direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C15×C4⋊1D4, C60⋊28D4, C42⋊9C30, C4⋊1(D4×C15), C12⋊6(C5×D4), C20⋊6(C3×D4), (C4×C20)⋊19C6, (C4×C60)⋊27C2, (C2×D4)⋊3C30, C2.9(D4×C30), (C4×C12)⋊13C10, (C6×D4)⋊12C10, (D4×C10)⋊12C6, (D4×C30)⋊30C2, C6.72(D4×C10), C10.72(C6×D4), C30.455(C2×D4), C23.5(C2×C30), (C2×C60).580C22, (C2×C30).462C23, (C22×C30).5C22, C22.17(C22×C30), (C2×C4).23(C2×C30), (C2×C20).125(C2×C6), (C22×C10).9(C2×C6), (C22×C6).4(C2×C10), (C2×C12).126(C2×C10), (C2×C10).82(C22×C6), (C2×C6).82(C22×C10), SmallGroup(480,932)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×C4⋊1D4
G = < a,b,c,d | a15=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 360 in 216 conjugacy classes, 104 normal (16 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, D4, C23, C10, C10, C12, C2×C6, C2×C6, C15, C42, C2×D4, C20, C2×C10, C2×C10, C2×C12, C3×D4, C22×C6, C30, C30, C4⋊1D4, C2×C20, C5×D4, C22×C10, C4×C12, C6×D4, C60, C2×C30, C2×C30, C4×C20, D4×C10, C3×C4⋊1D4, C2×C60, D4×C15, C22×C30, C5×C4⋊1D4, C4×C60, D4×C30, C15×C4⋊1D4
Quotients: C1, C2, C3, C22, C5, C6, D4, C23, C10, C2×C6, C15, C2×D4, C2×C10, C3×D4, C22×C6, C30, C4⋊1D4, C5×D4, C22×C10, C6×D4, C2×C30, D4×C10, C3×C4⋊1D4, D4×C15, C22×C30, C5×C4⋊1D4, D4×C30, C15×C4⋊1D4
►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)
(1 103 218 207)(2 104 219 208)(3 105 220 209)(4 91 221 210)(5 92 222 196)(6 93 223 197)(7 94 224 198)(8 95 225 199)(9 96 211 200)(10 97 212 201)(11 98 213 202)(12 99 214 203)(13 100 215 204)(14 101 216 205)(15 102 217 206)(16 234 45 84)(17 235 31 85)(18 236 32 86)(19 237 33 87)(20 238 34 88)(21 239 35 89)(22 240 36 90)(23 226 37 76)(24 227 38 77)(25 228 39 78)(26 229 40 79)(27 230 41 80)(28 231 42 81)(29 232 43 82)(30 233 44 83)(46 113 125 170)(47 114 126 171)(48 115 127 172)(49 116 128 173)(50 117 129 174)(51 118 130 175)(52 119 131 176)(53 120 132 177)(54 106 133 178)(55 107 134 179)(56 108 135 180)(57 109 121 166)(58 110 122 167)(59 111 123 168)(60 112 124 169)(61 151 182 141)(62 152 183 142)(63 153 184 143)(64 154 185 144)(65 155 186 145)(66 156 187 146)(67 157 188 147)(68 158 189 148)(69 159 190 149)(70 160 191 150)(71 161 192 136)(72 162 193 137)(73 163 194 138)(74 164 195 139)(75 165 181 140)
(1 174 234 65)(2 175 235 66)(3 176 236 67)(4 177 237 68)(5 178 238 69)(6 179 239 70)(7 180 240 71)(8 166 226 72)(9 167 227 73)(10 168 228 74)(11 169 229 75)(12 170 230 61)(13 171 231 62)(14 172 232 63)(15 173 233 64)(16 145 207 129)(17 146 208 130)(18 147 209 131)(19 148 210 132)(20 149 196 133)(21 150 197 134)(22 136 198 135)(23 137 199 121)(24 138 200 122)(25 139 201 123)(26 140 202 124)(27 141 203 125)(28 142 204 126)(29 143 205 127)(30 144 206 128)(31 156 104 51)(32 157 105 52)(33 158 91 53)(34 159 92 54)(35 160 93 55)(36 161 94 56)(37 162 95 57)(38 163 96 58)(39 164 97 59)(40 165 98 60)(41 151 99 46)(42 152 100 47)(43 153 101 48)(44 154 102 49)(45 155 103 50)(76 193 225 109)(77 194 211 110)(78 195 212 111)(79 181 213 112)(80 182 214 113)(81 183 215 114)(82 184 216 115)(83 185 217 116)(84 186 218 117)(85 187 219 118)(86 188 220 119)(87 189 221 120)(88 190 222 106)(89 191 223 107)(90 192 224 108)
(1 129)(2 130)(3 131)(4 132)(5 133)(6 134)(7 135)(8 121)(9 122)(10 123)(11 124)(12 125)(13 126)(14 127)(15 128)(16 65)(17 66)(18 67)(19 68)(20 69)(21 70)(22 71)(23 72)(24 73)(25 74)(26 75)(27 61)(28 62)(29 63)(30 64)(31 187)(32 188)(33 189)(34 190)(35 191)(36 192)(37 193)(38 194)(39 195)(40 181)(41 182)(42 183)(43 184)(44 185)(45 186)(46 214)(47 215)(48 216)(49 217)(50 218)(51 219)(52 220)(53 221)(54 222)(55 223)(56 224)(57 225)(58 211)(59 212)(60 213)(76 162)(77 163)(78 164)(79 165)(80 151)(81 152)(82 153)(83 154)(84 155)(85 156)(86 157)(87 158)(88 159)(89 160)(90 161)(91 120)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(99 113)(100 114)(101 115)(102 116)(103 117)(104 118)(105 119)(136 240)(137 226)(138 227)(139 228)(140 229)(141 230)(142 231)(143 232)(144 233)(145 234)(146 235)(147 236)(148 237)(149 238)(150 239)(166 199)(167 200)(168 201)(169 202)(170 203)(171 204)(172 205)(173 206)(174 207)(175 208)(176 209)(177 210)(178 196)(179 197)(180 198)
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), (1,103,218,207)(2,104,219,208)(3,105,220,209)(4,91,221,210)(5,92,222,196)(6,93,223,197)(7,94,224,198)(8,95,225,199)(9,96,211,200)(10,97,212,201)(11,98,213,202)(12,99,214,203)(13,100,215,204)(14,101,216,205)(15,102,217,206)(16,234,45,84)(17,235,31,85)(18,236,32,86)(19,237,33,87)(20,238,34,88)(21,239,35,89)(22,240,36,90)(23,226,37,76)(24,227,38,77)(25,228,39,78)(26,229,40,79)(27,230,41,80)(28,231,42,81)(29,232,43,82)(30,233,44,83)(46,113,125,170)(47,114,126,171)(48,115,127,172)(49,116,128,173)(50,117,129,174)(51,118,130,175)(52,119,131,176)(53,120,132,177)(54,106,133,178)(55,107,134,179)(56,108,135,180)(57,109,121,166)(58,110,122,167)(59,111,123,168)(60,112,124,169)(61,151,182,141)(62,152,183,142)(63,153,184,143)(64,154,185,144)(65,155,186,145)(66,156,187,146)(67,157,188,147)(68,158,189,148)(69,159,190,149)(70,160,191,150)(71,161,192,136)(72,162,193,137)(73,163,194,138)(74,164,195,139)(75,165,181,140), (1,174,234,65)(2,175,235,66)(3,176,236,67)(4,177,237,68)(5,178,238,69)(6,179,239,70)(7,180,240,71)(8,166,226,72)(9,167,227,73)(10,168,228,74)(11,169,229,75)(12,170,230,61)(13,171,231,62)(14,172,232,63)(15,173,233,64)(16,145,207,129)(17,146,208,130)(18,147,209,131)(19,148,210,132)(20,149,196,133)(21,150,197,134)(22,136,198,135)(23,137,199,121)(24,138,200,122)(25,139,201,123)(26,140,202,124)(27,141,203,125)(28,142,204,126)(29,143,205,127)(30,144,206,128)(31,156,104,51)(32,157,105,52)(33,158,91,53)(34,159,92,54)(35,160,93,55)(36,161,94,56)(37,162,95,57)(38,163,96,58)(39,164,97,59)(40,165,98,60)(41,151,99,46)(42,152,100,47)(43,153,101,48)(44,154,102,49)(45,155,103,50)(76,193,225,109)(77,194,211,110)(78,195,212,111)(79,181,213,112)(80,182,214,113)(81,183,215,114)(82,184,216,115)(83,185,217,116)(84,186,218,117)(85,187,219,118)(86,188,220,119)(87,189,221,120)(88,190,222,106)(89,191,223,107)(90,192,224,108), (1,129)(2,130)(3,131)(4,132)(5,133)(6,134)(7,135)(8,121)(9,122)(10,123)(11,124)(12,125)(13,126)(14,127)(15,128)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,61)(28,62)(29,63)(30,64)(31,187)(32,188)(33,189)(34,190)(35,191)(36,192)(37,193)(38,194)(39,195)(40,181)(41,182)(42,183)(43,184)(44,185)(45,186)(46,214)(47,215)(48,216)(49,217)(50,218)(51,219)(52,220)(53,221)(54,222)(55,223)(56,224)(57,225)(58,211)(59,212)(60,213)(76,162)(77,163)(78,164)(79,165)(80,151)(81,152)(82,153)(83,154)(84,155)(85,156)(86,157)(87,158)(88,159)(89,160)(90,161)(91,120)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(99,113)(100,114)(101,115)(102,116)(103,117)(104,118)(105,119)(136,240)(137,226)(138,227)(139,228)(140,229)(141,230)(142,231)(143,232)(144,233)(145,234)(146,235)(147,236)(148,237)(149,238)(150,239)(166,199)(167,200)(168,201)(169,202)(170,203)(171,204)(172,205)(173,206)(174,207)(175,208)(176,209)(177,210)(178,196)(179,197)(180,198)>;
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), (1,103,218,207)(2,104,219,208)(3,105,220,209)(4,91,221,210)(5,92,222,196)(6,93,223,197)(7,94,224,198)(8,95,225,199)(9,96,211,200)(10,97,212,201)(11,98,213,202)(12,99,214,203)(13,100,215,204)(14,101,216,205)(15,102,217,206)(16,234,45,84)(17,235,31,85)(18,236,32,86)(19,237,33,87)(20,238,34,88)(21,239,35,89)(22,240,36,90)(23,226,37,76)(24,227,38,77)(25,228,39,78)(26,229,40,79)(27,230,41,80)(28,231,42,81)(29,232,43,82)(30,233,44,83)(46,113,125,170)(47,114,126,171)(48,115,127,172)(49,116,128,173)(50,117,129,174)(51,118,130,175)(52,119,131,176)(53,120,132,177)(54,106,133,178)(55,107,134,179)(56,108,135,180)(57,109,121,166)(58,110,122,167)(59,111,123,168)(60,112,124,169)(61,151,182,141)(62,152,183,142)(63,153,184,143)(64,154,185,144)(65,155,186,145)(66,156,187,146)(67,157,188,147)(68,158,189,148)(69,159,190,149)(70,160,191,150)(71,161,192,136)(72,162,193,137)(73,163,194,138)(74,164,195,139)(75,165,181,140), (1,174,234,65)(2,175,235,66)(3,176,236,67)(4,177,237,68)(5,178,238,69)(6,179,239,70)(7,180,240,71)(8,166,226,72)(9,167,227,73)(10,168,228,74)(11,169,229,75)(12,170,230,61)(13,171,231,62)(14,172,232,63)(15,173,233,64)(16,145,207,129)(17,146,208,130)(18,147,209,131)(19,148,210,132)(20,149,196,133)(21,150,197,134)(22,136,198,135)(23,137,199,121)(24,138,200,122)(25,139,201,123)(26,140,202,124)(27,141,203,125)(28,142,204,126)(29,143,205,127)(30,144,206,128)(31,156,104,51)(32,157,105,52)(33,158,91,53)(34,159,92,54)(35,160,93,55)(36,161,94,56)(37,162,95,57)(38,163,96,58)(39,164,97,59)(40,165,98,60)(41,151,99,46)(42,152,100,47)(43,153,101,48)(44,154,102,49)(45,155,103,50)(76,193,225,109)(77,194,211,110)(78,195,212,111)(79,181,213,112)(80,182,214,113)(81,183,215,114)(82,184,216,115)(83,185,217,116)(84,186,218,117)(85,187,219,118)(86,188,220,119)(87,189,221,120)(88,190,222,106)(89,191,223,107)(90,192,224,108), (1,129)(2,130)(3,131)(4,132)(5,133)(6,134)(7,135)(8,121)(9,122)(10,123)(11,124)(12,125)(13,126)(14,127)(15,128)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,61)(28,62)(29,63)(30,64)(31,187)(32,188)(33,189)(34,190)(35,191)(36,192)(37,193)(38,194)(39,195)(40,181)(41,182)(42,183)(43,184)(44,185)(45,186)(46,214)(47,215)(48,216)(49,217)(50,218)(51,219)(52,220)(53,221)(54,222)(55,223)(56,224)(57,225)(58,211)(59,212)(60,213)(76,162)(77,163)(78,164)(79,165)(80,151)(81,152)(82,153)(83,154)(84,155)(85,156)(86,157)(87,158)(88,159)(89,160)(90,161)(91,120)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(99,113)(100,114)(101,115)(102,116)(103,117)(104,118)(105,119)(136,240)(137,226)(138,227)(139,228)(140,229)(141,230)(142,231)(143,232)(144,233)(145,234)(146,235)(147,236)(148,237)(149,238)(150,239)(166,199)(167,200)(168,201)(169,202)(170,203)(171,204)(172,205)(173,206)(174,207)(175,208)(176,209)(177,210)(178,196)(179,197)(180,198) );
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)], [(1,103,218,207),(2,104,219,208),(3,105,220,209),(4,91,221,210),(5,92,222,196),(6,93,223,197),(7,94,224,198),(8,95,225,199),(9,96,211,200),(10,97,212,201),(11,98,213,202),(12,99,214,203),(13,100,215,204),(14,101,216,205),(15,102,217,206),(16,234,45,84),(17,235,31,85),(18,236,32,86),(19,237,33,87),(20,238,34,88),(21,239,35,89),(22,240,36,90),(23,226,37,76),(24,227,38,77),(25,228,39,78),(26,229,40,79),(27,230,41,80),(28,231,42,81),(29,232,43,82),(30,233,44,83),(46,113,125,170),(47,114,126,171),(48,115,127,172),(49,116,128,173),(50,117,129,174),(51,118,130,175),(52,119,131,176),(53,120,132,177),(54,106,133,178),(55,107,134,179),(56,108,135,180),(57,109,121,166),(58,110,122,167),(59,111,123,168),(60,112,124,169),(61,151,182,141),(62,152,183,142),(63,153,184,143),(64,154,185,144),(65,155,186,145),(66,156,187,146),(67,157,188,147),(68,158,189,148),(69,159,190,149),(70,160,191,150),(71,161,192,136),(72,162,193,137),(73,163,194,138),(74,164,195,139),(75,165,181,140)], [(1,174,234,65),(2,175,235,66),(3,176,236,67),(4,177,237,68),(5,178,238,69),(6,179,239,70),(7,180,240,71),(8,166,226,72),(9,167,227,73),(10,168,228,74),(11,169,229,75),(12,170,230,61),(13,171,231,62),(14,172,232,63),(15,173,233,64),(16,145,207,129),(17,146,208,130),(18,147,209,131),(19,148,210,132),(20,149,196,133),(21,150,197,134),(22,136,198,135),(23,137,199,121),(24,138,200,122),(25,139,201,123),(26,140,202,124),(27,141,203,125),(28,142,204,126),(29,143,205,127),(30,144,206,128),(31,156,104,51),(32,157,105,52),(33,158,91,53),(34,159,92,54),(35,160,93,55),(36,161,94,56),(37,162,95,57),(38,163,96,58),(39,164,97,59),(40,165,98,60),(41,151,99,46),(42,152,100,47),(43,153,101,48),(44,154,102,49),(45,155,103,50),(76,193,225,109),(77,194,211,110),(78,195,212,111),(79,181,213,112),(80,182,214,113),(81,183,215,114),(82,184,216,115),(83,185,217,116),(84,186,218,117),(85,187,219,118),(86,188,220,119),(87,189,221,120),(88,190,222,106),(89,191,223,107),(90,192,224,108)], [(1,129),(2,130),(3,131),(4,132),(5,133),(6,134),(7,135),(8,121),(9,122),(10,123),(11,124),(12,125),(13,126),(14,127),(15,128),(16,65),(17,66),(18,67),(19,68),(20,69),(21,70),(22,71),(23,72),(24,73),(25,74),(26,75),(27,61),(28,62),(29,63),(30,64),(31,187),(32,188),(33,189),(34,190),(35,191),(36,192),(37,193),(38,194),(39,195),(40,181),(41,182),(42,183),(43,184),(44,185),(45,186),(46,214),(47,215),(48,216),(49,217),(50,218),(51,219),(52,220),(53,221),(54,222),(55,223),(56,224),(57,225),(58,211),(59,212),(60,213),(76,162),(77,163),(78,164),(79,165),(80,151),(81,152),(82,153),(83,154),(84,155),(85,156),(86,157),(87,158),(88,159),(89,160),(90,161),(91,120),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112),(99,113),(100,114),(101,115),(102,116),(103,117),(104,118),(105,119),(136,240),(137,226),(138,227),(139,228),(140,229),(141,230),(142,231),(143,232),(144,233),(145,234),(146,235),(147,236),(148,237),(149,238),(150,239),(166,199),(167,200),(168,201),(169,202),(170,203),(171,204),(172,205),(173,206),(174,207),(175,208),(176,209),(177,210),(178,196),(179,197),(180,198)]])
210 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | ··· | 4F | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 6G | ··· | 6N | 10A | ··· | 10L | 10M | ··· | 10AB | 12A | ··· | 12L | 15A | ··· | 15H | 20A | ··· | 20X | 30A | ··· | 30X | 30Y | ··· | 30BD | 60A | ··· | 60AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 |
210 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C3 | C5 | C6 | C6 | C10 | C10 | C15 | C30 | C30 | D4 | C3×D4 | C5×D4 | D4×C15 |
| kernel | C15×C4⋊1D4 | C4×C60 | D4×C30 | C5×C4⋊1D4 | C3×C4⋊1D4 | C4×C20 | D4×C10 | C4×C12 | C6×D4 | C4⋊1D4 | C42 | C2×D4 | C60 | C20 | C12 | C4 |
| # reps | 1 | 1 | 6 | 2 | 4 | 2 | 12 | 4 | 24 | 8 | 8 | 48 | 6 | 12 | 24 | 48 |
Matrix representation of C15×C4⋊1D4 ►in GL4(𝔽61) generated by
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 47 | 0 |
| 0 | 0 | 0 | 47 |
| 0 | 60 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 60 |
| 0 | 0 | 2 | 60 |
| 0 | 1 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 59 | 1 |
G:=sub<GL(4,GF(61))| [15,0,0,0,0,15,0,0,0,0,47,0,0,0,0,47],[0,1,0,0,60,0,0,0,0,0,1,2,0,0,60,60],[0,60,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,1,0,0,0,0,60,59,0,0,0,1] >;
C15×C4⋊1D4 in GAP, Magma, Sage, TeX
C_{15}\times C_4\rtimes_1D_4 % in TeX
G:=Group("C15xC4:1D4"); // GroupNames label
G:=SmallGroup(480,932);
// by ID
G=gap.SmallGroup(480,932);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-5,-2,-2,1709,848,5126,1276]);
// 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,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations