metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4.5D60, C42⋊7D15, C20.30D12, C12.30D20, C60.161D4, (C4×C60)⋊7C2, (C4×C20)⋊7S3, (C4×C12)⋊5D5, C2.6(C2×D60), (C2×D60).3C2, (C2×C4).64D30, C6.32(C2×D20), D30⋊3C4⋊1C2, C5⋊2(C42⋊7S3), (C2×Dic30)⋊4C2, (C2×C20).380D6, C30.261(C2×D4), C10.33(C2×D12), C3⋊2(C4.D20), C6.93(C4○D20), (C2×C12).395D10, C15⋊16(C4.4D4), C30.167(C4○D4), C10.93(C4○D12), (C2×C30).272C23, (C2×C60).461C22, C2.7(D60⋊11C2), (C2×Dic15).3C22, (C22×D15).2C22, C22.37(C22×D15), (C2×C6).268(C22×D5), (C2×C10).267(C22×S3), SmallGroup(480,840)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊7D15
G = < a,b,c,d | a4=b4=c15=d2=1, ab=ba, ac=ca, dad=ab2, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 1092 in 152 conjugacy classes, 55 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C42, C22⋊C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, Dic6, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, D15, C30, C30, C4.4D4, Dic10, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, D6⋊C4, C4×C12, C2×Dic6, C2×D12, Dic15, C60, C60, D30, C2×C30, D10⋊C4, C4×C20, C2×Dic10, C2×D20, C42⋊7S3, Dic30, D60, C2×Dic15, C2×C60, C2×C60, C22×D15, C4.D20, D30⋊3C4, C4×C60, C2×Dic30, C2×D60, C42⋊7D15
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, D12, C22×S3, D15, C4.4D4, D20, C22×D5, C2×D12, C4○D12, D30, C2×D20, C4○D20, C42⋊7S3, D60, C22×D15, C4.D20, C2×D60, D60⋊11C2, C42⋊7D15
►On 240 points
Generators in S240
(1 155 39 133)(2 156 40 134)(3 157 41 135)(4 158 42 121)(5 159 43 122)(6 160 44 123)(7 161 45 124)(8 162 31 125)(9 163 32 126)(10 164 33 127)(11 165 34 128)(12 151 35 129)(13 152 36 130)(14 153 37 131)(15 154 38 132)(16 170 54 144)(17 171 55 145)(18 172 56 146)(19 173 57 147)(20 174 58 148)(21 175 59 149)(22 176 60 150)(23 177 46 136)(24 178 47 137)(25 179 48 138)(26 180 49 139)(27 166 50 140)(28 167 51 141)(29 168 52 142)(30 169 53 143)(61 211 103 189)(62 212 104 190)(63 213 105 191)(64 214 91 192)(65 215 92 193)(66 216 93 194)(67 217 94 195)(68 218 95 181)(69 219 96 182)(70 220 97 183)(71 221 98 184)(72 222 99 185)(73 223 100 186)(74 224 101 187)(75 225 102 188)(76 237 109 196)(77 238 110 197)(78 239 111 198)(79 240 112 199)(80 226 113 200)(81 227 114 201)(82 228 115 202)(83 229 116 203)(84 230 117 204)(85 231 118 205)(86 232 119 206)(87 233 120 207)(88 234 106 208)(89 235 107 209)(90 236 108 210)
(1 77 20 65)(2 78 21 66)(3 79 22 67)(4 80 23 68)(5 81 24 69)(6 82 25 70)(7 83 26 71)(8 84 27 72)(9 85 28 73)(10 86 29 74)(11 87 30 75)(12 88 16 61)(13 89 17 62)(14 90 18 63)(15 76 19 64)(31 117 50 99)(32 118 51 100)(33 119 52 101)(34 120 53 102)(35 106 54 103)(36 107 55 104)(37 108 56 105)(38 109 57 91)(39 110 58 92)(40 111 59 93)(41 112 60 94)(42 113 46 95)(43 114 47 96)(44 115 48 97)(45 116 49 98)(121 200 136 181)(122 201 137 182)(123 202 138 183)(124 203 139 184)(125 204 140 185)(126 205 141 186)(127 206 142 187)(128 207 143 188)(129 208 144 189)(130 209 145 190)(131 210 146 191)(132 196 147 192)(133 197 148 193)(134 198 149 194)(135 199 150 195)(151 234 170 211)(152 235 171 212)(153 236 172 213)(154 237 173 214)(155 238 174 215)(156 239 175 216)(157 240 176 217)(158 226 177 218)(159 227 178 219)(160 228 179 220)(161 229 180 221)(162 230 166 222)(163 231 167 223)(164 232 168 224)(165 233 169 225)
(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)
(1 192)(2 191)(3 190)(4 189)(5 188)(6 187)(7 186)(8 185)(9 184)(10 183)(11 182)(12 181)(13 195)(14 194)(15 193)(16 200)(17 199)(18 198)(19 197)(20 196)(21 210)(22 209)(23 208)(24 207)(25 206)(26 205)(27 204)(28 203)(29 202)(30 201)(31 222)(32 221)(33 220)(34 219)(35 218)(36 217)(37 216)(38 215)(39 214)(40 213)(41 212)(42 211)(43 225)(44 224)(45 223)(46 234)(47 233)(48 232)(49 231)(50 230)(51 229)(52 228)(53 227)(54 226)(55 240)(56 239)(57 238)(58 237)(59 236)(60 235)(61 177)(62 176)(63 175)(64 174)(65 173)(66 172)(67 171)(68 170)(69 169)(70 168)(71 167)(72 166)(73 180)(74 179)(75 178)(76 155)(77 154)(78 153)(79 152)(80 151)(81 165)(82 164)(83 163)(84 162)(85 161)(86 160)(87 159)(88 158)(89 157)(90 156)(91 148)(92 147)(93 146)(94 145)(95 144)(96 143)(97 142)(98 141)(99 140)(100 139)(101 138)(102 137)(103 136)(104 150)(105 149)(106 121)(107 135)(108 134)(109 133)(110 132)(111 131)(112 130)(113 129)(114 128)(115 127)(116 126)(117 125)(118 124)(119 123)(120 122)
G:=sub<Sym(240)| (1,155,39,133)(2,156,40,134)(3,157,41,135)(4,158,42,121)(5,159,43,122)(6,160,44,123)(7,161,45,124)(8,162,31,125)(9,163,32,126)(10,164,33,127)(11,165,34,128)(12,151,35,129)(13,152,36,130)(14,153,37,131)(15,154,38,132)(16,170,54,144)(17,171,55,145)(18,172,56,146)(19,173,57,147)(20,174,58,148)(21,175,59,149)(22,176,60,150)(23,177,46,136)(24,178,47,137)(25,179,48,138)(26,180,49,139)(27,166,50,140)(28,167,51,141)(29,168,52,142)(30,169,53,143)(61,211,103,189)(62,212,104,190)(63,213,105,191)(64,214,91,192)(65,215,92,193)(66,216,93,194)(67,217,94,195)(68,218,95,181)(69,219,96,182)(70,220,97,183)(71,221,98,184)(72,222,99,185)(73,223,100,186)(74,224,101,187)(75,225,102,188)(76,237,109,196)(77,238,110,197)(78,239,111,198)(79,240,112,199)(80,226,113,200)(81,227,114,201)(82,228,115,202)(83,229,116,203)(84,230,117,204)(85,231,118,205)(86,232,119,206)(87,233,120,207)(88,234,106,208)(89,235,107,209)(90,236,108,210), (1,77,20,65)(2,78,21,66)(3,79,22,67)(4,80,23,68)(5,81,24,69)(6,82,25,70)(7,83,26,71)(8,84,27,72)(9,85,28,73)(10,86,29,74)(11,87,30,75)(12,88,16,61)(13,89,17,62)(14,90,18,63)(15,76,19,64)(31,117,50,99)(32,118,51,100)(33,119,52,101)(34,120,53,102)(35,106,54,103)(36,107,55,104)(37,108,56,105)(38,109,57,91)(39,110,58,92)(40,111,59,93)(41,112,60,94)(42,113,46,95)(43,114,47,96)(44,115,48,97)(45,116,49,98)(121,200,136,181)(122,201,137,182)(123,202,138,183)(124,203,139,184)(125,204,140,185)(126,205,141,186)(127,206,142,187)(128,207,143,188)(129,208,144,189)(130,209,145,190)(131,210,146,191)(132,196,147,192)(133,197,148,193)(134,198,149,194)(135,199,150,195)(151,234,170,211)(152,235,171,212)(153,236,172,213)(154,237,173,214)(155,238,174,215)(156,239,175,216)(157,240,176,217)(158,226,177,218)(159,227,178,219)(160,228,179,220)(161,229,180,221)(162,230,166,222)(163,231,167,223)(164,232,168,224)(165,233,169,225), (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), (1,192)(2,191)(3,190)(4,189)(5,188)(6,187)(7,186)(8,185)(9,184)(10,183)(11,182)(12,181)(13,195)(14,194)(15,193)(16,200)(17,199)(18,198)(19,197)(20,196)(21,210)(22,209)(23,208)(24,207)(25,206)(26,205)(27,204)(28,203)(29,202)(30,201)(31,222)(32,221)(33,220)(34,219)(35,218)(36,217)(37,216)(38,215)(39,214)(40,213)(41,212)(42,211)(43,225)(44,224)(45,223)(46,234)(47,233)(48,232)(49,231)(50,230)(51,229)(52,228)(53,227)(54,226)(55,240)(56,239)(57,238)(58,237)(59,236)(60,235)(61,177)(62,176)(63,175)(64,174)(65,173)(66,172)(67,171)(68,170)(69,169)(70,168)(71,167)(72,166)(73,180)(74,179)(75,178)(76,155)(77,154)(78,153)(79,152)(80,151)(81,165)(82,164)(83,163)(84,162)(85,161)(86,160)(87,159)(88,158)(89,157)(90,156)(91,148)(92,147)(93,146)(94,145)(95,144)(96,143)(97,142)(98,141)(99,140)(100,139)(101,138)(102,137)(103,136)(104,150)(105,149)(106,121)(107,135)(108,134)(109,133)(110,132)(111,131)(112,130)(113,129)(114,128)(115,127)(116,126)(117,125)(118,124)(119,123)(120,122)>;
G:=Group( (1,155,39,133)(2,156,40,134)(3,157,41,135)(4,158,42,121)(5,159,43,122)(6,160,44,123)(7,161,45,124)(8,162,31,125)(9,163,32,126)(10,164,33,127)(11,165,34,128)(12,151,35,129)(13,152,36,130)(14,153,37,131)(15,154,38,132)(16,170,54,144)(17,171,55,145)(18,172,56,146)(19,173,57,147)(20,174,58,148)(21,175,59,149)(22,176,60,150)(23,177,46,136)(24,178,47,137)(25,179,48,138)(26,180,49,139)(27,166,50,140)(28,167,51,141)(29,168,52,142)(30,169,53,143)(61,211,103,189)(62,212,104,190)(63,213,105,191)(64,214,91,192)(65,215,92,193)(66,216,93,194)(67,217,94,195)(68,218,95,181)(69,219,96,182)(70,220,97,183)(71,221,98,184)(72,222,99,185)(73,223,100,186)(74,224,101,187)(75,225,102,188)(76,237,109,196)(77,238,110,197)(78,239,111,198)(79,240,112,199)(80,226,113,200)(81,227,114,201)(82,228,115,202)(83,229,116,203)(84,230,117,204)(85,231,118,205)(86,232,119,206)(87,233,120,207)(88,234,106,208)(89,235,107,209)(90,236,108,210), (1,77,20,65)(2,78,21,66)(3,79,22,67)(4,80,23,68)(5,81,24,69)(6,82,25,70)(7,83,26,71)(8,84,27,72)(9,85,28,73)(10,86,29,74)(11,87,30,75)(12,88,16,61)(13,89,17,62)(14,90,18,63)(15,76,19,64)(31,117,50,99)(32,118,51,100)(33,119,52,101)(34,120,53,102)(35,106,54,103)(36,107,55,104)(37,108,56,105)(38,109,57,91)(39,110,58,92)(40,111,59,93)(41,112,60,94)(42,113,46,95)(43,114,47,96)(44,115,48,97)(45,116,49,98)(121,200,136,181)(122,201,137,182)(123,202,138,183)(124,203,139,184)(125,204,140,185)(126,205,141,186)(127,206,142,187)(128,207,143,188)(129,208,144,189)(130,209,145,190)(131,210,146,191)(132,196,147,192)(133,197,148,193)(134,198,149,194)(135,199,150,195)(151,234,170,211)(152,235,171,212)(153,236,172,213)(154,237,173,214)(155,238,174,215)(156,239,175,216)(157,240,176,217)(158,226,177,218)(159,227,178,219)(160,228,179,220)(161,229,180,221)(162,230,166,222)(163,231,167,223)(164,232,168,224)(165,233,169,225), (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), (1,192)(2,191)(3,190)(4,189)(5,188)(6,187)(7,186)(8,185)(9,184)(10,183)(11,182)(12,181)(13,195)(14,194)(15,193)(16,200)(17,199)(18,198)(19,197)(20,196)(21,210)(22,209)(23,208)(24,207)(25,206)(26,205)(27,204)(28,203)(29,202)(30,201)(31,222)(32,221)(33,220)(34,219)(35,218)(36,217)(37,216)(38,215)(39,214)(40,213)(41,212)(42,211)(43,225)(44,224)(45,223)(46,234)(47,233)(48,232)(49,231)(50,230)(51,229)(52,228)(53,227)(54,226)(55,240)(56,239)(57,238)(58,237)(59,236)(60,235)(61,177)(62,176)(63,175)(64,174)(65,173)(66,172)(67,171)(68,170)(69,169)(70,168)(71,167)(72,166)(73,180)(74,179)(75,178)(76,155)(77,154)(78,153)(79,152)(80,151)(81,165)(82,164)(83,163)(84,162)(85,161)(86,160)(87,159)(88,158)(89,157)(90,156)(91,148)(92,147)(93,146)(94,145)(95,144)(96,143)(97,142)(98,141)(99,140)(100,139)(101,138)(102,137)(103,136)(104,150)(105,149)(106,121)(107,135)(108,134)(109,133)(110,132)(111,131)(112,130)(113,129)(114,128)(115,127)(116,126)(117,125)(118,124)(119,123)(120,122) );
G=PermutationGroup([[(1,155,39,133),(2,156,40,134),(3,157,41,135),(4,158,42,121),(5,159,43,122),(6,160,44,123),(7,161,45,124),(8,162,31,125),(9,163,32,126),(10,164,33,127),(11,165,34,128),(12,151,35,129),(13,152,36,130),(14,153,37,131),(15,154,38,132),(16,170,54,144),(17,171,55,145),(18,172,56,146),(19,173,57,147),(20,174,58,148),(21,175,59,149),(22,176,60,150),(23,177,46,136),(24,178,47,137),(25,179,48,138),(26,180,49,139),(27,166,50,140),(28,167,51,141),(29,168,52,142),(30,169,53,143),(61,211,103,189),(62,212,104,190),(63,213,105,191),(64,214,91,192),(65,215,92,193),(66,216,93,194),(67,217,94,195),(68,218,95,181),(69,219,96,182),(70,220,97,183),(71,221,98,184),(72,222,99,185),(73,223,100,186),(74,224,101,187),(75,225,102,188),(76,237,109,196),(77,238,110,197),(78,239,111,198),(79,240,112,199),(80,226,113,200),(81,227,114,201),(82,228,115,202),(83,229,116,203),(84,230,117,204),(85,231,118,205),(86,232,119,206),(87,233,120,207),(88,234,106,208),(89,235,107,209),(90,236,108,210)], [(1,77,20,65),(2,78,21,66),(3,79,22,67),(4,80,23,68),(5,81,24,69),(6,82,25,70),(7,83,26,71),(8,84,27,72),(9,85,28,73),(10,86,29,74),(11,87,30,75),(12,88,16,61),(13,89,17,62),(14,90,18,63),(15,76,19,64),(31,117,50,99),(32,118,51,100),(33,119,52,101),(34,120,53,102),(35,106,54,103),(36,107,55,104),(37,108,56,105),(38,109,57,91),(39,110,58,92),(40,111,59,93),(41,112,60,94),(42,113,46,95),(43,114,47,96),(44,115,48,97),(45,116,49,98),(121,200,136,181),(122,201,137,182),(123,202,138,183),(124,203,139,184),(125,204,140,185),(126,205,141,186),(127,206,142,187),(128,207,143,188),(129,208,144,189),(130,209,145,190),(131,210,146,191),(132,196,147,192),(133,197,148,193),(134,198,149,194),(135,199,150,195),(151,234,170,211),(152,235,171,212),(153,236,172,213),(154,237,173,214),(155,238,174,215),(156,239,175,216),(157,240,176,217),(158,226,177,218),(159,227,178,219),(160,228,179,220),(161,229,180,221),(162,230,166,222),(163,231,167,223),(164,232,168,224),(165,233,169,225)], [(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)], [(1,192),(2,191),(3,190),(4,189),(5,188),(6,187),(7,186),(8,185),(9,184),(10,183),(11,182),(12,181),(13,195),(14,194),(15,193),(16,200),(17,199),(18,198),(19,197),(20,196),(21,210),(22,209),(23,208),(24,207),(25,206),(26,205),(27,204),(28,203),(29,202),(30,201),(31,222),(32,221),(33,220),(34,219),(35,218),(36,217),(37,216),(38,215),(39,214),(40,213),(41,212),(42,211),(43,225),(44,224),(45,223),(46,234),(47,233),(48,232),(49,231),(50,230),(51,229),(52,228),(53,227),(54,226),(55,240),(56,239),(57,238),(58,237),(59,236),(60,235),(61,177),(62,176),(63,175),(64,174),(65,173),(66,172),(67,171),(68,170),(69,169),(70,168),(71,167),(72,166),(73,180),(74,179),(75,178),(76,155),(77,154),(78,153),(79,152),(80,151),(81,165),(82,164),(83,163),(84,162),(85,161),(86,160),(87,159),(88,158),(89,157),(90,156),(91,148),(92,147),(93,146),(94,145),(95,144),(96,143),(97,142),(98,141),(99,140),(100,139),(101,138),(102,137),(103,136),(104,150),(105,149),(106,121),(107,135),(108,134),(109,133),(110,132),(111,131),(112,130),(113,129),(114,128),(115,127),(116,126),(117,125),(118,124),(119,123),(120,122)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4F | 4G | 4H | 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 | 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 | 60 | 2 | 2 | ··· | 2 | 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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | C4○D4 | D10 | D12 | D15 | D20 | C4○D12 | D30 | C4○D20 | D60 | D60⋊11C2 |
| kernel | C42⋊7D15 | D30⋊3C4 | C4×C60 | C2×Dic30 | C2×D60 | C4×C20 | C60 | C4×C12 | C2×C20 | C30 | C2×C12 | C20 | C42 | C12 | C10 | C2×C4 | C6 | C4 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 4 | 6 | 4 | 4 | 8 | 8 | 12 | 16 | 16 | 32 |
Matrix representation of C42⋊7D15 ►in GL4(𝔽61) generated by
| 14 | 45 | 0 | 0 |
| 16 | 47 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 50 |
| 50 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 |
| 0 | 0 | 14 | 45 |
| 0 | 0 | 16 | 47 |
| 37 | 14 | 0 | 0 |
| 47 | 31 | 0 | 0 |
| 0 | 0 | 25 | 30 |
| 0 | 0 | 31 | 47 |
| 29 | 2 | 0 | 0 |
| 7 | 32 | 0 | 0 |
| 0 | 0 | 20 | 53 |
| 0 | 0 | 27 | 41 |
G:=sub<GL(4,GF(61))| [14,16,0,0,45,47,0,0,0,0,50,0,0,0,0,50],[50,0,0,0,0,50,0,0,0,0,14,16,0,0,45,47],[37,47,0,0,14,31,0,0,0,0,25,31,0,0,30,47],[29,7,0,0,2,32,0,0,0,0,20,27,0,0,53,41] >;
C42⋊7D15 in GAP, Magma, Sage, TeX
C_4^2\rtimes_7D_{15} % in TeX
G:=Group("C4^2:7D15"); // GroupNames label
G:=SmallGroup(480,840);
// by ID
G=gap.SmallGroup(480,840);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,64,254,100,2693,18822]);
// 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,d*c*d=c^-1>;
// generators/relations