metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.49C23, C30.24C24, C15⋊42- 1+4, Dic10.29D6, Dic6.29D10, D30.10C23, Dic30.19C22, Dic15.12C23, D15⋊Q8⋊4C2, C5⋊3(Q8○D12), C5⋊D4.1D6, D4⋊2D5⋊4S3, D4⋊2S3⋊4D5, (D5×Dic6)⋊4C2, D4.19(S3×D5), (C4×D5).15D6, (C5×D4).24D6, C3⋊D4.1D10, C15⋊7D4.C22, D4⋊2D15⋊4C2, C15⋊Q8.4C22, (S3×Dic10)⋊4C2, (C4×S3).15D10, (C3×D4).24D10, (C2×C30).1C23, C6.24(C23×D5), Dic5.D6⋊2C2, Dic3.D10⋊2C2, C30.C23⋊2C2, D6.D10⋊1C2, C10.24(S3×C23), C20.49(C22×S3), (C2×Dic5).70D6, (C6×D5).10C23, D6.10(C22×D5), C12.49(C22×D5), C3⋊D20.2C22, C15⋊D4.2C22, C5⋊D12.2C22, C3⋊3(D4.10D10), (S3×C20).17C22, (S3×C10).10C23, (C2×Dic3).69D10, (D5×C12).17C22, (D4×C15).19C22, D10.10(C22×S3), (C4×D15).17C22, D30.C2.3C22, (S3×Dic5).3C22, (D5×Dic3).3C22, (C3×Dic5).44C23, (C5×Dic6).19C22, Dic5.12(C22×S3), (C5×Dic3).12C23, Dic3.26(C22×D5), (C3×Dic10).19C22, (C6×Dic5).131C22, (C10×Dic3).131C22, (C2×Dic15).152C22, C4.49(C2×S3×D5), (C2×C15⋊Q8)⋊22C2, C22.1(C2×S3×D5), (C5×C3⋊D4).C22, (C3×C5⋊D4).C22, (C5×D4⋊2S3)⋊4C2, (C3×D4⋊2D5)⋊4C2, C2.27(C22×S3×D5), (C2×C6).1(C22×D5), (C2×C10).1(C22×S3), SmallGroup(480,1096)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15⋊2- 1+4
G = < a,b,c,d,e | a15=b4=c2=1, d2=e2=b2, bab-1=dad-1=a4, ac=ca, eae-1=a11, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >
Subgroups: 1308 in 292 conjugacy classes, 108 normal (50 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, D4, Q8, D5, C10, C10, Dic3, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C15, C2×Q8, C4○D4, Dic5, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, Dic6, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, C5×S3, C3×D5, D15, C30, C30, 2- 1+4, Dic10, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C5⋊D4, C2×C20, C5×D4, C5×D4, C5×Q8, C2×Dic6, C4○D12, D4⋊2S3, D4⋊2S3, S3×Q8, C3×C4○D4, C5×Dic3, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, Dic15, C60, C6×D5, S3×C10, D30, C2×C30, C2×Dic10, C4○D20, D4⋊2D5, D4⋊2D5, Q8×D5, C5×C4○D4, Q8○D12, D5×Dic3, S3×Dic5, D30.C2, C15⋊D4, C3⋊D20, C5⋊D12, C15⋊Q8, C15⋊Q8, C3×Dic10, D5×C12, C6×Dic5, C3×C5⋊D4, C5×Dic6, S3×C20, C10×Dic3, C5×C3⋊D4, Dic30, C4×D15, C2×Dic15, C15⋊7D4, D4×C15, D4.10D10, D5×Dic6, S3×Dic10, D15⋊Q8, D6.D10, Dic5.D6, C30.C23, Dic3.D10, C2×C15⋊Q8, C3×D4⋊2D5, C5×D4⋊2S3, D4⋊2D15, C15⋊2- 1+4
Quotients: C1, C2, C22, S3, C23, D5, D6, C24, D10, C22×S3, 2- 1+4, C22×D5, S3×C23, S3×D5, C23×D5, Q8○D12, C2×S3×D5, D4.10D10, C22×S3×D5, C15⋊2- 1+4
►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)
(1 132 16 149)(2 121 17 138)(3 125 18 142)(4 129 19 146)(5 133 20 150)(6 122 21 139)(7 126 22 143)(8 130 23 147)(9 134 24 136)(10 123 25 140)(11 127 26 144)(12 131 27 148)(13 135 28 137)(14 124 29 141)(15 128 30 145)(31 166 58 158)(32 170 59 162)(33 174 60 151)(34 178 46 155)(35 167 47 159)(36 171 48 163)(37 175 49 152)(38 179 50 156)(39 168 51 160)(40 172 52 164)(41 176 53 153)(42 180 54 157)(43 169 55 161)(44 173 56 165)(45 177 57 154)(61 183 88 200)(62 187 89 204)(63 191 90 208)(64 195 76 197)(65 184 77 201)(66 188 78 205)(67 192 79 209)(68 181 80 198)(69 185 81 202)(70 189 82 206)(71 193 83 210)(72 182 84 199)(73 186 85 203)(74 190 86 207)(75 194 87 196)(91 240 106 220)(92 229 107 224)(93 233 108 213)(94 237 109 217)(95 226 110 221)(96 230 111 225)(97 234 112 214)(98 238 113 218)(99 227 114 222)(100 231 115 211)(101 235 116 215)(102 239 117 219)(103 228 118 223)(104 232 119 212)(105 236 120 216)
(31 58)(32 59)(33 60)(34 46)(35 47)(36 48)(37 49)(38 50)(39 51)(40 52)(41 53)(42 54)(43 55)(44 56)(45 57)(91 106)(92 107)(93 108)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)(105 120)(121 138)(122 139)(123 140)(124 141)(125 142)(126 143)(127 144)(128 145)(129 146)(130 147)(131 148)(132 149)(133 150)(134 136)(135 137)(181 198)(182 199)(183 200)(184 201)(185 202)(186 203)(187 204)(188 205)(189 206)(190 207)(191 208)(192 209)(193 210)(194 196)(195 197)
(1 166 16 158)(2 170 17 162)(3 174 18 151)(4 178 19 155)(5 167 20 159)(6 171 21 163)(7 175 22 152)(8 179 23 156)(9 168 24 160)(10 172 25 164)(11 176 26 153)(12 180 27 157)(13 169 28 161)(14 173 29 165)(15 177 30 154)(31 132 58 149)(32 121 59 138)(33 125 60 142)(34 129 46 146)(35 133 47 150)(36 122 48 139)(37 126 49 143)(38 130 50 147)(39 134 51 136)(40 123 52 140)(41 127 53 144)(42 131 54 148)(43 135 55 137)(44 124 56 141)(45 128 57 145)(61 217 88 237)(62 221 89 226)(63 225 90 230)(64 214 76 234)(65 218 77 238)(66 222 78 227)(67 211 79 231)(68 215 80 235)(69 219 81 239)(70 223 82 228)(71 212 83 232)(72 216 84 236)(73 220 85 240)(74 224 86 229)(75 213 87 233)(91 203 106 186)(92 207 107 190)(93 196 108 194)(94 200 109 183)(95 204 110 187)(96 208 111 191)(97 197 112 195)(98 201 113 184)(99 205 114 188)(100 209 115 192)(101 198 116 181)(102 202 117 185)(103 206 118 189)(104 210 119 193)(105 199 120 182)
(1 87 16 75)(2 83 17 71)(3 79 18 67)(4 90 19 63)(5 86 20 74)(6 82 21 70)(7 78 22 66)(8 89 23 62)(9 85 24 73)(10 81 25 69)(11 77 26 65)(12 88 27 61)(13 84 28 72)(14 80 29 68)(15 76 30 64)(31 108 58 93)(32 119 59 104)(33 115 60 100)(34 111 46 96)(35 107 47 92)(36 118 48 103)(37 114 49 99)(38 110 50 95)(39 106 51 91)(40 117 52 102)(41 113 53 98)(42 109 54 94)(43 120 55 105)(44 116 56 101)(45 112 57 97)(121 210 138 193)(122 206 139 189)(123 202 140 185)(124 198 141 181)(125 209 142 192)(126 205 143 188)(127 201 144 184)(128 197 145 195)(129 208 146 191)(130 204 147 187)(131 200 148 183)(132 196 149 194)(133 207 150 190)(134 203 136 186)(135 199 137 182)(151 231 174 211)(152 227 175 222)(153 238 176 218)(154 234 177 214)(155 230 178 225)(156 226 179 221)(157 237 180 217)(158 233 166 213)(159 229 167 224)(160 240 168 220)(161 236 169 216)(162 232 170 212)(163 228 171 223)(164 239 172 219)(165 235 173 215)
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), (1,132,16,149)(2,121,17,138)(3,125,18,142)(4,129,19,146)(5,133,20,150)(6,122,21,139)(7,126,22,143)(8,130,23,147)(9,134,24,136)(10,123,25,140)(11,127,26,144)(12,131,27,148)(13,135,28,137)(14,124,29,141)(15,128,30,145)(31,166,58,158)(32,170,59,162)(33,174,60,151)(34,178,46,155)(35,167,47,159)(36,171,48,163)(37,175,49,152)(38,179,50,156)(39,168,51,160)(40,172,52,164)(41,176,53,153)(42,180,54,157)(43,169,55,161)(44,173,56,165)(45,177,57,154)(61,183,88,200)(62,187,89,204)(63,191,90,208)(64,195,76,197)(65,184,77,201)(66,188,78,205)(67,192,79,209)(68,181,80,198)(69,185,81,202)(70,189,82,206)(71,193,83,210)(72,182,84,199)(73,186,85,203)(74,190,86,207)(75,194,87,196)(91,240,106,220)(92,229,107,224)(93,233,108,213)(94,237,109,217)(95,226,110,221)(96,230,111,225)(97,234,112,214)(98,238,113,218)(99,227,114,222)(100,231,115,211)(101,235,116,215)(102,239,117,219)(103,228,118,223)(104,232,119,212)(105,236,120,216), (31,58)(32,59)(33,60)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(43,55)(44,56)(45,57)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120)(121,138)(122,139)(123,140)(124,141)(125,142)(126,143)(127,144)(128,145)(129,146)(130,147)(131,148)(132,149)(133,150)(134,136)(135,137)(181,198)(182,199)(183,200)(184,201)(185,202)(186,203)(187,204)(188,205)(189,206)(190,207)(191,208)(192,209)(193,210)(194,196)(195,197), (1,166,16,158)(2,170,17,162)(3,174,18,151)(4,178,19,155)(5,167,20,159)(6,171,21,163)(7,175,22,152)(8,179,23,156)(9,168,24,160)(10,172,25,164)(11,176,26,153)(12,180,27,157)(13,169,28,161)(14,173,29,165)(15,177,30,154)(31,132,58,149)(32,121,59,138)(33,125,60,142)(34,129,46,146)(35,133,47,150)(36,122,48,139)(37,126,49,143)(38,130,50,147)(39,134,51,136)(40,123,52,140)(41,127,53,144)(42,131,54,148)(43,135,55,137)(44,124,56,141)(45,128,57,145)(61,217,88,237)(62,221,89,226)(63,225,90,230)(64,214,76,234)(65,218,77,238)(66,222,78,227)(67,211,79,231)(68,215,80,235)(69,219,81,239)(70,223,82,228)(71,212,83,232)(72,216,84,236)(73,220,85,240)(74,224,86,229)(75,213,87,233)(91,203,106,186)(92,207,107,190)(93,196,108,194)(94,200,109,183)(95,204,110,187)(96,208,111,191)(97,197,112,195)(98,201,113,184)(99,205,114,188)(100,209,115,192)(101,198,116,181)(102,202,117,185)(103,206,118,189)(104,210,119,193)(105,199,120,182), (1,87,16,75)(2,83,17,71)(3,79,18,67)(4,90,19,63)(5,86,20,74)(6,82,21,70)(7,78,22,66)(8,89,23,62)(9,85,24,73)(10,81,25,69)(11,77,26,65)(12,88,27,61)(13,84,28,72)(14,80,29,68)(15,76,30,64)(31,108,58,93)(32,119,59,104)(33,115,60,100)(34,111,46,96)(35,107,47,92)(36,118,48,103)(37,114,49,99)(38,110,50,95)(39,106,51,91)(40,117,52,102)(41,113,53,98)(42,109,54,94)(43,120,55,105)(44,116,56,101)(45,112,57,97)(121,210,138,193)(122,206,139,189)(123,202,140,185)(124,198,141,181)(125,209,142,192)(126,205,143,188)(127,201,144,184)(128,197,145,195)(129,208,146,191)(130,204,147,187)(131,200,148,183)(132,196,149,194)(133,207,150,190)(134,203,136,186)(135,199,137,182)(151,231,174,211)(152,227,175,222)(153,238,176,218)(154,234,177,214)(155,230,178,225)(156,226,179,221)(157,237,180,217)(158,233,166,213)(159,229,167,224)(160,240,168,220)(161,236,169,216)(162,232,170,212)(163,228,171,223)(164,239,172,219)(165,235,173,215)>;
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), (1,132,16,149)(2,121,17,138)(3,125,18,142)(4,129,19,146)(5,133,20,150)(6,122,21,139)(7,126,22,143)(8,130,23,147)(9,134,24,136)(10,123,25,140)(11,127,26,144)(12,131,27,148)(13,135,28,137)(14,124,29,141)(15,128,30,145)(31,166,58,158)(32,170,59,162)(33,174,60,151)(34,178,46,155)(35,167,47,159)(36,171,48,163)(37,175,49,152)(38,179,50,156)(39,168,51,160)(40,172,52,164)(41,176,53,153)(42,180,54,157)(43,169,55,161)(44,173,56,165)(45,177,57,154)(61,183,88,200)(62,187,89,204)(63,191,90,208)(64,195,76,197)(65,184,77,201)(66,188,78,205)(67,192,79,209)(68,181,80,198)(69,185,81,202)(70,189,82,206)(71,193,83,210)(72,182,84,199)(73,186,85,203)(74,190,86,207)(75,194,87,196)(91,240,106,220)(92,229,107,224)(93,233,108,213)(94,237,109,217)(95,226,110,221)(96,230,111,225)(97,234,112,214)(98,238,113,218)(99,227,114,222)(100,231,115,211)(101,235,116,215)(102,239,117,219)(103,228,118,223)(104,232,119,212)(105,236,120,216), (31,58)(32,59)(33,60)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(43,55)(44,56)(45,57)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120)(121,138)(122,139)(123,140)(124,141)(125,142)(126,143)(127,144)(128,145)(129,146)(130,147)(131,148)(132,149)(133,150)(134,136)(135,137)(181,198)(182,199)(183,200)(184,201)(185,202)(186,203)(187,204)(188,205)(189,206)(190,207)(191,208)(192,209)(193,210)(194,196)(195,197), (1,166,16,158)(2,170,17,162)(3,174,18,151)(4,178,19,155)(5,167,20,159)(6,171,21,163)(7,175,22,152)(8,179,23,156)(9,168,24,160)(10,172,25,164)(11,176,26,153)(12,180,27,157)(13,169,28,161)(14,173,29,165)(15,177,30,154)(31,132,58,149)(32,121,59,138)(33,125,60,142)(34,129,46,146)(35,133,47,150)(36,122,48,139)(37,126,49,143)(38,130,50,147)(39,134,51,136)(40,123,52,140)(41,127,53,144)(42,131,54,148)(43,135,55,137)(44,124,56,141)(45,128,57,145)(61,217,88,237)(62,221,89,226)(63,225,90,230)(64,214,76,234)(65,218,77,238)(66,222,78,227)(67,211,79,231)(68,215,80,235)(69,219,81,239)(70,223,82,228)(71,212,83,232)(72,216,84,236)(73,220,85,240)(74,224,86,229)(75,213,87,233)(91,203,106,186)(92,207,107,190)(93,196,108,194)(94,200,109,183)(95,204,110,187)(96,208,111,191)(97,197,112,195)(98,201,113,184)(99,205,114,188)(100,209,115,192)(101,198,116,181)(102,202,117,185)(103,206,118,189)(104,210,119,193)(105,199,120,182), (1,87,16,75)(2,83,17,71)(3,79,18,67)(4,90,19,63)(5,86,20,74)(6,82,21,70)(7,78,22,66)(8,89,23,62)(9,85,24,73)(10,81,25,69)(11,77,26,65)(12,88,27,61)(13,84,28,72)(14,80,29,68)(15,76,30,64)(31,108,58,93)(32,119,59,104)(33,115,60,100)(34,111,46,96)(35,107,47,92)(36,118,48,103)(37,114,49,99)(38,110,50,95)(39,106,51,91)(40,117,52,102)(41,113,53,98)(42,109,54,94)(43,120,55,105)(44,116,56,101)(45,112,57,97)(121,210,138,193)(122,206,139,189)(123,202,140,185)(124,198,141,181)(125,209,142,192)(126,205,143,188)(127,201,144,184)(128,197,145,195)(129,208,146,191)(130,204,147,187)(131,200,148,183)(132,196,149,194)(133,207,150,190)(134,203,136,186)(135,199,137,182)(151,231,174,211)(152,227,175,222)(153,238,176,218)(154,234,177,214)(155,230,178,225)(156,226,179,221)(157,237,180,217)(158,233,166,213)(159,229,167,224)(160,240,168,220)(161,236,169,216)(162,232,170,212)(163,228,171,223)(164,239,172,219)(165,235,173,215) );
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)], [(1,132,16,149),(2,121,17,138),(3,125,18,142),(4,129,19,146),(5,133,20,150),(6,122,21,139),(7,126,22,143),(8,130,23,147),(9,134,24,136),(10,123,25,140),(11,127,26,144),(12,131,27,148),(13,135,28,137),(14,124,29,141),(15,128,30,145),(31,166,58,158),(32,170,59,162),(33,174,60,151),(34,178,46,155),(35,167,47,159),(36,171,48,163),(37,175,49,152),(38,179,50,156),(39,168,51,160),(40,172,52,164),(41,176,53,153),(42,180,54,157),(43,169,55,161),(44,173,56,165),(45,177,57,154),(61,183,88,200),(62,187,89,204),(63,191,90,208),(64,195,76,197),(65,184,77,201),(66,188,78,205),(67,192,79,209),(68,181,80,198),(69,185,81,202),(70,189,82,206),(71,193,83,210),(72,182,84,199),(73,186,85,203),(74,190,86,207),(75,194,87,196),(91,240,106,220),(92,229,107,224),(93,233,108,213),(94,237,109,217),(95,226,110,221),(96,230,111,225),(97,234,112,214),(98,238,113,218),(99,227,114,222),(100,231,115,211),(101,235,116,215),(102,239,117,219),(103,228,118,223),(104,232,119,212),(105,236,120,216)], [(31,58),(32,59),(33,60),(34,46),(35,47),(36,48),(37,49),(38,50),(39,51),(40,52),(41,53),(42,54),(43,55),(44,56),(45,57),(91,106),(92,107),(93,108),(94,109),(95,110),(96,111),(97,112),(98,113),(99,114),(100,115),(101,116),(102,117),(103,118),(104,119),(105,120),(121,138),(122,139),(123,140),(124,141),(125,142),(126,143),(127,144),(128,145),(129,146),(130,147),(131,148),(132,149),(133,150),(134,136),(135,137),(181,198),(182,199),(183,200),(184,201),(185,202),(186,203),(187,204),(188,205),(189,206),(190,207),(191,208),(192,209),(193,210),(194,196),(195,197)], [(1,166,16,158),(2,170,17,162),(3,174,18,151),(4,178,19,155),(5,167,20,159),(6,171,21,163),(7,175,22,152),(8,179,23,156),(9,168,24,160),(10,172,25,164),(11,176,26,153),(12,180,27,157),(13,169,28,161),(14,173,29,165),(15,177,30,154),(31,132,58,149),(32,121,59,138),(33,125,60,142),(34,129,46,146),(35,133,47,150),(36,122,48,139),(37,126,49,143),(38,130,50,147),(39,134,51,136),(40,123,52,140),(41,127,53,144),(42,131,54,148),(43,135,55,137),(44,124,56,141),(45,128,57,145),(61,217,88,237),(62,221,89,226),(63,225,90,230),(64,214,76,234),(65,218,77,238),(66,222,78,227),(67,211,79,231),(68,215,80,235),(69,219,81,239),(70,223,82,228),(71,212,83,232),(72,216,84,236),(73,220,85,240),(74,224,86,229),(75,213,87,233),(91,203,106,186),(92,207,107,190),(93,196,108,194),(94,200,109,183),(95,204,110,187),(96,208,111,191),(97,197,112,195),(98,201,113,184),(99,205,114,188),(100,209,115,192),(101,198,116,181),(102,202,117,185),(103,206,118,189),(104,210,119,193),(105,199,120,182)], [(1,87,16,75),(2,83,17,71),(3,79,18,67),(4,90,19,63),(5,86,20,74),(6,82,21,70),(7,78,22,66),(8,89,23,62),(9,85,24,73),(10,81,25,69),(11,77,26,65),(12,88,27,61),(13,84,28,72),(14,80,29,68),(15,76,30,64),(31,108,58,93),(32,119,59,104),(33,115,60,100),(34,111,46,96),(35,107,47,92),(36,118,48,103),(37,114,49,99),(38,110,50,95),(39,106,51,91),(40,117,52,102),(41,113,53,98),(42,109,54,94),(43,120,55,105),(44,116,56,101),(45,112,57,97),(121,210,138,193),(122,206,139,189),(123,202,140,185),(124,198,141,181),(125,209,142,192),(126,205,143,188),(127,201,144,184),(128,197,145,195),(129,208,146,191),(130,204,147,187),(131,200,148,183),(132,196,149,194),(133,207,150,190),(134,203,136,186),(135,199,137,182),(151,231,174,211),(152,227,175,222),(153,238,176,218),(154,234,177,214),(155,230,178,225),(156,226,179,221),(157,237,180,217),(158,233,166,213),(159,229,167,224),(160,240,168,220),(161,236,169,216),(162,232,170,212),(163,228,171,223),(164,239,172,219),(165,235,173,215)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | 6D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 30A | 30B | 30C | 30D | 30E | 30F | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 30 | 30 | 60 | 60 |
| size | 1 | 1 | 2 | 2 | 6 | 10 | 30 | 2 | 2 | 6 | 6 | 6 | 10 | 10 | 10 | 30 | 30 | 30 | 2 | 2 | 2 | 4 | 4 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 4 | 10 | 10 | 20 | 20 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | D6 | D6 | D10 | D10 | D10 | D10 | D10 | 2- 1+4 | S3×D5 | Q8○D12 | C2×S3×D5 | C2×S3×D5 | D4.10D10 | C15⋊2- 1+4 |
| kernel | C15⋊2- 1+4 | D5×Dic6 | S3×Dic10 | D15⋊Q8 | D6.D10 | Dic5.D6 | C30.C23 | Dic3.D10 | C2×C15⋊Q8 | C3×D4⋊2D5 | C5×D4⋊2S3 | D4⋊2D15 | D4⋊2D5 | D4⋊2S3 | Dic10 | C4×D5 | C2×Dic5 | C5⋊D4 | C5×D4 | Dic6 | C4×S3 | C2×Dic3 | C3⋊D4 | C3×D4 | C15 | D4 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of C15⋊2- 1+4 ►in GL6(𝔽61)
| 59 | 5 | 0 | 0 | 0 | 0 |
| 36 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 43 | 0 | 0 |
| 0 | 0 | 17 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 43 |
| 0 | 0 | 0 | 0 | 18 | 18 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 53 | 58 | 17 | 53 |
| 0 | 0 | 42 | 8 | 36 | 30 |
| 0 | 0 | 0 | 0 | 56 | 29 |
| 0 | 0 | 0 | 0 | 58 | 5 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 47 | 47 | 60 | 0 |
| 0 | 0 | 14 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 3 | 0 | 0 |
| 0 | 0 | 19 | 53 | 0 | 0 |
| 0 | 0 | 0 | 0 | 56 | 29 |
| 0 | 0 | 0 | 0 | 58 | 5 |
| 58 | 2 | 0 | 0 | 0 | 0 |
| 57 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 57 | 0 | 0 |
| 0 | 0 | 58 | 29 | 0 | 0 |
| 0 | 0 | 21 | 16 | 36 | 4 |
| 0 | 0 | 21 | 5 | 57 | 25 |
G:=sub<GL(6,GF(61))| [59,36,0,0,0,0,5,1,0,0,0,0,0,0,0,17,0,0,0,0,43,17,0,0,0,0,0,0,60,18,0,0,0,0,43,18],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,53,42,0,0,0,0,58,8,0,0,0,0,17,36,56,58,0,0,53,30,29,5],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,47,14,0,0,0,1,47,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,19,0,0,0,0,3,53,0,0,0,0,0,0,56,58,0,0,0,0,29,5],[58,57,0,0,0,0,2,3,0,0,0,0,0,0,32,58,21,21,0,0,57,29,16,5,0,0,0,0,36,57,0,0,0,0,4,25] >;
C15⋊2- 1+4 in GAP, Magma, Sage, TeX
C_{15}\rtimes 2_-^{1+4} % in TeX
G:=Group("C15:ES-(2,2)"); // GroupNames label
G:=SmallGroup(480,1096);
// by ID
G=gap.SmallGroup(480,1096);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,219,100,675,185,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^15=b^4=c^2=1,d^2=e^2=b^2,b*a*b^-1=d*a*d^-1=a^4,a*c=c*a,e*a*e^-1=a^11,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations