metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊3D15, (C4×C60)⋊3C2, (C4×C20)⋊5S3, (C4×C12)⋊3D5, (C2×C4).65D30, C5⋊3(C42⋊3S3), (C2×C20).381D6, C3⋊3(C42⋊2D5), C30.4Q8⋊1C2, D30⋊3C4.1C2, C6.94(C4○D20), (C2×C12).380D10, C15⋊15(C42⋊2C2), C10.94(C4○D12), C30.168(C4○D4), (C2×C60).462C22, (C2×C30).273C23, C2.8(D60⋊11C2), (C2×Dic15).4C22, (C22×D15).3C22, C22.38(C22×D15), (C2×C6).269(C22×D5), (C2×C10).268(C22×S3), SmallGroup(480,841)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊3D15
G = < a,b,c,d | a4=b4=c15=d2=1, ab=ba, ac=ca, dad=ab2, bc=cb, dbd=a2b-1, dcd=c-1 >
Subgroups: 708 in 120 conjugacy classes, 47 normal (14 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C23, D5, C10, Dic3, C12, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, Dic5, C20, D10, C2×C10, C2×Dic3, C2×C12, C22×S3, D15, C30, C42⋊2C2, C2×Dic5, C2×C20, C22×D5, Dic3⋊C4, D6⋊C4, C4×C12, Dic15, C60, D30, C2×C30, C10.D4, D10⋊C4, C4×C20, C42⋊3S3, C2×Dic15, C2×C60, C22×D15, C42⋊2D5, C30.4Q8, D30⋊3C4, C4×C60, C42⋊3D15
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, D10, C22×S3, D15, C42⋊2C2, C22×D5, C4○D12, D30, C4○D20, C42⋊3S3, C22×D15, C42⋊2D5, D60⋊11C2, C42⋊3D15
►On 240 points
Generators in S240
(1 163 32 133)(2 164 33 134)(3 165 34 135)(4 151 35 121)(5 152 36 122)(6 153 37 123)(7 154 38 124)(8 155 39 125)(9 156 40 126)(10 157 41 127)(11 158 42 128)(12 159 43 129)(13 160 44 130)(14 161 45 131)(15 162 31 132)(16 167 46 137)(17 168 47 138)(18 169 48 139)(19 170 49 140)(20 171 50 141)(21 172 51 142)(22 173 52 143)(23 174 53 144)(24 175 54 145)(25 176 55 146)(26 177 56 147)(27 178 57 148)(28 179 58 149)(29 180 59 150)(30 166 60 136)(61 211 99 189)(62 212 100 190)(63 213 101 191)(64 214 102 192)(65 215 103 193)(66 216 104 194)(67 217 105 195)(68 218 91 181)(69 219 92 182)(70 220 93 183)(71 221 94 184)(72 222 95 185)(73 223 96 186)(74 224 97 187)(75 225 98 188)(76 230 106 200)(77 231 107 201)(78 232 108 202)(79 233 109 203)(80 234 110 204)(81 235 111 205)(82 236 112 206)(83 237 113 207)(84 238 114 208)(85 239 115 209)(86 240 116 210)(87 226 117 196)(88 227 118 197)(89 228 119 198)(90 229 120 199)
(1 84 16 61)(2 85 17 62)(3 86 18 63)(4 87 19 64)(5 88 20 65)(6 89 21 66)(7 90 22 67)(8 76 23 68)(9 77 24 69)(10 78 25 70)(11 79 26 71)(12 80 27 72)(13 81 28 73)(14 82 29 74)(15 83 30 75)(31 113 60 98)(32 114 46 99)(33 115 47 100)(34 116 48 101)(35 117 49 102)(36 118 50 103)(37 119 51 104)(38 120 52 105)(39 106 53 91)(40 107 54 92)(41 108 55 93)(42 109 56 94)(43 110 57 95)(44 111 58 96)(45 112 59 97)(121 196 140 192)(122 197 141 193)(123 198 142 194)(124 199 143 195)(125 200 144 181)(126 201 145 182)(127 202 146 183)(128 203 147 184)(129 204 148 185)(130 205 149 186)(131 206 150 187)(132 207 136 188)(133 208 137 189)(134 209 138 190)(135 210 139 191)(151 226 170 214)(152 227 171 215)(153 228 172 216)(154 229 173 217)(155 230 174 218)(156 231 175 219)(157 232 176 220)(158 233 177 221)(159 234 178 222)(160 235 179 223)(161 236 180 224)(162 237 166 225)(163 238 167 211)(164 239 168 212)(165 240 169 213)
(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 32)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(61 113)(62 112)(63 111)(64 110)(65 109)(66 108)(67 107)(68 106)(69 120)(70 119)(71 118)(72 117)(73 116)(74 115)(75 114)(76 91)(77 105)(78 104)(79 103)(80 102)(81 101)(82 100)(83 99)(84 98)(85 97)(86 96)(87 95)(88 94)(89 93)(90 92)(121 148)(122 147)(123 146)(124 145)(125 144)(126 143)(127 142)(128 141)(129 140)(130 139)(131 138)(132 137)(133 136)(134 150)(135 149)(151 178)(152 177)(153 176)(154 175)(155 174)(156 173)(157 172)(158 171)(159 170)(160 169)(161 168)(162 167)(163 166)(164 180)(165 179)(181 218)(182 217)(183 216)(184 215)(185 214)(186 213)(187 212)(188 211)(189 225)(190 224)(191 223)(192 222)(193 221)(194 220)(195 219)(196 234)(197 233)(198 232)(199 231)(200 230)(201 229)(202 228)(203 227)(204 226)(205 240)(206 239)(207 238)(208 237)(209 236)(210 235)
G:=sub<Sym(240)| (1,163,32,133)(2,164,33,134)(3,165,34,135)(4,151,35,121)(5,152,36,122)(6,153,37,123)(7,154,38,124)(8,155,39,125)(9,156,40,126)(10,157,41,127)(11,158,42,128)(12,159,43,129)(13,160,44,130)(14,161,45,131)(15,162,31,132)(16,167,46,137)(17,168,47,138)(18,169,48,139)(19,170,49,140)(20,171,50,141)(21,172,51,142)(22,173,52,143)(23,174,53,144)(24,175,54,145)(25,176,55,146)(26,177,56,147)(27,178,57,148)(28,179,58,149)(29,180,59,150)(30,166,60,136)(61,211,99,189)(62,212,100,190)(63,213,101,191)(64,214,102,192)(65,215,103,193)(66,216,104,194)(67,217,105,195)(68,218,91,181)(69,219,92,182)(70,220,93,183)(71,221,94,184)(72,222,95,185)(73,223,96,186)(74,224,97,187)(75,225,98,188)(76,230,106,200)(77,231,107,201)(78,232,108,202)(79,233,109,203)(80,234,110,204)(81,235,111,205)(82,236,112,206)(83,237,113,207)(84,238,114,208)(85,239,115,209)(86,240,116,210)(87,226,117,196)(88,227,118,197)(89,228,119,198)(90,229,120,199), (1,84,16,61)(2,85,17,62)(3,86,18,63)(4,87,19,64)(5,88,20,65)(6,89,21,66)(7,90,22,67)(8,76,23,68)(9,77,24,69)(10,78,25,70)(11,79,26,71)(12,80,27,72)(13,81,28,73)(14,82,29,74)(15,83,30,75)(31,113,60,98)(32,114,46,99)(33,115,47,100)(34,116,48,101)(35,117,49,102)(36,118,50,103)(37,119,51,104)(38,120,52,105)(39,106,53,91)(40,107,54,92)(41,108,55,93)(42,109,56,94)(43,110,57,95)(44,111,58,96)(45,112,59,97)(121,196,140,192)(122,197,141,193)(123,198,142,194)(124,199,143,195)(125,200,144,181)(126,201,145,182)(127,202,146,183)(128,203,147,184)(129,204,148,185)(130,205,149,186)(131,206,150,187)(132,207,136,188)(133,208,137,189)(134,209,138,190)(135,210,139,191)(151,226,170,214)(152,227,171,215)(153,228,172,216)(154,229,173,217)(155,230,174,218)(156,231,175,219)(157,232,176,220)(158,233,177,221)(159,234,178,222)(160,235,179,223)(161,236,180,224)(162,237,166,225)(163,238,167,211)(164,239,168,212)(165,240,169,213), (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)(31,32)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(61,113)(62,112)(63,111)(64,110)(65,109)(66,108)(67,107)(68,106)(69,120)(70,119)(71,118)(72,117)(73,116)(74,115)(75,114)(76,91)(77,105)(78,104)(79,103)(80,102)(81,101)(82,100)(83,99)(84,98)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92)(121,148)(122,147)(123,146)(124,145)(125,144)(126,143)(127,142)(128,141)(129,140)(130,139)(131,138)(132,137)(133,136)(134,150)(135,149)(151,178)(152,177)(153,176)(154,175)(155,174)(156,173)(157,172)(158,171)(159,170)(160,169)(161,168)(162,167)(163,166)(164,180)(165,179)(181,218)(182,217)(183,216)(184,215)(185,214)(186,213)(187,212)(188,211)(189,225)(190,224)(191,223)(192,222)(193,221)(194,220)(195,219)(196,234)(197,233)(198,232)(199,231)(200,230)(201,229)(202,228)(203,227)(204,226)(205,240)(206,239)(207,238)(208,237)(209,236)(210,235)>;
G:=Group( (1,163,32,133)(2,164,33,134)(3,165,34,135)(4,151,35,121)(5,152,36,122)(6,153,37,123)(7,154,38,124)(8,155,39,125)(9,156,40,126)(10,157,41,127)(11,158,42,128)(12,159,43,129)(13,160,44,130)(14,161,45,131)(15,162,31,132)(16,167,46,137)(17,168,47,138)(18,169,48,139)(19,170,49,140)(20,171,50,141)(21,172,51,142)(22,173,52,143)(23,174,53,144)(24,175,54,145)(25,176,55,146)(26,177,56,147)(27,178,57,148)(28,179,58,149)(29,180,59,150)(30,166,60,136)(61,211,99,189)(62,212,100,190)(63,213,101,191)(64,214,102,192)(65,215,103,193)(66,216,104,194)(67,217,105,195)(68,218,91,181)(69,219,92,182)(70,220,93,183)(71,221,94,184)(72,222,95,185)(73,223,96,186)(74,224,97,187)(75,225,98,188)(76,230,106,200)(77,231,107,201)(78,232,108,202)(79,233,109,203)(80,234,110,204)(81,235,111,205)(82,236,112,206)(83,237,113,207)(84,238,114,208)(85,239,115,209)(86,240,116,210)(87,226,117,196)(88,227,118,197)(89,228,119,198)(90,229,120,199), (1,84,16,61)(2,85,17,62)(3,86,18,63)(4,87,19,64)(5,88,20,65)(6,89,21,66)(7,90,22,67)(8,76,23,68)(9,77,24,69)(10,78,25,70)(11,79,26,71)(12,80,27,72)(13,81,28,73)(14,82,29,74)(15,83,30,75)(31,113,60,98)(32,114,46,99)(33,115,47,100)(34,116,48,101)(35,117,49,102)(36,118,50,103)(37,119,51,104)(38,120,52,105)(39,106,53,91)(40,107,54,92)(41,108,55,93)(42,109,56,94)(43,110,57,95)(44,111,58,96)(45,112,59,97)(121,196,140,192)(122,197,141,193)(123,198,142,194)(124,199,143,195)(125,200,144,181)(126,201,145,182)(127,202,146,183)(128,203,147,184)(129,204,148,185)(130,205,149,186)(131,206,150,187)(132,207,136,188)(133,208,137,189)(134,209,138,190)(135,210,139,191)(151,226,170,214)(152,227,171,215)(153,228,172,216)(154,229,173,217)(155,230,174,218)(156,231,175,219)(157,232,176,220)(158,233,177,221)(159,234,178,222)(160,235,179,223)(161,236,180,224)(162,237,166,225)(163,238,167,211)(164,239,168,212)(165,240,169,213), (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)(31,32)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(61,113)(62,112)(63,111)(64,110)(65,109)(66,108)(67,107)(68,106)(69,120)(70,119)(71,118)(72,117)(73,116)(74,115)(75,114)(76,91)(77,105)(78,104)(79,103)(80,102)(81,101)(82,100)(83,99)(84,98)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92)(121,148)(122,147)(123,146)(124,145)(125,144)(126,143)(127,142)(128,141)(129,140)(130,139)(131,138)(132,137)(133,136)(134,150)(135,149)(151,178)(152,177)(153,176)(154,175)(155,174)(156,173)(157,172)(158,171)(159,170)(160,169)(161,168)(162,167)(163,166)(164,180)(165,179)(181,218)(182,217)(183,216)(184,215)(185,214)(186,213)(187,212)(188,211)(189,225)(190,224)(191,223)(192,222)(193,221)(194,220)(195,219)(196,234)(197,233)(198,232)(199,231)(200,230)(201,229)(202,228)(203,227)(204,226)(205,240)(206,239)(207,238)(208,237)(209,236)(210,235) );
G=PermutationGroup([[(1,163,32,133),(2,164,33,134),(3,165,34,135),(4,151,35,121),(5,152,36,122),(6,153,37,123),(7,154,38,124),(8,155,39,125),(9,156,40,126),(10,157,41,127),(11,158,42,128),(12,159,43,129),(13,160,44,130),(14,161,45,131),(15,162,31,132),(16,167,46,137),(17,168,47,138),(18,169,48,139),(19,170,49,140),(20,171,50,141),(21,172,51,142),(22,173,52,143),(23,174,53,144),(24,175,54,145),(25,176,55,146),(26,177,56,147),(27,178,57,148),(28,179,58,149),(29,180,59,150),(30,166,60,136),(61,211,99,189),(62,212,100,190),(63,213,101,191),(64,214,102,192),(65,215,103,193),(66,216,104,194),(67,217,105,195),(68,218,91,181),(69,219,92,182),(70,220,93,183),(71,221,94,184),(72,222,95,185),(73,223,96,186),(74,224,97,187),(75,225,98,188),(76,230,106,200),(77,231,107,201),(78,232,108,202),(79,233,109,203),(80,234,110,204),(81,235,111,205),(82,236,112,206),(83,237,113,207),(84,238,114,208),(85,239,115,209),(86,240,116,210),(87,226,117,196),(88,227,118,197),(89,228,119,198),(90,229,120,199)], [(1,84,16,61),(2,85,17,62),(3,86,18,63),(4,87,19,64),(5,88,20,65),(6,89,21,66),(7,90,22,67),(8,76,23,68),(9,77,24,69),(10,78,25,70),(11,79,26,71),(12,80,27,72),(13,81,28,73),(14,82,29,74),(15,83,30,75),(31,113,60,98),(32,114,46,99),(33,115,47,100),(34,116,48,101),(35,117,49,102),(36,118,50,103),(37,119,51,104),(38,120,52,105),(39,106,53,91),(40,107,54,92),(41,108,55,93),(42,109,56,94),(43,110,57,95),(44,111,58,96),(45,112,59,97),(121,196,140,192),(122,197,141,193),(123,198,142,194),(124,199,143,195),(125,200,144,181),(126,201,145,182),(127,202,146,183),(128,203,147,184),(129,204,148,185),(130,205,149,186),(131,206,150,187),(132,207,136,188),(133,208,137,189),(134,209,138,190),(135,210,139,191),(151,226,170,214),(152,227,171,215),(153,228,172,216),(154,229,173,217),(155,230,174,218),(156,231,175,219),(157,232,176,220),(158,233,177,221),(159,234,178,222),(160,235,179,223),(161,236,180,224),(162,237,166,225),(163,238,167,211),(164,239,168,212),(165,240,169,213)], [(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),(31,32),(33,45),(34,44),(35,43),(36,42),(37,41),(38,40),(46,60),(47,59),(48,58),(49,57),(50,56),(51,55),(52,54),(61,113),(62,112),(63,111),(64,110),(65,109),(66,108),(67,107),(68,106),(69,120),(70,119),(71,118),(72,117),(73,116),(74,115),(75,114),(76,91),(77,105),(78,104),(79,103),(80,102),(81,101),(82,100),(83,99),(84,98),(85,97),(86,96),(87,95),(88,94),(89,93),(90,92),(121,148),(122,147),(123,146),(124,145),(125,144),(126,143),(127,142),(128,141),(129,140),(130,139),(131,138),(132,137),(133,136),(134,150),(135,149),(151,178),(152,177),(153,176),(154,175),(155,174),(156,173),(157,172),(158,171),(159,170),(160,169),(161,168),(162,167),(163,166),(164,180),(165,179),(181,218),(182,217),(183,216),(184,215),(185,214),(186,213),(187,212),(188,211),(189,225),(190,224),(191,223),(192,222),(193,221),(194,220),(195,219),(196,234),(197,233),(198,232),(199,231),(200,230),(201,229),(202,228),(203,227),(204,226),(205,240),(206,239),(207,238),(208,237),(209,236),(210,235)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 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 | 3 | 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 | 60 | 2 | 2 | ··· | 2 | 60 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | S3 | D5 | D6 | C4○D4 | D10 | D15 | C4○D12 | D30 | C4○D20 | D60⋊11C2 |
| kernel | C42⋊3D15 | C30.4Q8 | D30⋊3C4 | C4×C60 | C4×C20 | C4×C12 | C2×C20 | C30 | C2×C12 | C42 | C10 | C2×C4 | C6 | C2 |
| # reps | 1 | 3 | 3 | 1 | 1 | 2 | 3 | 6 | 6 | 4 | 12 | 12 | 24 | 48 |
Matrix representation of C42⋊3D15 ►in GL4(𝔽61) generated by
| 29 | 7 | 0 | 0 |
| 54 | 32 | 0 | 0 |
| 0 | 0 | 14 | 28 |
| 0 | 0 | 17 | 47 |
| 11 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 32 | 3 |
| 0 | 0 | 4 | 29 |
| 44 | 17 | 0 | 0 |
| 44 | 60 | 0 | 0 |
| 0 | 0 | 28 | 59 |
| 0 | 0 | 38 | 30 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 37 | 12 |
| 0 | 0 | 8 | 24 |
G:=sub<GL(4,GF(61))| [29,54,0,0,7,32,0,0,0,0,14,17,0,0,28,47],[11,0,0,0,0,11,0,0,0,0,32,4,0,0,3,29],[44,44,0,0,17,60,0,0,0,0,28,38,0,0,59,30],[0,1,0,0,1,0,0,0,0,0,37,8,0,0,12,24] >;
C42⋊3D15 in GAP, Magma, Sage, TeX
C_4^2\rtimes_3D_{15} % in TeX
G:=Group("C4^2:3D15"); // GroupNames label
G:=SmallGroup(480,841);
// by ID
G=gap.SmallGroup(480,841);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,64,590,100,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^15=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a*b^2,b*c=c*b,d*b*d=a^2*b^-1,d*c*d=c^-1>;
// generators/relations