direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C15×C4⋊D4, C60⋊35D4, C4⋊C4⋊2C30, C4⋊2(D4×C15), C12⋊9(C5×D4), C20⋊9(C3×D4), (C2×D4)⋊2C30, (C2×C30)⋊22D4, C2.5(D4×C30), (D4×C30)⋊29C2, (D4×C10)⋊11C6, (C6×D4)⋊11C10, C22⋊C4⋊3C30, (C22×C4)⋊6C30, C6.68(D4×C10), C10.68(C6×D4), C22⋊2(D4×C15), (C22×C20)⋊15C6, (C22×C60)⋊25C2, C30.451(C2×D4), C23.2(C2×C30), (C22×C12)⋊11C10, C30.277(C4○D4), (C2×C60).578C22, (C2×C30).456C23, (C22×C30).2C22, C22.11(C22×C30), (C2×C6)⋊4(C5×D4), (C5×C4⋊C4)⋊11C6, (C2×C10)⋊7(C3×D4), (C15×C4⋊C4)⋊29C2, (C3×C4⋊C4)⋊11C10, (C2×C4).3(C2×C30), C2.4(C15×C4○D4), C6.41(C5×C4○D4), (C5×C22⋊C4)⋊11C6, (C2×C20).65(C2×C6), C10.41(C3×C4○D4), (C15×C22⋊C4)⋊27C2, (C3×C22⋊C4)⋊11C10, (C2×C12).65(C2×C10), (C22×C10).35(C2×C6), (C22×C6).27(C2×C10), (C2×C10).76(C22×C6), (C2×C6).76(C22×C10), SmallGroup(480,926)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×C4⋊D4
G = < a,b,c,d | a15=b4=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 296 in 188 conjugacy classes, 96 normal (48 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C5, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C23, C10, C10, C12, C12, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C20, C20, C2×C10, C2×C10, C2×C10, C2×C12, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C30, C30, C4⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×D4, C60, C60, C2×C30, C2×C30, C2×C30, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, D4×C10, C3×C4⋊D4, C2×C60, C2×C60, C2×C60, D4×C15, C22×C30, C22×C30, C5×C4⋊D4, C15×C22⋊C4, C15×C4⋊C4, C22×C60, D4×C30, D4×C30, C15×C4⋊D4
Quotients: C1, C2, C3, C22, C5, C6, D4, C23, C10, C2×C6, C15, C2×D4, C4○D4, C2×C10, C3×D4, C22×C6, C30, C4⋊D4, C5×D4, C22×C10, C6×D4, C3×C4○D4, C2×C30, D4×C10, C5×C4○D4, C3×C4⋊D4, D4×C15, C22×C30, C5×C4⋊D4, D4×C30, C15×C4○D4, C15×C4⋊D4
►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 66 158 24)(2 67 159 25)(3 68 160 26)(4 69 161 27)(5 70 162 28)(6 71 163 29)(7 72 164 30)(8 73 165 16)(9 74 151 17)(10 75 152 18)(11 61 153 19)(12 62 154 20)(13 63 155 21)(14 64 156 22)(15 65 157 23)(31 169 210 129)(32 170 196 130)(33 171 197 131)(34 172 198 132)(35 173 199 133)(36 174 200 134)(37 175 201 135)(38 176 202 121)(39 177 203 122)(40 178 204 123)(41 179 205 124)(42 180 206 125)(43 166 207 126)(44 167 208 127)(45 168 209 128)(46 106 147 190)(47 107 148 191)(48 108 149 192)(49 109 150 193)(50 110 136 194)(51 111 137 195)(52 112 138 181)(53 113 139 182)(54 114 140 183)(55 115 141 184)(56 116 142 185)(57 117 143 186)(58 118 144 187)(59 119 145 188)(60 120 146 189)(76 213 104 231)(77 214 105 232)(78 215 91 233)(79 216 92 234)(80 217 93 235)(81 218 94 236)(82 219 95 237)(83 220 96 238)(84 221 97 239)(85 222 98 240)(86 223 99 226)(87 224 100 227)(88 225 101 228)(89 211 102 229)(90 212 103 230)
(1 31 77 193)(2 32 78 194)(3 33 79 195)(4 34 80 181)(5 35 81 182)(6 36 82 183)(7 37 83 184)(8 38 84 185)(9 39 85 186)(10 40 86 187)(11 41 87 188)(12 42 88 189)(13 43 89 190)(14 44 90 191)(15 45 76 192)(16 176 239 56)(17 177 240 57)(18 178 226 58)(19 179 227 59)(20 180 228 60)(21 166 229 46)(22 167 230 47)(23 168 231 48)(24 169 232 49)(25 170 233 50)(26 171 234 51)(27 172 235 52)(28 173 236 53)(29 174 237 54)(30 175 238 55)(61 124 224 145)(62 125 225 146)(63 126 211 147)(64 127 212 148)(65 128 213 149)(66 129 214 150)(67 130 215 136)(68 131 216 137)(69 132 217 138)(70 133 218 139)(71 134 219 140)(72 135 220 141)(73 121 221 142)(74 122 222 143)(75 123 223 144)(91 110 159 196)(92 111 160 197)(93 112 161 198)(94 113 162 199)(95 114 163 200)(96 115 164 201)(97 116 165 202)(98 117 151 203)(99 118 152 204)(100 119 153 205)(101 120 154 206)(102 106 155 207)(103 107 156 208)(104 108 157 209)(105 109 158 210)
(1 193)(2 194)(3 195)(4 181)(5 182)(6 183)(7 184)(8 185)(9 186)(10 187)(11 188)(12 189)(13 190)(14 191)(15 192)(16 56)(17 57)(18 58)(19 59)(20 60)(21 46)(22 47)(23 48)(24 49)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 77)(32 78)(33 79)(34 80)(35 81)(36 82)(37 83)(38 84)(39 85)(40 86)(41 87)(42 88)(43 89)(44 90)(45 76)(61 145)(62 146)(63 147)(64 148)(65 149)(66 150)(67 136)(68 137)(69 138)(70 139)(71 140)(72 141)(73 142)(74 143)(75 144)(91 196)(92 197)(93 198)(94 199)(95 200)(96 201)(97 202)(98 203)(99 204)(100 205)(101 206)(102 207)(103 208)(104 209)(105 210)(106 155)(107 156)(108 157)(109 158)(110 159)(111 160)(112 161)(113 162)(114 163)(115 164)(116 165)(117 151)(118 152)(119 153)(120 154)(121 221)(122 222)(123 223)(124 224)(125 225)(126 211)(127 212)(128 213)(129 214)(130 215)(131 216)(132 217)(133 218)(134 219)(135 220)(166 229)(167 230)(168 231)(169 232)(170 233)(171 234)(172 235)(173 236)(174 237)(175 238)(176 239)(177 240)(178 226)(179 227)(180 228)
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), (1,66,158,24)(2,67,159,25)(3,68,160,26)(4,69,161,27)(5,70,162,28)(6,71,163,29)(7,72,164,30)(8,73,165,16)(9,74,151,17)(10,75,152,18)(11,61,153,19)(12,62,154,20)(13,63,155,21)(14,64,156,22)(15,65,157,23)(31,169,210,129)(32,170,196,130)(33,171,197,131)(34,172,198,132)(35,173,199,133)(36,174,200,134)(37,175,201,135)(38,176,202,121)(39,177,203,122)(40,178,204,123)(41,179,205,124)(42,180,206,125)(43,166,207,126)(44,167,208,127)(45,168,209,128)(46,106,147,190)(47,107,148,191)(48,108,149,192)(49,109,150,193)(50,110,136,194)(51,111,137,195)(52,112,138,181)(53,113,139,182)(54,114,140,183)(55,115,141,184)(56,116,142,185)(57,117,143,186)(58,118,144,187)(59,119,145,188)(60,120,146,189)(76,213,104,231)(77,214,105,232)(78,215,91,233)(79,216,92,234)(80,217,93,235)(81,218,94,236)(82,219,95,237)(83,220,96,238)(84,221,97,239)(85,222,98,240)(86,223,99,226)(87,224,100,227)(88,225,101,228)(89,211,102,229)(90,212,103,230), (1,31,77,193)(2,32,78,194)(3,33,79,195)(4,34,80,181)(5,35,81,182)(6,36,82,183)(7,37,83,184)(8,38,84,185)(9,39,85,186)(10,40,86,187)(11,41,87,188)(12,42,88,189)(13,43,89,190)(14,44,90,191)(15,45,76,192)(16,176,239,56)(17,177,240,57)(18,178,226,58)(19,179,227,59)(20,180,228,60)(21,166,229,46)(22,167,230,47)(23,168,231,48)(24,169,232,49)(25,170,233,50)(26,171,234,51)(27,172,235,52)(28,173,236,53)(29,174,237,54)(30,175,238,55)(61,124,224,145)(62,125,225,146)(63,126,211,147)(64,127,212,148)(65,128,213,149)(66,129,214,150)(67,130,215,136)(68,131,216,137)(69,132,217,138)(70,133,218,139)(71,134,219,140)(72,135,220,141)(73,121,221,142)(74,122,222,143)(75,123,223,144)(91,110,159,196)(92,111,160,197)(93,112,161,198)(94,113,162,199)(95,114,163,200)(96,115,164,201)(97,116,165,202)(98,117,151,203)(99,118,152,204)(100,119,153,205)(101,120,154,206)(102,106,155,207)(103,107,156,208)(104,108,157,209)(105,109,158,210), (1,193)(2,194)(3,195)(4,181)(5,182)(6,183)(7,184)(8,185)(9,186)(10,187)(11,188)(12,189)(13,190)(14,191)(15,192)(16,56)(17,57)(18,58)(19,59)(20,60)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,77)(32,78)(33,79)(34,80)(35,81)(36,82)(37,83)(38,84)(39,85)(40,86)(41,87)(42,88)(43,89)(44,90)(45,76)(61,145)(62,146)(63,147)(64,148)(65,149)(66,150)(67,136)(68,137)(69,138)(70,139)(71,140)(72,141)(73,142)(74,143)(75,144)(91,196)(92,197)(93,198)(94,199)(95,200)(96,201)(97,202)(98,203)(99,204)(100,205)(101,206)(102,207)(103,208)(104,209)(105,210)(106,155)(107,156)(108,157)(109,158)(110,159)(111,160)(112,161)(113,162)(114,163)(115,164)(116,165)(117,151)(118,152)(119,153)(120,154)(121,221)(122,222)(123,223)(124,224)(125,225)(126,211)(127,212)(128,213)(129,214)(130,215)(131,216)(132,217)(133,218)(134,219)(135,220)(166,229)(167,230)(168,231)(169,232)(170,233)(171,234)(172,235)(173,236)(174,237)(175,238)(176,239)(177,240)(178,226)(179,227)(180,228)>;
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), (1,66,158,24)(2,67,159,25)(3,68,160,26)(4,69,161,27)(5,70,162,28)(6,71,163,29)(7,72,164,30)(8,73,165,16)(9,74,151,17)(10,75,152,18)(11,61,153,19)(12,62,154,20)(13,63,155,21)(14,64,156,22)(15,65,157,23)(31,169,210,129)(32,170,196,130)(33,171,197,131)(34,172,198,132)(35,173,199,133)(36,174,200,134)(37,175,201,135)(38,176,202,121)(39,177,203,122)(40,178,204,123)(41,179,205,124)(42,180,206,125)(43,166,207,126)(44,167,208,127)(45,168,209,128)(46,106,147,190)(47,107,148,191)(48,108,149,192)(49,109,150,193)(50,110,136,194)(51,111,137,195)(52,112,138,181)(53,113,139,182)(54,114,140,183)(55,115,141,184)(56,116,142,185)(57,117,143,186)(58,118,144,187)(59,119,145,188)(60,120,146,189)(76,213,104,231)(77,214,105,232)(78,215,91,233)(79,216,92,234)(80,217,93,235)(81,218,94,236)(82,219,95,237)(83,220,96,238)(84,221,97,239)(85,222,98,240)(86,223,99,226)(87,224,100,227)(88,225,101,228)(89,211,102,229)(90,212,103,230), (1,31,77,193)(2,32,78,194)(3,33,79,195)(4,34,80,181)(5,35,81,182)(6,36,82,183)(7,37,83,184)(8,38,84,185)(9,39,85,186)(10,40,86,187)(11,41,87,188)(12,42,88,189)(13,43,89,190)(14,44,90,191)(15,45,76,192)(16,176,239,56)(17,177,240,57)(18,178,226,58)(19,179,227,59)(20,180,228,60)(21,166,229,46)(22,167,230,47)(23,168,231,48)(24,169,232,49)(25,170,233,50)(26,171,234,51)(27,172,235,52)(28,173,236,53)(29,174,237,54)(30,175,238,55)(61,124,224,145)(62,125,225,146)(63,126,211,147)(64,127,212,148)(65,128,213,149)(66,129,214,150)(67,130,215,136)(68,131,216,137)(69,132,217,138)(70,133,218,139)(71,134,219,140)(72,135,220,141)(73,121,221,142)(74,122,222,143)(75,123,223,144)(91,110,159,196)(92,111,160,197)(93,112,161,198)(94,113,162,199)(95,114,163,200)(96,115,164,201)(97,116,165,202)(98,117,151,203)(99,118,152,204)(100,119,153,205)(101,120,154,206)(102,106,155,207)(103,107,156,208)(104,108,157,209)(105,109,158,210), (1,193)(2,194)(3,195)(4,181)(5,182)(6,183)(7,184)(8,185)(9,186)(10,187)(11,188)(12,189)(13,190)(14,191)(15,192)(16,56)(17,57)(18,58)(19,59)(20,60)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,77)(32,78)(33,79)(34,80)(35,81)(36,82)(37,83)(38,84)(39,85)(40,86)(41,87)(42,88)(43,89)(44,90)(45,76)(61,145)(62,146)(63,147)(64,148)(65,149)(66,150)(67,136)(68,137)(69,138)(70,139)(71,140)(72,141)(73,142)(74,143)(75,144)(91,196)(92,197)(93,198)(94,199)(95,200)(96,201)(97,202)(98,203)(99,204)(100,205)(101,206)(102,207)(103,208)(104,209)(105,210)(106,155)(107,156)(108,157)(109,158)(110,159)(111,160)(112,161)(113,162)(114,163)(115,164)(116,165)(117,151)(118,152)(119,153)(120,154)(121,221)(122,222)(123,223)(124,224)(125,225)(126,211)(127,212)(128,213)(129,214)(130,215)(131,216)(132,217)(133,218)(134,219)(135,220)(166,229)(167,230)(168,231)(169,232)(170,233)(171,234)(172,235)(173,236)(174,237)(175,238)(176,239)(177,240)(178,226)(179,227)(180,228) );
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)], [(1,66,158,24),(2,67,159,25),(3,68,160,26),(4,69,161,27),(5,70,162,28),(6,71,163,29),(7,72,164,30),(8,73,165,16),(9,74,151,17),(10,75,152,18),(11,61,153,19),(12,62,154,20),(13,63,155,21),(14,64,156,22),(15,65,157,23),(31,169,210,129),(32,170,196,130),(33,171,197,131),(34,172,198,132),(35,173,199,133),(36,174,200,134),(37,175,201,135),(38,176,202,121),(39,177,203,122),(40,178,204,123),(41,179,205,124),(42,180,206,125),(43,166,207,126),(44,167,208,127),(45,168,209,128),(46,106,147,190),(47,107,148,191),(48,108,149,192),(49,109,150,193),(50,110,136,194),(51,111,137,195),(52,112,138,181),(53,113,139,182),(54,114,140,183),(55,115,141,184),(56,116,142,185),(57,117,143,186),(58,118,144,187),(59,119,145,188),(60,120,146,189),(76,213,104,231),(77,214,105,232),(78,215,91,233),(79,216,92,234),(80,217,93,235),(81,218,94,236),(82,219,95,237),(83,220,96,238),(84,221,97,239),(85,222,98,240),(86,223,99,226),(87,224,100,227),(88,225,101,228),(89,211,102,229),(90,212,103,230)], [(1,31,77,193),(2,32,78,194),(3,33,79,195),(4,34,80,181),(5,35,81,182),(6,36,82,183),(7,37,83,184),(8,38,84,185),(9,39,85,186),(10,40,86,187),(11,41,87,188),(12,42,88,189),(13,43,89,190),(14,44,90,191),(15,45,76,192),(16,176,239,56),(17,177,240,57),(18,178,226,58),(19,179,227,59),(20,180,228,60),(21,166,229,46),(22,167,230,47),(23,168,231,48),(24,169,232,49),(25,170,233,50),(26,171,234,51),(27,172,235,52),(28,173,236,53),(29,174,237,54),(30,175,238,55),(61,124,224,145),(62,125,225,146),(63,126,211,147),(64,127,212,148),(65,128,213,149),(66,129,214,150),(67,130,215,136),(68,131,216,137),(69,132,217,138),(70,133,218,139),(71,134,219,140),(72,135,220,141),(73,121,221,142),(74,122,222,143),(75,123,223,144),(91,110,159,196),(92,111,160,197),(93,112,161,198),(94,113,162,199),(95,114,163,200),(96,115,164,201),(97,116,165,202),(98,117,151,203),(99,118,152,204),(100,119,153,205),(101,120,154,206),(102,106,155,207),(103,107,156,208),(104,108,157,209),(105,109,158,210)], [(1,193),(2,194),(3,195),(4,181),(5,182),(6,183),(7,184),(8,185),(9,186),(10,187),(11,188),(12,189),(13,190),(14,191),(15,192),(16,56),(17,57),(18,58),(19,59),(20,60),(21,46),(22,47),(23,48),(24,49),(25,50),(26,51),(27,52),(28,53),(29,54),(30,55),(31,77),(32,78),(33,79),(34,80),(35,81),(36,82),(37,83),(38,84),(39,85),(40,86),(41,87),(42,88),(43,89),(44,90),(45,76),(61,145),(62,146),(63,147),(64,148),(65,149),(66,150),(67,136),(68,137),(69,138),(70,139),(71,140),(72,141),(73,142),(74,143),(75,144),(91,196),(92,197),(93,198),(94,199),(95,200),(96,201),(97,202),(98,203),(99,204),(100,205),(101,206),(102,207),(103,208),(104,209),(105,210),(106,155),(107,156),(108,157),(109,158),(110,159),(111,160),(112,161),(113,162),(114,163),(115,164),(116,165),(117,151),(118,152),(119,153),(120,154),(121,221),(122,222),(123,223),(124,224),(125,225),(126,211),(127,212),(128,213),(129,214),(130,215),(131,216),(132,217),(133,218),(134,219),(135,220),(166,229),(167,230),(168,231),(169,232),(170,233),(171,234),(172,235),(173,236),(174,237),(175,238),(176,239),(177,240),(178,226),(179,227),(180,228)]])
210 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AB | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 15A | ··· | 15H | 20A | ··· | 20P | 20Q | ··· | 20X | 30A | ··· | 30X | 30Y | ··· | 30AN | 30AO | ··· | 30BD | 60A | ··· | 60AF | 60AG | ··· | 60AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
210 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C5 | C6 | C6 | C6 | C6 | C10 | C10 | C10 | C10 | C15 | C30 | C30 | C30 | C30 | D4 | D4 | C4○D4 | C3×D4 | C3×D4 | C5×D4 | C5×D4 | C3×C4○D4 | C5×C4○D4 | D4×C15 | D4×C15 | C15×C4○D4 |
| kernel | C15×C4⋊D4 | C15×C22⋊C4 | C15×C4⋊C4 | C22×C60 | D4×C30 | C5×C4⋊D4 | C3×C4⋊D4 | C5×C22⋊C4 | C5×C4⋊C4 | C22×C20 | D4×C10 | C3×C22⋊C4 | C3×C4⋊C4 | C22×C12 | C6×D4 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C60 | C2×C30 | C30 | C20 | C2×C10 | C12 | C2×C6 | C10 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 3 | 2 | 4 | 4 | 2 | 2 | 6 | 8 | 4 | 4 | 12 | 8 | 16 | 8 | 8 | 24 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 8 | 16 | 16 | 16 |
Matrix representation of C15×C4⋊D4 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 25 | 0 |
| 0 | 0 | 0 | 25 |
| 11 | 0 | 0 | 0 |
| 8 | 50 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 34 | 59 | 0 | 0 |
| 60 | 27 | 0 | 0 |
| 0 | 0 | 17 | 1 |
| 0 | 0 | 15 | 44 |
| 34 | 59 | 0 | 0 |
| 59 | 27 | 0 | 0 |
| 0 | 0 | 17 | 1 |
| 0 | 0 | 17 | 44 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,25,0,0,0,0,25],[11,8,0,0,0,50,0,0,0,0,1,0,0,0,0,1],[34,60,0,0,59,27,0,0,0,0,17,15,0,0,1,44],[34,59,0,0,59,27,0,0,0,0,17,17,0,0,1,44] >;
C15×C4⋊D4 in GAP, Magma, Sage, TeX
C_{15}\times C_4\rtimes D_4 % in TeX
G:=Group("C15xC4:D4"); // GroupNames label
G:=SmallGroup(480,926);
// by ID
G=gap.SmallGroup(480,926);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-5,-2,-2,1709,848,5126]);
// Polycyclic
G:=Group<a,b,c,d|a^15=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations