direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C15×D4⋊C4, D4⋊1C60, C30.58D8, C60.245D4, C30.40SD16, C4⋊C4⋊1C30, (C2×C8)⋊2C30, (C2×C40)⋊4C6, (C2×C120)⋊8C2, (C2×C24)⋊4C10, (C3×D4)⋊4C20, (C5×D4)⋊7C12, C4.1(C2×C60), C2.1(C15×D8), C6.13(C5×D8), (D4×C15)⋊16C4, (D4×C10).9C6, (C2×D4).3C30, (C6×D4).9C10, C4.11(D4×C15), C20.60(C3×D4), C10.13(C3×D8), C12.60(C5×D4), C6.9(C5×SD16), C60.225(C2×C4), C20.50(C2×C12), C12.29(C2×C20), (D4×C30).21C2, (C2×C30).189D4, C2.1(C15×SD16), C10.9(C3×SD16), C22.8(D4×C15), (C2×C60).569C22, C30.128(C22⋊C4), (C5×C4⋊C4)⋊10C6, (C3×C4⋊C4)⋊10C10, (C15×C4⋊C4)⋊28C2, (C2×C6).46(C5×D4), (C2×C4).16(C2×C30), (C2×C10).46(C3×D4), C6.24(C5×C22⋊C4), C2.6(C15×C22⋊C4), (C2×C20).117(C2×C6), C10.35(C3×C22⋊C4), (C2×C12).120(C2×C10), SmallGroup(480,205)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×D4⋊C4
G = < a,b,c,d | a15=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >
Subgroups: 184 in 100 conjugacy classes, 56 normal (48 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, D4, C23, C10, C10, C12, C12, C2×C6, C2×C6, C15, C4⋊C4, C2×C8, C2×D4, C20, C20, C2×C10, C2×C10, C24, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, C30, C30, D4⋊C4, C40, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, C3×C4⋊C4, C2×C24, C6×D4, C60, C60, C2×C30, C2×C30, C5×C4⋊C4, C2×C40, D4×C10, C3×D4⋊C4, C120, C2×C60, C2×C60, D4×C15, D4×C15, C22×C30, C5×D4⋊C4, C15×C4⋊C4, C2×C120, D4×C30, C15×D4⋊C4
Quotients: C1, C2, C3, C4, C22, C5, C6, C2×C4, D4, C10, C12, C2×C6, C15, C22⋊C4, D8, SD16, C20, C2×C10, C2×C12, C3×D4, C30, D4⋊C4, C2×C20, C5×D4, C3×C22⋊C4, C3×D8, C3×SD16, C60, C2×C30, C5×C22⋊C4, C5×D8, C5×SD16, C3×D4⋊C4, C2×C60, D4×C15, C5×D4⋊C4, C15×C22⋊C4, C15×D8, C15×SD16, C15×D4⋊C4
►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)
(1 238 126 45)(2 239 127 31)(3 240 128 32)(4 226 129 33)(5 227 130 34)(6 228 131 35)(7 229 132 36)(8 230 133 37)(9 231 134 38)(10 232 135 39)(11 233 121 40)(12 234 122 41)(13 235 123 42)(14 236 124 43)(15 237 125 44)(16 116 152 138)(17 117 153 139)(18 118 154 140)(19 119 155 141)(20 120 156 142)(21 106 157 143)(22 107 158 144)(23 108 159 145)(24 109 160 146)(25 110 161 147)(26 111 162 148)(27 112 163 149)(28 113 164 150)(29 114 165 136)(30 115 151 137)(46 216 81 172)(47 217 82 173)(48 218 83 174)(49 219 84 175)(50 220 85 176)(51 221 86 177)(52 222 87 178)(53 223 88 179)(54 224 89 180)(55 225 90 166)(56 211 76 167)(57 212 77 168)(58 213 78 169)(59 214 79 170)(60 215 80 171)(61 205 189 94)(62 206 190 95)(63 207 191 96)(64 208 192 97)(65 209 193 98)(66 210 194 99)(67 196 195 100)(68 197 181 101)(69 198 182 102)(70 199 183 103)(71 200 184 104)(72 201 185 105)(73 202 186 91)(74 203 187 92)(75 204 188 93)
(1 174)(2 175)(3 176)(4 177)(5 178)(6 179)(7 180)(8 166)(9 167)(10 168)(11 169)(12 170)(13 171)(14 172)(15 173)(16 182)(17 183)(18 184)(19 185)(20 186)(21 187)(22 188)(23 189)(24 190)(25 191)(26 192)(27 193)(28 194)(29 195)(30 181)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 46)(44 47)(45 48)(61 159)(62 160)(63 161)(64 162)(65 163)(66 164)(67 165)(68 151)(69 152)(70 153)(71 154)(72 155)(73 156)(74 157)(75 158)(76 231)(77 232)(78 233)(79 234)(80 235)(81 236)(82 237)(83 238)(84 239)(85 240)(86 226)(87 227)(88 228)(89 229)(90 230)(91 142)(92 143)(93 144)(94 145)(95 146)(96 147)(97 148)(98 149)(99 150)(100 136)(101 137)(102 138)(103 139)(104 140)(105 141)(106 203)(107 204)(108 205)(109 206)(110 207)(111 208)(112 209)(113 210)(114 196)(115 197)(116 198)(117 199)(118 200)(119 201)(120 202)(121 213)(122 214)(123 215)(124 216)(125 217)(126 218)(127 219)(128 220)(129 221)(130 222)(131 223)(132 224)(133 225)(134 211)(135 212)
(1 104 48 118)(2 105 49 119)(3 91 50 120)(4 92 51 106)(5 93 52 107)(6 94 53 108)(7 95 54 109)(8 96 55 110)(9 97 56 111)(10 98 57 112)(11 99 58 113)(12 100 59 114)(13 101 60 115)(14 102 46 116)(15 103 47 117)(16 236 182 216)(17 237 183 217)(18 238 184 218)(19 239 185 219)(20 240 186 220)(21 226 187 221)(22 227 188 222)(23 228 189 223)(24 229 190 224)(25 230 191 225)(26 231 192 211)(27 232 193 212)(28 233 194 213)(29 234 195 214)(30 235 181 215)(31 72 175 155)(32 73 176 156)(33 74 177 157)(34 75 178 158)(35 61 179 159)(36 62 180 160)(37 63 166 161)(38 64 167 162)(39 65 168 163)(40 66 169 164)(41 67 170 165)(42 68 171 151)(43 69 172 152)(44 70 173 153)(45 71 174 154)(76 148 134 208)(77 149 135 209)(78 150 121 210)(79 136 122 196)(80 137 123 197)(81 138 124 198)(82 139 125 199)(83 140 126 200)(84 141 127 201)(85 142 128 202)(86 143 129 203)(87 144 130 204)(88 145 131 205)(89 146 132 206)(90 147 133 207)
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), (1,238,126,45)(2,239,127,31)(3,240,128,32)(4,226,129,33)(5,227,130,34)(6,228,131,35)(7,229,132,36)(8,230,133,37)(9,231,134,38)(10,232,135,39)(11,233,121,40)(12,234,122,41)(13,235,123,42)(14,236,124,43)(15,237,125,44)(16,116,152,138)(17,117,153,139)(18,118,154,140)(19,119,155,141)(20,120,156,142)(21,106,157,143)(22,107,158,144)(23,108,159,145)(24,109,160,146)(25,110,161,147)(26,111,162,148)(27,112,163,149)(28,113,164,150)(29,114,165,136)(30,115,151,137)(46,216,81,172)(47,217,82,173)(48,218,83,174)(49,219,84,175)(50,220,85,176)(51,221,86,177)(52,222,87,178)(53,223,88,179)(54,224,89,180)(55,225,90,166)(56,211,76,167)(57,212,77,168)(58,213,78,169)(59,214,79,170)(60,215,80,171)(61,205,189,94)(62,206,190,95)(63,207,191,96)(64,208,192,97)(65,209,193,98)(66,210,194,99)(67,196,195,100)(68,197,181,101)(69,198,182,102)(70,199,183,103)(71,200,184,104)(72,201,185,105)(73,202,186,91)(74,203,187,92)(75,204,188,93), (1,174)(2,175)(3,176)(4,177)(5,178)(6,179)(7,180)(8,166)(9,167)(10,168)(11,169)(12,170)(13,171)(14,172)(15,173)(16,182)(17,183)(18,184)(19,185)(20,186)(21,187)(22,188)(23,189)(24,190)(25,191)(26,192)(27,193)(28,194)(29,195)(30,181)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,46)(44,47)(45,48)(61,159)(62,160)(63,161)(64,162)(65,163)(66,164)(67,165)(68,151)(69,152)(70,153)(71,154)(72,155)(73,156)(74,157)(75,158)(76,231)(77,232)(78,233)(79,234)(80,235)(81,236)(82,237)(83,238)(84,239)(85,240)(86,226)(87,227)(88,228)(89,229)(90,230)(91,142)(92,143)(93,144)(94,145)(95,146)(96,147)(97,148)(98,149)(99,150)(100,136)(101,137)(102,138)(103,139)(104,140)(105,141)(106,203)(107,204)(108,205)(109,206)(110,207)(111,208)(112,209)(113,210)(114,196)(115,197)(116,198)(117,199)(118,200)(119,201)(120,202)(121,213)(122,214)(123,215)(124,216)(125,217)(126,218)(127,219)(128,220)(129,221)(130,222)(131,223)(132,224)(133,225)(134,211)(135,212), (1,104,48,118)(2,105,49,119)(3,91,50,120)(4,92,51,106)(5,93,52,107)(6,94,53,108)(7,95,54,109)(8,96,55,110)(9,97,56,111)(10,98,57,112)(11,99,58,113)(12,100,59,114)(13,101,60,115)(14,102,46,116)(15,103,47,117)(16,236,182,216)(17,237,183,217)(18,238,184,218)(19,239,185,219)(20,240,186,220)(21,226,187,221)(22,227,188,222)(23,228,189,223)(24,229,190,224)(25,230,191,225)(26,231,192,211)(27,232,193,212)(28,233,194,213)(29,234,195,214)(30,235,181,215)(31,72,175,155)(32,73,176,156)(33,74,177,157)(34,75,178,158)(35,61,179,159)(36,62,180,160)(37,63,166,161)(38,64,167,162)(39,65,168,163)(40,66,169,164)(41,67,170,165)(42,68,171,151)(43,69,172,152)(44,70,173,153)(45,71,174,154)(76,148,134,208)(77,149,135,209)(78,150,121,210)(79,136,122,196)(80,137,123,197)(81,138,124,198)(82,139,125,199)(83,140,126,200)(84,141,127,201)(85,142,128,202)(86,143,129,203)(87,144,130,204)(88,145,131,205)(89,146,132,206)(90,147,133,207)>;
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), (1,238,126,45)(2,239,127,31)(3,240,128,32)(4,226,129,33)(5,227,130,34)(6,228,131,35)(7,229,132,36)(8,230,133,37)(9,231,134,38)(10,232,135,39)(11,233,121,40)(12,234,122,41)(13,235,123,42)(14,236,124,43)(15,237,125,44)(16,116,152,138)(17,117,153,139)(18,118,154,140)(19,119,155,141)(20,120,156,142)(21,106,157,143)(22,107,158,144)(23,108,159,145)(24,109,160,146)(25,110,161,147)(26,111,162,148)(27,112,163,149)(28,113,164,150)(29,114,165,136)(30,115,151,137)(46,216,81,172)(47,217,82,173)(48,218,83,174)(49,219,84,175)(50,220,85,176)(51,221,86,177)(52,222,87,178)(53,223,88,179)(54,224,89,180)(55,225,90,166)(56,211,76,167)(57,212,77,168)(58,213,78,169)(59,214,79,170)(60,215,80,171)(61,205,189,94)(62,206,190,95)(63,207,191,96)(64,208,192,97)(65,209,193,98)(66,210,194,99)(67,196,195,100)(68,197,181,101)(69,198,182,102)(70,199,183,103)(71,200,184,104)(72,201,185,105)(73,202,186,91)(74,203,187,92)(75,204,188,93), (1,174)(2,175)(3,176)(4,177)(5,178)(6,179)(7,180)(8,166)(9,167)(10,168)(11,169)(12,170)(13,171)(14,172)(15,173)(16,182)(17,183)(18,184)(19,185)(20,186)(21,187)(22,188)(23,189)(24,190)(25,191)(26,192)(27,193)(28,194)(29,195)(30,181)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,46)(44,47)(45,48)(61,159)(62,160)(63,161)(64,162)(65,163)(66,164)(67,165)(68,151)(69,152)(70,153)(71,154)(72,155)(73,156)(74,157)(75,158)(76,231)(77,232)(78,233)(79,234)(80,235)(81,236)(82,237)(83,238)(84,239)(85,240)(86,226)(87,227)(88,228)(89,229)(90,230)(91,142)(92,143)(93,144)(94,145)(95,146)(96,147)(97,148)(98,149)(99,150)(100,136)(101,137)(102,138)(103,139)(104,140)(105,141)(106,203)(107,204)(108,205)(109,206)(110,207)(111,208)(112,209)(113,210)(114,196)(115,197)(116,198)(117,199)(118,200)(119,201)(120,202)(121,213)(122,214)(123,215)(124,216)(125,217)(126,218)(127,219)(128,220)(129,221)(130,222)(131,223)(132,224)(133,225)(134,211)(135,212), (1,104,48,118)(2,105,49,119)(3,91,50,120)(4,92,51,106)(5,93,52,107)(6,94,53,108)(7,95,54,109)(8,96,55,110)(9,97,56,111)(10,98,57,112)(11,99,58,113)(12,100,59,114)(13,101,60,115)(14,102,46,116)(15,103,47,117)(16,236,182,216)(17,237,183,217)(18,238,184,218)(19,239,185,219)(20,240,186,220)(21,226,187,221)(22,227,188,222)(23,228,189,223)(24,229,190,224)(25,230,191,225)(26,231,192,211)(27,232,193,212)(28,233,194,213)(29,234,195,214)(30,235,181,215)(31,72,175,155)(32,73,176,156)(33,74,177,157)(34,75,178,158)(35,61,179,159)(36,62,180,160)(37,63,166,161)(38,64,167,162)(39,65,168,163)(40,66,169,164)(41,67,170,165)(42,68,171,151)(43,69,172,152)(44,70,173,153)(45,71,174,154)(76,148,134,208)(77,149,135,209)(78,150,121,210)(79,136,122,196)(80,137,123,197)(81,138,124,198)(82,139,125,199)(83,140,126,200)(84,141,127,201)(85,142,128,202)(86,143,129,203)(87,144,130,204)(88,145,131,205)(89,146,132,206)(90,147,133,207) );
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)], [(1,238,126,45),(2,239,127,31),(3,240,128,32),(4,226,129,33),(5,227,130,34),(6,228,131,35),(7,229,132,36),(8,230,133,37),(9,231,134,38),(10,232,135,39),(11,233,121,40),(12,234,122,41),(13,235,123,42),(14,236,124,43),(15,237,125,44),(16,116,152,138),(17,117,153,139),(18,118,154,140),(19,119,155,141),(20,120,156,142),(21,106,157,143),(22,107,158,144),(23,108,159,145),(24,109,160,146),(25,110,161,147),(26,111,162,148),(27,112,163,149),(28,113,164,150),(29,114,165,136),(30,115,151,137),(46,216,81,172),(47,217,82,173),(48,218,83,174),(49,219,84,175),(50,220,85,176),(51,221,86,177),(52,222,87,178),(53,223,88,179),(54,224,89,180),(55,225,90,166),(56,211,76,167),(57,212,77,168),(58,213,78,169),(59,214,79,170),(60,215,80,171),(61,205,189,94),(62,206,190,95),(63,207,191,96),(64,208,192,97),(65,209,193,98),(66,210,194,99),(67,196,195,100),(68,197,181,101),(69,198,182,102),(70,199,183,103),(71,200,184,104),(72,201,185,105),(73,202,186,91),(74,203,187,92),(75,204,188,93)], [(1,174),(2,175),(3,176),(4,177),(5,178),(6,179),(7,180),(8,166),(9,167),(10,168),(11,169),(12,170),(13,171),(14,172),(15,173),(16,182),(17,183),(18,184),(19,185),(20,186),(21,187),(22,188),(23,189),(24,190),(25,191),(26,192),(27,193),(28,194),(29,195),(30,181),(31,49),(32,50),(33,51),(34,52),(35,53),(36,54),(37,55),(38,56),(39,57),(40,58),(41,59),(42,60),(43,46),(44,47),(45,48),(61,159),(62,160),(63,161),(64,162),(65,163),(66,164),(67,165),(68,151),(69,152),(70,153),(71,154),(72,155),(73,156),(74,157),(75,158),(76,231),(77,232),(78,233),(79,234),(80,235),(81,236),(82,237),(83,238),(84,239),(85,240),(86,226),(87,227),(88,228),(89,229),(90,230),(91,142),(92,143),(93,144),(94,145),(95,146),(96,147),(97,148),(98,149),(99,150),(100,136),(101,137),(102,138),(103,139),(104,140),(105,141),(106,203),(107,204),(108,205),(109,206),(110,207),(111,208),(112,209),(113,210),(114,196),(115,197),(116,198),(117,199),(118,200),(119,201),(120,202),(121,213),(122,214),(123,215),(124,216),(125,217),(126,218),(127,219),(128,220),(129,221),(130,222),(131,223),(132,224),(133,225),(134,211),(135,212)], [(1,104,48,118),(2,105,49,119),(3,91,50,120),(4,92,51,106),(5,93,52,107),(6,94,53,108),(7,95,54,109),(8,96,55,110),(9,97,56,111),(10,98,57,112),(11,99,58,113),(12,100,59,114),(13,101,60,115),(14,102,46,116),(15,103,47,117),(16,236,182,216),(17,237,183,217),(18,238,184,218),(19,239,185,219),(20,240,186,220),(21,226,187,221),(22,227,188,222),(23,228,189,223),(24,229,190,224),(25,230,191,225),(26,231,192,211),(27,232,193,212),(28,233,194,213),(29,234,195,214),(30,235,181,215),(31,72,175,155),(32,73,176,156),(33,74,177,157),(34,75,178,158),(35,61,179,159),(36,62,180,160),(37,63,166,161),(38,64,167,162),(39,65,168,163),(40,66,169,164),(41,67,170,165),(42,68,171,151),(43,69,172,152),(44,70,173,153),(45,71,174,154),(76,148,134,208),(77,149,135,209),(78,150,121,210),(79,136,122,196),(80,137,123,197),(81,138,124,198),(82,139,125,199),(83,140,126,200),(84,141,127,201),(85,142,128,202),(86,143,129,203),(87,144,130,204),(88,145,131,205),(89,146,132,206),(90,147,133,207)]])
210 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | ··· | 15H | 20A | ··· | 20H | 20I | ··· | 20P | 24A | ··· | 24H | 30A | ··· | 30X | 30Y | ··· | 30AN | 40A | ··· | 40P | 60A | ··· | 60P | 60Q | ··· | 60AF | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
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 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C5 | C6 | C6 | C6 | C10 | C10 | C10 | C12 | C15 | C20 | C30 | C30 | C30 | C60 | D4 | D4 | D8 | SD16 | C3×D4 | C3×D4 | C5×D4 | C5×D4 | C3×D8 | C3×SD16 | C5×D8 | C5×SD16 | D4×C15 | D4×C15 | C15×D8 | C15×SD16 |
| kernel | C15×D4⋊C4 | C15×C4⋊C4 | C2×C120 | D4×C30 | C5×D4⋊C4 | D4×C15 | C3×D4⋊C4 | C5×C4⋊C4 | C2×C40 | D4×C10 | C3×C4⋊C4 | C2×C24 | C6×D4 | C5×D4 | D4⋊C4 | C3×D4 | C4⋊C4 | C2×C8 | C2×D4 | D4 | C60 | C2×C30 | C30 | C30 | C20 | C2×C10 | C12 | C2×C6 | C10 | C10 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 16 | 8 | 8 | 8 | 32 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 16 | 16 |
Matrix representation of C15×D4⋊C4 ►in GL3(𝔽241) generated by
| 98 | 0 | 0 |
| 0 | 94 | 0 |
| 0 | 0 | 94 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 240 | 0 |
| 1 | 0 | 0 |
| 0 | 0 | 240 |
| 0 | 240 | 0 |
| 64 | 0 | 0 |
| 0 | 19 | 222 |
| 0 | 222 | 222 |
G:=sub<GL(3,GF(241))| [98,0,0,0,94,0,0,0,94],[1,0,0,0,0,240,0,1,0],[1,0,0,0,0,240,0,240,0],[64,0,0,0,19,222,0,222,222] >;
C15×D4⋊C4 in GAP, Magma, Sage, TeX
C_{15}\times D_4\rtimes C_4 % in TeX
G:=Group("C15xD4:C4"); // GroupNames label
G:=SmallGroup(480,205);
// by ID
G=gap.SmallGroup(480,205);
# by ID
G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,840,869,10504,5261,172]);
// Polycyclic
G:=Group<a,b,c,d|a^15=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b*c>;
// generators/relations