183),(72,196,87,184),(73,197,88,185),(74,198,89,186),(75,199,90,187),(91,234,114,222),(92,235,115,223),(93,236,116,224),(94,237,117,225),(95,238,118,211),(96,239,119,212),(97,240,120,213),(98,226,106,214),(99,227,107,215),(100,228,108,216),(101,229,109,217),(102,230,110,218),(103,231,111,219),(104,232,112,220),(105,233,113,221)], [(1,99,31,73),(2,100,32,74),(3,101,33,75),(4,102,34,61),(5,103,35,62),(6,104,36,63),(7,105,37,64),(8,91,38,65),(9,92,39,66),(10,93,40,67),(11,94,41,68),(12,95,42,69),(13,96,43,70),(14,97,44,71),(15,98,45,72),(16,110,57,76),(17,111,58,77),(18,112,59,78),(19,113,60,79),(20,114,46,80),(21,115,47,81),(22,116,48,82),(23,117,49,83),(24,118,50,84),(25,119,51,85),(26,120,52,86),(27,106,53,87),(28,107,54,88),(29,108,55,89),(30,109,56,90),(121,218,155,188),(122,219,156,189),(123,220,157,190),(124,221,158,191),(125,222,159,192),(126,223,160,193),(127,224,161,194),(128,225,162,195),(129,211,163,181),(130,212,164,182),(131,213,165,183),(132,214,151,184),(133,215,152,185),(134,216,153,186),(135,217,154,187),(136,234,174,204),(137,235,175,205),(138,236,176,206),(139,237,177,207),(140,238,178,208),(141,239,179,209),(142,240,180,210),(143,226,166,196),(144,227,167,197),(145,228,168,198),(146,229,169,199),(147,230,170,200),(148,231,171,201),(149,232,172,202),(150,233,173,203)], [(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)], [(1,45),(2,44),(3,43),(4,42),(5,41),(6,40),(7,39),(8,38),(9,37),(10,36),(11,35),(12,34),(13,33),(14,32),(15,31),(16,50),(17,49),(18,48),(19,47),(20,46),(21,60),(22,59),(23,58),(24,57),(25,56),(26,55),(27,54),(28,53),(29,52),(30,51),(61,118),(62,117),(63,116),(64,115),(65,114),(66,113),(67,112),(68,111),(69,110),(70,109),(71,108),(72,107),(73,106),(74,120),(75,119),(76,95),(77,94),(78,93),(79,92),(80,91),(81,105),(82,104),(83,103),(84,102),(85,101),(86,100),(87,99),(88,98),(89,97),(90,96),(121,163),(122,162),(123,161),(124,160),(125,159),(126,158),(127,157),(128,156),(129,155),(130,154),(131,153),(132,152),(133,151),(134,165),(135,164),(136,174),(137,173),(138,172),(139,171),(140,170),(141,169),(142,168),(143,167),(144,166),(145,180),(146,179),(147,178),(148,177),(149,176),(150,175),(181,230),(182,229),(183,228),(184,227),(185,226),(186,240),(187,239),(188,238),(189,237),(190,236),(191,235),(192,234),(193,233),(194,232),(195,231),(196,215),(197,214),(198,213),(199,212),(200,211),(201,225),(202,224),(203,223),(204,222),(205,221),(206,220),(207,219),(208,218),(209,217),(210,216)]])
132 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | ··· | 12L | 15A | 15B | 15C | 15D | 20A | ··· | 20X | 30A | ··· | 30L | 60A | ··· | 60AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 30 | 30 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 30 | ··· | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D5 | D6 | C4○D4 | D10 | C4×S3 | D15 | C4×D5 | C4○D12 | D30 | C4○D20 | C4×D15 | D60⋊11C2 |
| kernel | C42⋊2D15 | C4×Dic15 | C30.4Q8 | D30⋊3C4 | C4×C60 | C2×C4×D15 | C4×D15 | C4×C20 | C4×C12 | C2×C20 | C30 | C2×C12 | C20 | C42 | C12 | C10 | C2×C4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 8 | 1 | 2 | 3 | 4 | 6 | 4 | 4 | 8 | 8 | 12 | 16 | 16 | 32 |
Matrix representation of C42⋊2D15 ►in GL3(𝔽61) generated by
| 60 | 0 | 0 |
| 0 | 50 | 0 |
| 0 | 0 | 50 |
| 11 | 0 | 0 |
| 0 | 29 | 4 |
| 0 | 3 | 32 |
| 1 | 0 | 0 |
| 0 | 33 | 56 |
| 0 | 42 | 14 |
| 1 | 0 | 0 |
| 0 | 25 | 16 |
| 0 | 22 | 36 |
G:=sub<GL(3,GF(61))| [60,0,0,0,50,0,0,0,50],[11,0,0,0,29,3,0,4,32],[1,0,0,0,33,42,0,56,14],[1,0,0,0,25,22,0,16,36] >;
C42⋊2D15 in GAP, Magma, Sage, TeX
C_4^2\rtimes_2D_{15} % in TeX
G:=Group("C4^2:2D15"); // GroupNames label
G:=SmallGroup(480,837);
// by ID
G=gap.SmallGroup(480,837);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,422,58,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^15=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations