direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×C15⋊D4, C60⋊13D4, D6⋊5(C4×D5), C15⋊15(C4×D4), D10⋊7(C4×S3), C20⋊11(C3⋊D4), C12⋊11(C5⋊D4), D6⋊Dic5⋊39C2, (C2×C20).339D6, C30.140(C2×D4), Dic15⋊17(C2×C4), (C4×Dic15)⋊34C2, C6.34(C4○D20), C30.75(C4○D4), (C2×C12).343D10, C30.58(C22×C4), Dic15⋊5C4⋊43C2, (C22×D5).85D6, C10.37(C4○D12), D10⋊Dic3⋊39C2, (C2×C60).241C22, (C2×C30).129C23, (C2×Dic5).178D6, (C22×S3).69D10, (C2×Dic3).151D10, C2.4(D6.D10), (C6×Dic5).203C22, (C2×Dic15).209C22, (C10×Dic3).183C22, (S3×C2×C4)⋊8D5, (C2×C4×D5)⋊8S3, C3⋊5(C4×C5⋊D4), C5⋊6(C4×C3⋊D4), (S3×C2×C20)⋊7C2, (D5×C2×C12)⋊7C2, C2.28(C4×S3×D5), C6.26(C2×C4×D5), C10.59(S3×C2×C4), (C6×D5)⋊17(C2×C4), C2.1(C2×C15⋊D4), C6.85(C2×C5⋊D4), (S3×C10)⋊18(C2×C4), C22.61(C2×S3×D5), (C2×C4).244(S3×D5), C10.86(C2×C3⋊D4), (C2×C15⋊D4).11C2, (S3×C2×C10).84C22, (D5×C2×C6).101C22, (C2×C6).141(C22×D5), (C2×C10).141(C22×S3), SmallGroup(480,515)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×C15⋊D4
G = < a,b,c,d | a4=b15=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=b4, dcd=c-1 >
Subgroups: 796 in 188 conjugacy classes, 66 normal (44 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C5×S3, C3×D5, C30, C4×D4, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, S3×C2×C4, C2×C3⋊D4, C22×C12, C5×Dic3, C3×Dic5, Dic15, Dic15, C60, C6×D5, C6×D5, S3×C10, S3×C10, C2×C30, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C4×D5, C2×C5⋊D4, C22×C20, C4×C3⋊D4, C15⋊D4, D5×C12, C6×Dic5, S3×C20, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, S3×C2×C10, C4×C5⋊D4, D10⋊Dic3, D6⋊Dic5, Dic15⋊5C4, C4×Dic15, C2×C15⋊D4, D5×C2×C12, S3×C2×C20, C4×C15⋊D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, D6, C22×C4, C2×D4, C4○D4, D10, C4×S3, C3⋊D4, C22×S3, C4×D4, C4×D5, C5⋊D4, C22×D5, S3×C2×C4, C4○D12, C2×C3⋊D4, S3×D5, C2×C4×D5, C4○D20, C2×C5⋊D4, C4×C3⋊D4, C15⋊D4, C2×S3×D5, C4×C5⋊D4, D6.D10, C4×S3×D5, C2×C15⋊D4, C4×C15⋊D4
►On 240 points
Generators in S240
(1 92 32 64)(2 93 33 65)(3 94 34 66)(4 95 35 67)(5 96 36 68)(6 97 37 69)(7 98 38 70)(8 99 39 71)(9 100 40 72)(10 101 41 73)(11 102 42 74)(12 103 43 75)(13 104 44 61)(14 105 45 62)(15 91 31 63)(16 111 52 86)(17 112 53 87)(18 113 54 88)(19 114 55 89)(20 115 56 90)(21 116 57 76)(22 117 58 77)(23 118 59 78)(24 119 60 79)(25 120 46 80)(26 106 47 81)(27 107 48 82)(28 108 49 83)(29 109 50 84)(30 110 51 85)(121 211 159 183)(122 212 160 184)(123 213 161 185)(124 214 162 186)(125 215 163 187)(126 216 164 188)(127 217 165 189)(128 218 151 190)(129 219 152 191)(130 220 153 192)(131 221 154 193)(132 222 155 194)(133 223 156 195)(134 224 157 181)(135 225 158 182)(136 235 177 196)(137 236 178 197)(138 237 179 198)(139 238 180 199)(140 239 166 200)(141 240 167 201)(142 226 168 202)(143 227 169 203)(144 228 170 204)(145 229 171 205)(146 230 172 206)(147 231 173 207)(148 232 174 208)(149 233 175 209)(150 234 176 210)
(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)
(1 151 24 175)(2 165 25 174)(3 164 26 173)(4 163 27 172)(5 162 28 171)(6 161 29 170)(7 160 30 169)(8 159 16 168)(9 158 17 167)(10 157 18 166)(11 156 19 180)(12 155 20 179)(13 154 21 178)(14 153 22 177)(15 152 23 176)(31 129 59 150)(32 128 60 149)(33 127 46 148)(34 126 47 147)(35 125 48 146)(36 124 49 145)(37 123 50 144)(38 122 51 143)(39 121 52 142)(40 135 53 141)(41 134 54 140)(42 133 55 139)(43 132 56 138)(44 131 57 137)(45 130 58 136)(61 221 76 236)(62 220 77 235)(63 219 78 234)(64 218 79 233)(65 217 80 232)(66 216 81 231)(67 215 82 230)(68 214 83 229)(69 213 84 228)(70 212 85 227)(71 211 86 226)(72 225 87 240)(73 224 88 239)(74 223 89 238)(75 222 90 237)(91 191 118 210)(92 190 119 209)(93 189 120 208)(94 188 106 207)(95 187 107 206)(96 186 108 205)(97 185 109 204)(98 184 110 203)(99 183 111 202)(100 182 112 201)(101 181 113 200)(102 195 114 199)(103 194 115 198)(104 193 116 197)(105 192 117 196)
(1 32)(2 36)(3 40)(4 44)(5 33)(6 37)(7 41)(8 45)(9 34)(10 38)(11 42)(12 31)(13 35)(14 39)(15 43)(16 58)(17 47)(18 51)(19 55)(20 59)(21 48)(22 52)(23 56)(24 60)(25 49)(26 53)(27 57)(28 46)(29 50)(30 54)(61 95)(62 99)(63 103)(64 92)(65 96)(66 100)(67 104)(68 93)(69 97)(70 101)(71 105)(72 94)(73 98)(74 102)(75 91)(76 107)(77 111)(78 115)(79 119)(80 108)(81 112)(82 116)(83 120)(84 109)(85 113)(86 117)(87 106)(88 110)(89 114)(90 118)(121 177)(122 166)(123 170)(124 174)(125 178)(126 167)(127 171)(128 175)(129 179)(130 168)(131 172)(132 176)(133 180)(134 169)(135 173)(136 159)(137 163)(138 152)(139 156)(140 160)(141 164)(142 153)(143 157)(144 161)(145 165)(146 154)(147 158)(148 162)(149 151)(150 155)(181 227)(182 231)(183 235)(184 239)(185 228)(186 232)(187 236)(188 240)(189 229)(190 233)(191 237)(192 226)(193 230)(194 234)(195 238)(196 211)(197 215)(198 219)(199 223)(200 212)(201 216)(202 220)(203 224)(204 213)(205 217)(206 221)(207 225)(208 214)(209 218)(210 222)
G:=sub<Sym(240)| (1,92,32,64)(2,93,33,65)(3,94,34,66)(4,95,35,67)(5,96,36,68)(6,97,37,69)(7,98,38,70)(8,99,39,71)(9,100,40,72)(10,101,41,73)(11,102,42,74)(12,103,43,75)(13,104,44,61)(14,105,45,62)(15,91,31,63)(16,111,52,86)(17,112,53,87)(18,113,54,88)(19,114,55,89)(20,115,56,90)(21,116,57,76)(22,117,58,77)(23,118,59,78)(24,119,60,79)(25,120,46,80)(26,106,47,81)(27,107,48,82)(28,108,49,83)(29,109,50,84)(30,110,51,85)(121,211,159,183)(122,212,160,184)(123,213,161,185)(124,214,162,186)(125,215,163,187)(126,216,164,188)(127,217,165,189)(128,218,151,190)(129,219,152,191)(130,220,153,192)(131,221,154,193)(132,222,155,194)(133,223,156,195)(134,224,157,181)(135,225,158,182)(136,235,177,196)(137,236,178,197)(138,237,179,198)(139,238,180,199)(140,239,166,200)(141,240,167,201)(142,226,168,202)(143,227,169,203)(144,228,170,204)(145,229,171,205)(146,230,172,206)(147,231,173,207)(148,232,174,208)(149,233,175,209)(150,234,176,210), (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), (1,151,24,175)(2,165,25,174)(3,164,26,173)(4,163,27,172)(5,162,28,171)(6,161,29,170)(7,160,30,169)(8,159,16,168)(9,158,17,167)(10,157,18,166)(11,156,19,180)(12,155,20,179)(13,154,21,178)(14,153,22,177)(15,152,23,176)(31,129,59,150)(32,128,60,149)(33,127,46,148)(34,126,47,147)(35,125,48,146)(36,124,49,145)(37,123,50,144)(38,122,51,143)(39,121,52,142)(40,135,53,141)(41,134,54,140)(42,133,55,139)(43,132,56,138)(44,131,57,137)(45,130,58,136)(61,221,76,236)(62,220,77,235)(63,219,78,234)(64,218,79,233)(65,217,80,232)(66,216,81,231)(67,215,82,230)(68,214,83,229)(69,213,84,228)(70,212,85,227)(71,211,86,226)(72,225,87,240)(73,224,88,239)(74,223,89,238)(75,222,90,237)(91,191,118,210)(92,190,119,209)(93,189,120,208)(94,188,106,207)(95,187,107,206)(96,186,108,205)(97,185,109,204)(98,184,110,203)(99,183,111,202)(100,182,112,201)(101,181,113,200)(102,195,114,199)(103,194,115,198)(104,193,116,197)(105,192,117,196), (1,32)(2,36)(3,40)(4,44)(5,33)(6,37)(7,41)(8,45)(9,34)(10,38)(11,42)(12,31)(13,35)(14,39)(15,43)(16,58)(17,47)(18,51)(19,55)(20,59)(21,48)(22,52)(23,56)(24,60)(25,49)(26,53)(27,57)(28,46)(29,50)(30,54)(61,95)(62,99)(63,103)(64,92)(65,96)(66,100)(67,104)(68,93)(69,97)(70,101)(71,105)(72,94)(73,98)(74,102)(75,91)(76,107)(77,111)(78,115)(79,119)(80,108)(81,112)(82,116)(83,120)(84,109)(85,113)(86,117)(87,106)(88,110)(89,114)(90,118)(121,177)(122,166)(123,170)(124,174)(125,178)(126,167)(127,171)(128,175)(129,179)(130,168)(131,172)(132,176)(133,180)(134,169)(135,173)(136,159)(137,163)(138,152)(139,156)(140,160)(141,164)(142,153)(143,157)(144,161)(145,165)(146,154)(147,158)(148,162)(149,151)(150,155)(181,227)(182,231)(183,235)(184,239)(185,228)(186,232)(187,236)(188,240)(189,229)(190,233)(191,237)(192,226)(193,230)(194,234)(195,238)(196,211)(197,215)(198,219)(199,223)(200,212)(201,216)(202,220)(203,224)(204,213)(205,217)(206,221)(207,225)(208,214)(209,218)(210,222)>;
G:=Group( (1,92,32,64)(2,93,33,65)(3,94,34,66)(4,95,35,67)(5,96,36,68)(6,97,37,69)(7,98,38,70)(8,99,39,71)(9,100,40,72)(10,101,41,73)(11,102,42,74)(12,103,43,75)(13,104,44,61)(14,105,45,62)(15,91,31,63)(16,111,52,86)(17,112,53,87)(18,113,54,88)(19,114,55,89)(20,115,56,90)(21,116,57,76)(22,117,58,77)(23,118,59,78)(24,119,60,79)(25,120,46,80)(26,106,47,81)(27,107,48,82)(28,108,49,83)(29,109,50,84)(30,110,51,85)(121,211,159,183)(122,212,160,184)(123,213,161,185)(124,214,162,186)(125,215,163,187)(126,216,164,188)(127,217,165,189)(128,218,151,190)(129,219,152,191)(130,220,153,192)(131,221,154,193)(132,222,155,194)(133,223,156,195)(134,224,157,181)(135,225,158,182)(136,235,177,196)(137,236,178,197)(138,237,179,198)(139,238,180,199)(140,239,166,200)(141,240,167,201)(142,226,168,202)(143,227,169,203)(144,228,170,204)(145,229,171,205)(146,230,172,206)(147,231,173,207)(148,232,174,208)(149,233,175,209)(150,234,176,210), (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), (1,151,24,175)(2,165,25,174)(3,164,26,173)(4,163,27,172)(5,162,28,171)(6,161,29,170)(7,160,30,169)(8,159,16,168)(9,158,17,167)(10,157,18,166)(11,156,19,180)(12,155,20,179)(13,154,21,178)(14,153,22,177)(15,152,23,176)(31,129,59,150)(32,128,60,149)(33,127,46,148)(34,126,47,147)(35,125,48,146)(36,124,49,145)(37,123,50,144)(38,122,51,143)(39,121,52,142)(40,135,53,141)(41,134,54,140)(42,133,55,139)(43,132,56,138)(44,131,57,137)(45,130,58,136)(61,221,76,236)(62,220,77,235)(63,219,78,234)(64,218,79,233)(65,217,80,232)(66,216,81,231)(67,215,82,230)(68,214,83,229)(69,213,84,228)(70,212,85,227)(71,211,86,226)(72,225,87,240)(73,224,88,239)(74,223,89,238)(75,222,90,237)(91,191,118,210)(92,190,119,209)(93,189,120,208)(94,188,106,207)(95,187,107,206)(96,186,108,205)(97,185,109,204)(98,184,110,203)(99,183,111,202)(100,182,112,201)(101,181,113,200)(102,195,114,199)(103,194,115,198)(104,193,116,197)(105,192,117,196), (1,32)(2,36)(3,40)(4,44)(5,33)(6,37)(7,41)(8,45)(9,34)(10,38)(11,42)(12,31)(13,35)(14,39)(15,43)(16,58)(17,47)(18,51)(19,55)(20,59)(21,48)(22,52)(23,56)(24,60)(25,49)(26,53)(27,57)(28,46)(29,50)(30,54)(61,95)(62,99)(63,103)(64,92)(65,96)(66,100)(67,104)(68,93)(69,97)(70,101)(71,105)(72,94)(73,98)(74,102)(75,91)(76,107)(77,111)(78,115)(79,119)(80,108)(81,112)(82,116)(83,120)(84,109)(85,113)(86,117)(87,106)(88,110)(89,114)(90,118)(121,177)(122,166)(123,170)(124,174)(125,178)(126,167)(127,171)(128,175)(129,179)(130,168)(131,172)(132,176)(133,180)(134,169)(135,173)(136,159)(137,163)(138,152)(139,156)(140,160)(141,164)(142,153)(143,157)(144,161)(145,165)(146,154)(147,158)(148,162)(149,151)(150,155)(181,227)(182,231)(183,235)(184,239)(185,228)(186,232)(187,236)(188,240)(189,229)(190,233)(191,237)(192,226)(193,230)(194,234)(195,238)(196,211)(197,215)(198,219)(199,223)(200,212)(201,216)(202,220)(203,224)(204,213)(205,217)(206,221)(207,225)(208,214)(209,218)(210,222) );
G=PermutationGroup([[(1,92,32,64),(2,93,33,65),(3,94,34,66),(4,95,35,67),(5,96,36,68),(6,97,37,69),(7,98,38,70),(8,99,39,71),(9,100,40,72),(10,101,41,73),(11,102,42,74),(12,103,43,75),(13,104,44,61),(14,105,45,62),(15,91,31,63),(16,111,52,86),(17,112,53,87),(18,113,54,88),(19,114,55,89),(20,115,56,90),(21,116,57,76),(22,117,58,77),(23,118,59,78),(24,119,60,79),(25,120,46,80),(26,106,47,81),(27,107,48,82),(28,108,49,83),(29,109,50,84),(30,110,51,85),(121,211,159,183),(122,212,160,184),(123,213,161,185),(124,214,162,186),(125,215,163,187),(126,216,164,188),(127,217,165,189),(128,218,151,190),(129,219,152,191),(130,220,153,192),(131,221,154,193),(132,222,155,194),(133,223,156,195),(134,224,157,181),(135,225,158,182),(136,235,177,196),(137,236,178,197),(138,237,179,198),(139,238,180,199),(140,239,166,200),(141,240,167,201),(142,226,168,202),(143,227,169,203),(144,228,170,204),(145,229,171,205),(146,230,172,206),(147,231,173,207),(148,232,174,208),(149,233,175,209),(150,234,176,210)], [(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)], [(1,151,24,175),(2,165,25,174),(3,164,26,173),(4,163,27,172),(5,162,28,171),(6,161,29,170),(7,160,30,169),(8,159,16,168),(9,158,17,167),(10,157,18,166),(11,156,19,180),(12,155,20,179),(13,154,21,178),(14,153,22,177),(15,152,23,176),(31,129,59,150),(32,128,60,149),(33,127,46,148),(34,126,47,147),(35,125,48,146),(36,124,49,145),(37,123,50,144),(38,122,51,143),(39,121,52,142),(40,135,53,141),(41,134,54,140),(42,133,55,139),(43,132,56,138),(44,131,57,137),(45,130,58,136),(61,221,76,236),(62,220,77,235),(63,219,78,234),(64,218,79,233),(65,217,80,232),(66,216,81,231),(67,215,82,230),(68,214,83,229),(69,213,84,228),(70,212,85,227),(71,211,86,226),(72,225,87,240),(73,224,88,239),(74,223,89,238),(75,222,90,237),(91,191,118,210),(92,190,119,209),(93,189,120,208),(94,188,106,207),(95,187,107,206),(96,186,108,205),(97,185,109,204),(98,184,110,203),(99,183,111,202),(100,182,112,201),(101,181,113,200),(102,195,114,199),(103,194,115,198),(104,193,116,197),(105,192,117,196)], [(1,32),(2,36),(3,40),(4,44),(5,33),(6,37),(7,41),(8,45),(9,34),(10,38),(11,42),(12,31),(13,35),(14,39),(15,43),(16,58),(17,47),(18,51),(19,55),(20,59),(21,48),(22,52),(23,56),(24,60),(25,49),(26,53),(27,57),(28,46),(29,50),(30,54),(61,95),(62,99),(63,103),(64,92),(65,96),(66,100),(67,104),(68,93),(69,97),(70,101),(71,105),(72,94),(73,98),(74,102),(75,91),(76,107),(77,111),(78,115),(79,119),(80,108),(81,112),(82,116),(83,120),(84,109),(85,113),(86,117),(87,106),(88,110),(89,114),(90,118),(121,177),(122,166),(123,170),(124,174),(125,178),(126,167),(127,171),(128,175),(129,179),(130,168),(131,172),(132,176),(133,180),(134,169),(135,173),(136,159),(137,163),(138,152),(139,156),(140,160),(141,164),(142,153),(143,157),(144,161),(145,165),(146,154),(147,158),(148,162),(149,151),(150,155),(181,227),(182,231),(183,235),(184,239),(185,228),(186,232),(187,236),(188,240),(189,229),(190,233),(191,237),(192,226),(193,230),(194,234),(195,238),(196,211),(197,215),(198,219),(199,223),(200,212),(201,216),(202,220),(203,224),(204,213),(205,217),(206,221),(207,225),(208,214),(209,218),(210,222)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20P | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 10 | 10 | 2 | 1 | 1 | 1 | 1 | 6 | 6 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D10 | C3⋊D4 | C4×S3 | C5⋊D4 | C4×D5 | C4○D12 | C4○D20 | S3×D5 | C15⋊D4 | C2×S3×D5 | D6.D10 | C4×S3×D5 |
| kernel | C4×C15⋊D4 | D10⋊Dic3 | D6⋊Dic5 | Dic15⋊5C4 | C4×Dic15 | C2×C15⋊D4 | D5×C2×C12 | S3×C2×C20 | C15⋊D4 | C2×C4×D5 | C60 | S3×C2×C4 | C2×Dic5 | C2×C20 | C22×D5 | C30 | C2×Dic3 | C2×C12 | C22×S3 | C20 | D10 | C12 | D6 | C10 | C6 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 8 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of C4×C15⋊D4 ►in GL4(𝔽61) generated by
| 50 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 13 | 0 | 0 | 0 |
| 0 | 47 | 0 | 0 |
| 0 | 0 | 0 | 44 |
| 0 | 0 | 18 | 43 |
| 0 | 1 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 53 | 47 |
| 0 | 0 | 9 | 8 |
| 60 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 43 | 1 |
| 0 | 0 | 43 | 18 |
G:=sub<GL(4,GF(61))| [50,0,0,0,0,50,0,0,0,0,1,0,0,0,0,1],[13,0,0,0,0,47,0,0,0,0,0,18,0,0,44,43],[0,60,0,0,1,0,0,0,0,0,53,9,0,0,47,8],[60,0,0,0,0,1,0,0,0,0,43,43,0,0,1,18] >;
C4×C15⋊D4 in GAP, Magma, Sage, TeX
C_4\times C_{15}\rtimes D_4 % in TeX
G:=Group("C4xC15:D4"); // GroupNames label
G:=SmallGroup(480,515);
// by ID
G=gap.SmallGroup(480,515);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,58,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^15=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d=b^4,d*c*d=c^-1>;
// generators/relations