?
G = C15⋊2- (1+4) order 480 = 25·3·5
The semidirect product of C15 and 2- (1+4) acting via 2- (1+4)/D4=C22
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C30.24C24, C60.49C23, C15⋊42- (1+4), Dic6.29D10, Dic10.29D6, D30.10C23, Dic30.19C22, Dic15.12C23, D15⋊Q8⋊4C2, C5⋊3(Q8○D12), C5⋊D4.1D6, D4⋊2S3⋊4D5, D4⋊2D5⋊4S3, (D5×Dic6)⋊4C2, (C5×D4).24D6, (C4×D5).15D6, D4.19(S3×D5), C3⋊D4.1D10, C15⋊7D4.C22, D4⋊2D15⋊4C2, C15⋊Q8.4C22, (S3×Dic10)⋊4C2, (C3×D4).24D10, (C4×S3).15D10, (C2×C30).1C23, C6.24(C23×D5), D6.D10⋊1C2, Dic3.D10⋊2C2, C30.C23⋊2C2, Dic5.D6⋊2C2, C20.49(C22×S3), C10.24(S3×C23), (C2×Dic5).70D6, (C6×D5).10C23, D6.10(C22×D5), C12.49(C22×D5), C15⋊D4.2C22, C3⋊D20.2C22, C5⋊D12.2C22, C3⋊3(D4.10D10), (S3×C20).17C22, (S3×C10).10C23, (C2×Dic3).69D10, (C4×D15).17C22, (D4×C15).19C22, D10.10(C22×S3), (D5×C12).17C22, D30.C2.3C22, (D5×Dic3).3C22, (S3×Dic5).3C22, Dic3.26(C22×D5), (C5×Dic3).12C23, Dic5.12(C22×S3), (C3×Dic5).44C23, (C5×Dic6).19C22, (C6×Dic5).131C22, (C3×Dic10).19C22, (C2×Dic15).152C22, (C10×Dic3).131C22, 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
Subgroups: 1308 in 292 conjugacy classes, 108 normal (50 characteristic)
C1, C2, C2 [×5], C3, C4, C4 [×9], C22 [×2], C22 [×3], C5, S3 [×2], C6, C6 [×3], C2×C4 [×15], D4, D4 [×9], Q8 [×10], D5 [×2], C10, C10 [×3], Dic3, Dic3 [×2], Dic3 [×3], C12, C12 [×3], D6, D6, C2×C6 [×2], C2×C6, C15, C2×Q8 [×5], C4○D4 [×10], Dic5, Dic5 [×2], Dic5 [×3], C20, C20 [×3], D10, D10, C2×C10 [×2], C2×C10, Dic6, Dic6 [×8], C4×S3, C4×S3 [×5], D12, C2×Dic3 [×2], C2×Dic3 [×4], C3⋊D4 [×2], C3⋊D4 [×4], C2×C12 [×3], C3×D4, C3×D4 [×2], C3×Q8, C5×S3, C3×D5, D15, C30, C30 [×2], 2- (1+4), Dic10, Dic10 [×8], C4×D5, C4×D5 [×5], D20, C2×Dic5 [×2], C2×Dic5 [×4], C5⋊D4 [×2], C5⋊D4 [×4], C2×C20 [×3], C5×D4, C5×D4 [×2], C5×Q8, C2×Dic6 [×3], C4○D12 [×3], D4⋊2S3, D4⋊2S3 [×5], S3×Q8 [×2], C3×C4○D4, C5×Dic3, C5×Dic3 [×2], C3×Dic5, C3×Dic5 [×2], Dic15, Dic15 [×2], C60, C6×D5, S3×C10, D30, C2×C30 [×2], C2×Dic10 [×3], C4○D20 [×3], D4⋊2D5, D4⋊2D5 [×5], Q8×D5 [×2], C5×C4○D4, Q8○D12, D5×Dic3 [×2], S3×Dic5 [×2], D30.C2 [×2], C15⋊D4, C3⋊D20, C5⋊D12, C15⋊Q8, C15⋊Q8 [×6], C3×Dic10, D5×C12, C6×Dic5 [×2], C3×C5⋊D4 [×2], C5×Dic6, S3×C20, C10×Dic3 [×2], C5×C3⋊D4 [×2], Dic30, C4×D15, C2×Dic15 [×2], C15⋊7D4 [×2], D4×C15, D4.10D10, D5×Dic6, S3×Dic10, D15⋊Q8, D6.D10, Dic5.D6 [×2], C30.C23 [×2], Dic3.D10 [×2], C2×C15⋊Q8 [×2], C3×D4⋊2D5, C5×D4⋊2S3, D4⋊2D15, C15⋊2- (1+4)
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C24, D10 [×7], C22×S3 [×7], 2- (1+4), C22×D5 [×7], S3×C23, S3×D5, C23×D5, Q8○D12, C2×S3×D5 [×3], D4.10D10, C22×S3×D5, C15⋊2- (1+4)
Generators and relations
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 >
►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)
(1 121 28 136)(2 125 29 140)(3 129 30 144)(4 133 16 148)(5 122 17 137)(6 126 18 141)(7 130 19 145)(8 134 20 149)(9 123 21 138)(10 127 22 142)(11 131 23 146)(12 135 24 150)(13 124 25 139)(14 128 26 143)(15 132 27 147)(31 173 57 158)(32 177 58 162)(33 166 59 151)(34 170 60 155)(35 174 46 159)(36 178 47 163)(37 167 48 152)(38 171 49 156)(39 175 50 160)(40 179 51 164)(41 168 52 153)(42 172 53 157)(43 176 54 161)(44 180 55 165)(45 169 56 154)(61 193 80 196)(62 182 81 200)(63 186 82 204)(64 190 83 208)(65 194 84 197)(66 183 85 201)(67 187 86 205)(68 191 87 209)(69 195 88 198)(70 184 89 202)(71 188 90 206)(72 192 76 210)(73 181 77 199)(74 185 78 203)(75 189 79 207)(91 231 106 225)(92 235 107 214)(93 239 108 218)(94 228 109 222)(95 232 110 211)(96 236 111 215)(97 240 112 219)(98 229 113 223)(99 233 114 212)(100 237 115 216)(101 226 116 220)(102 230 117 224)(103 234 118 213)(104 238 119 217)(105 227 120 221)
(31 57)(32 58)(33 59)(34 60)(35 46)(36 47)(37 48)(38 49)(39 50)(40 51)(41 52)(42 53)(43 54)(44 55)(45 56)(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)(121 136)(122 137)(123 138)(124 139)(125 140)(126 141)(127 142)(128 143)(129 144)(130 145)(131 146)(132 147)(133 148)(134 149)(135 150)(181 199)(182 200)(183 201)(184 202)(185 203)(186 204)(187 205)(188 206)(189 207)(190 208)(191 209)(192 210)(193 196)(194 197)(195 198)
(1 177 28 162)(2 166 29 151)(3 170 30 155)(4 174 16 159)(5 178 17 163)(6 167 18 152)(7 171 19 156)(8 175 20 160)(9 179 21 164)(10 168 22 153)(11 172 23 157)(12 176 24 161)(13 180 25 165)(14 169 26 154)(15 173 27 158)(31 132 57 147)(32 121 58 136)(33 125 59 140)(34 129 60 144)(35 133 46 148)(36 122 47 137)(37 126 48 141)(38 130 49 145)(39 134 50 149)(40 123 51 138)(41 127 52 142)(42 131 53 146)(43 135 54 150)(44 124 55 139)(45 128 56 143)(61 217 80 238)(62 221 81 227)(63 225 82 231)(64 214 83 235)(65 218 84 239)(66 222 85 228)(67 211 86 232)(68 215 87 236)(69 219 88 240)(70 223 89 229)(71 212 90 233)(72 216 76 237)(73 220 77 226)(74 224 78 230)(75 213 79 234)(91 204 106 186)(92 208 107 190)(93 197 108 194)(94 201 109 183)(95 205 110 187)(96 209 111 191)(97 198 112 195)(98 202 113 184)(99 206 114 188)(100 210 115 192)(101 199 116 181)(102 203 117 185)(103 207 118 189)(104 196 119 193)(105 200 120 182)
(1 78 28 74)(2 89 29 70)(3 85 30 66)(4 81 16 62)(5 77 17 73)(6 88 18 69)(7 84 19 65)(8 80 20 61)(9 76 21 72)(10 87 22 68)(11 83 23 64)(12 79 24 75)(13 90 25 71)(14 86 26 67)(15 82 27 63)(31 106 57 91)(32 117 58 102)(33 113 59 98)(34 109 60 94)(35 120 46 105)(36 116 47 101)(37 112 48 97)(38 108 49 93)(39 119 50 104)(40 115 51 100)(41 111 52 96)(42 107 53 92)(43 118 54 103)(44 114 55 99)(45 110 56 95)(121 203 136 185)(122 199 137 181)(123 210 138 192)(124 206 139 188)(125 202 140 184)(126 198 141 195)(127 209 142 191)(128 205 143 187)(129 201 144 183)(130 197 145 194)(131 208 146 190)(132 204 147 186)(133 200 148 182)(134 196 149 193)(135 207 150 189)(151 229 166 223)(152 240 167 219)(153 236 168 215)(154 232 169 211)(155 228 170 222)(156 239 171 218)(157 235 172 214)(158 231 173 225)(159 227 174 221)(160 238 175 217)(161 234 176 213)(162 230 177 224)(163 226 178 220)(164 237 179 216)(165 233 180 212)
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), (1,121,28,136)(2,125,29,140)(3,129,30,144)(4,133,16,148)(5,122,17,137)(6,126,18,141)(7,130,19,145)(8,134,20,149)(9,123,21,138)(10,127,22,142)(11,131,23,146)(12,135,24,150)(13,124,25,139)(14,128,26,143)(15,132,27,147)(31,173,57,158)(32,177,58,162)(33,166,59,151)(34,170,60,155)(35,174,46,159)(36,178,47,163)(37,167,48,152)(38,171,49,156)(39,175,50,160)(40,179,51,164)(41,168,52,153)(42,172,53,157)(43,176,54,161)(44,180,55,165)(45,169,56,154)(61,193,80,196)(62,182,81,200)(63,186,82,204)(64,190,83,208)(65,194,84,197)(66,183,85,201)(67,187,86,205)(68,191,87,209)(69,195,88,198)(70,184,89,202)(71,188,90,206)(72,192,76,210)(73,181,77,199)(74,185,78,203)(75,189,79,207)(91,231,106,225)(92,235,107,214)(93,239,108,218)(94,228,109,222)(95,232,110,211)(96,236,111,215)(97,240,112,219)(98,229,113,223)(99,233,114,212)(100,237,115,216)(101,226,116,220)(102,230,117,224)(103,234,118,213)(104,238,119,217)(105,227,120,221), (31,57)(32,58)(33,59)(34,60)(35,46)(36,47)(37,48)(38,49)(39,50)(40,51)(41,52)(42,53)(43,54)(44,55)(45,56)(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)(121,136)(122,137)(123,138)(124,139)(125,140)(126,141)(127,142)(128,143)(129,144)(130,145)(131,146)(132,147)(133,148)(134,149)(135,150)(181,199)(182,200)(183,201)(184,202)(185,203)(186,204)(187,205)(188,206)(189,207)(190,208)(191,209)(192,210)(193,196)(194,197)(195,198), (1,177,28,162)(2,166,29,151)(3,170,30,155)(4,174,16,159)(5,178,17,163)(6,167,18,152)(7,171,19,156)(8,175,20,160)(9,179,21,164)(10,168,22,153)(11,172,23,157)(12,176,24,161)(13,180,25,165)(14,169,26,154)(15,173,27,158)(31,132,57,147)(32,121,58,136)(33,125,59,140)(34,129,60,144)(35,133,46,148)(36,122,47,137)(37,126,48,141)(38,130,49,145)(39,134,50,149)(40,123,51,138)(41,127,52,142)(42,131,53,146)(43,135,54,150)(44,124,55,139)(45,128,56,143)(61,217,80,238)(62,221,81,227)(63,225,82,231)(64,214,83,235)(65,218,84,239)(66,222,85,228)(67,211,86,232)(68,215,87,236)(69,219,88,240)(70,223,89,229)(71,212,90,233)(72,216,76,237)(73,220,77,226)(74,224,78,230)(75,213,79,234)(91,204,106,186)(92,208,107,190)(93,197,108,194)(94,201,109,183)(95,205,110,187)(96,209,111,191)(97,198,112,195)(98,202,113,184)(99,206,114,188)(100,210,115,192)(101,199,116,181)(102,203,117,185)(103,207,118,189)(104,196,119,193)(105,200,120,182), (1,78,28,74)(2,89,29,70)(3,85,30,66)(4,81,16,62)(5,77,17,73)(6,88,18,69)(7,84,19,65)(8,80,20,61)(9,76,21,72)(10,87,22,68)(11,83,23,64)(12,79,24,75)(13,90,25,71)(14,86,26,67)(15,82,27,63)(31,106,57,91)(32,117,58,102)(33,113,59,98)(34,109,60,94)(35,120,46,105)(36,116,47,101)(37,112,48,97)(38,108,49,93)(39,119,50,104)(40,115,51,100)(41,111,52,96)(42,107,53,92)(43,118,54,103)(44,114,55,99)(45,110,56,95)(121,203,136,185)(122,199,137,181)(123,210,138,192)(124,206,139,188)(125,202,140,184)(126,198,141,195)(127,209,142,191)(128,205,143,187)(129,201,144,183)(130,197,145,194)(131,208,146,190)(132,204,147,186)(133,200,148,182)(134,196,149,193)(135,207,150,189)(151,229,166,223)(152,240,167,219)(153,236,168,215)(154,232,169,211)(155,228,170,222)(156,239,171,218)(157,235,172,214)(158,231,173,225)(159,227,174,221)(160,238,175,217)(161,234,176,213)(162,230,177,224)(163,226,178,220)(164,237,179,216)(165,233,180,212)>;
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), (1,121,28,136)(2,125,29,140)(3,129,30,144)(4,133,16,148)(5,122,17,137)(6,126,18,141)(7,130,19,145)(8,134,20,149)(9,123,21,138)(10,127,22,142)(11,131,23,146)(12,135,24,150)(13,124,25,139)(14,128,26,143)(15,132,27,147)(31,173,57,158)(32,177,58,162)(33,166,59,151)(34,170,60,155)(35,174,46,159)(36,178,47,163)(37,167,48,152)(38,171,49,156)(39,175,50,160)(40,179,51,164)(41,168,52,153)(42,172,53,157)(43,176,54,161)(44,180,55,165)(45,169,56,154)(61,193,80,196)(62,182,81,200)(63,186,82,204)(64,190,83,208)(65,194,84,197)(66,183,85,201)(67,187,86,205)(68,191,87,209)(69,195,88,198)(70,184,89,202)(71,188,90,206)(72,192,76,210)(73,181,77,199)(74,185,78,203)(75,189,79,207)(91,231,106,225)(92,235,107,214)(93,239,108,218)(94,228,109,222)(95,232,110,211)(96,236,111,215)(97,240,112,219)(98,229,113,223)(99,233,114,212)(100,237,115,216)(101,226,116,220)(102,230,117,224)(103,234,118,213)(104,238,119,217)(105,227,120,221), (31,57)(32,58)(33,59)(34,60)(35,46)(36,47)(37,48)(38,49)(39,50)(40,51)(41,52)(42,53)(43,54)(44,55)(45,56)(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)(121,136)(122,137)(123,138)(124,139)(125,140)(126,141)(127,142)(128,143)(129,144)(130,145)(131,146)(132,147)(133,148)(134,149)(135,150)(181,199)(182,200)(183,201)(184,202)(185,203)(186,204)(187,205)(188,206)(189,207)(190,208)(191,209)(192,210)(193,196)(194,197)(195,198), (1,177,28,162)(2,166,29,151)(3,170,30,155)(4,174,16,159)(5,178,17,163)(6,167,18,152)(7,171,19,156)(8,175,20,160)(9,179,21,164)(10,168,22,153)(11,172,23,157)(12,176,24,161)(13,180,25,165)(14,169,26,154)(15,173,27,158)(31,132,57,147)(32,121,58,136)(33,125,59,140)(34,129,60,144)(35,133,46,148)(36,122,47,137)(37,126,48,141)(38,130,49,145)(39,134,50,149)(40,123,51,138)(41,127,52,142)(42,131,53,146)(43,135,54,150)(44,124,55,139)(45,128,56,143)(61,217,80,238)(62,221,81,227)(63,225,82,231)(64,214,83,235)(65,218,84,239)(66,222,85,228)(67,211,86,232)(68,215,87,236)(69,219,88,240)(70,223,89,229)(71,212,90,233)(72,216,76,237)(73,220,77,226)(74,224,78,230)(75,213,79,234)(91,204,106,186)(92,208,107,190)(93,197,108,194)(94,201,109,183)(95,205,110,187)(96,209,111,191)(97,198,112,195)(98,202,113,184)(99,206,114,188)(100,210,115,192)(101,199,116,181)(102,203,117,185)(103,207,118,189)(104,196,119,193)(105,200,120,182), (1,78,28,74)(2,89,29,70)(3,85,30,66)(4,81,16,62)(5,77,17,73)(6,88,18,69)(7,84,19,65)(8,80,20,61)(9,76,21,72)(10,87,22,68)(11,83,23,64)(12,79,24,75)(13,90,25,71)(14,86,26,67)(15,82,27,63)(31,106,57,91)(32,117,58,102)(33,113,59,98)(34,109,60,94)(35,120,46,105)(36,116,47,101)(37,112,48,97)(38,108,49,93)(39,119,50,104)(40,115,51,100)(41,111,52,96)(42,107,53,92)(43,118,54,103)(44,114,55,99)(45,110,56,95)(121,203,136,185)(122,199,137,181)(123,210,138,192)(124,206,139,188)(125,202,140,184)(126,198,141,195)(127,209,142,191)(128,205,143,187)(129,201,144,183)(130,197,145,194)(131,208,146,190)(132,204,147,186)(133,200,148,182)(134,196,149,193)(135,207,150,189)(151,229,166,223)(152,240,167,219)(153,236,168,215)(154,232,169,211)(155,228,170,222)(156,239,171,218)(157,235,172,214)(158,231,173,225)(159,227,174,221)(160,238,175,217)(161,234,176,213)(162,230,177,224)(163,226,178,220)(164,237,179,216)(165,233,180,212) );
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)], [(1,121,28,136),(2,125,29,140),(3,129,30,144),(4,133,16,148),(5,122,17,137),(6,126,18,141),(7,130,19,145),(8,134,20,149),(9,123,21,138),(10,127,22,142),(11,131,23,146),(12,135,24,150),(13,124,25,139),(14,128,26,143),(15,132,27,147),(31,173,57,158),(32,177,58,162),(33,166,59,151),(34,170,60,155),(35,174,46,159),(36,178,47,163),(37,167,48,152),(38,171,49,156),(39,175,50,160),(40,179,51,164),(41,168,52,153),(42,172,53,157),(43,176,54,161),(44,180,55,165),(45,169,56,154),(61,193,80,196),(62,182,81,200),(63,186,82,204),(64,190,83,208),(65,194,84,197),(66,183,85,201),(67,187,86,205),(68,191,87,209),(69,195,88,198),(70,184,89,202),(71,188,90,206),(72,192,76,210),(73,181,77,199),(74,185,78,203),(75,189,79,207),(91,231,106,225),(92,235,107,214),(93,239,108,218),(94,228,109,222),(95,232,110,211),(96,236,111,215),(97,240,112,219),(98,229,113,223),(99,233,114,212),(100,237,115,216),(101,226,116,220),(102,230,117,224),(103,234,118,213),(104,238,119,217),(105,227,120,221)], [(31,57),(32,58),(33,59),(34,60),(35,46),(36,47),(37,48),(38,49),(39,50),(40,51),(41,52),(42,53),(43,54),(44,55),(45,56),(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),(121,136),(122,137),(123,138),(124,139),(125,140),(126,141),(127,142),(128,143),(129,144),(130,145),(131,146),(132,147),(133,148),(134,149),(135,150),(181,199),(182,200),(183,201),(184,202),(185,203),(186,204),(187,205),(188,206),(189,207),(190,208),(191,209),(192,210),(193,196),(194,197),(195,198)], [(1,177,28,162),(2,166,29,151),(3,170,30,155),(4,174,16,159),(5,178,17,163),(6,167,18,152),(7,171,19,156),(8,175,20,160),(9,179,21,164),(10,168,22,153),(11,172,23,157),(12,176,24,161),(13,180,25,165),(14,169,26,154),(15,173,27,158),(31,132,57,147),(32,121,58,136),(33,125,59,140),(34,129,60,144),(35,133,46,148),(36,122,47,137),(37,126,48,141),(38,130,49,145),(39,134,50,149),(40,123,51,138),(41,127,52,142),(42,131,53,146),(43,135,54,150),(44,124,55,139),(45,128,56,143),(61,217,80,238),(62,221,81,227),(63,225,82,231),(64,214,83,235),(65,218,84,239),(66,222,85,228),(67,211,86,232),(68,215,87,236),(69,219,88,240),(70,223,89,229),(71,212,90,233),(72,216,76,237),(73,220,77,226),(74,224,78,230),(75,213,79,234),(91,204,106,186),(92,208,107,190),(93,197,108,194),(94,201,109,183),(95,205,110,187),(96,209,111,191),(97,198,112,195),(98,202,113,184),(99,206,114,188),(100,210,115,192),(101,199,116,181),(102,203,117,185),(103,207,118,189),(104,196,119,193),(105,200,120,182)], [(1,78,28,74),(2,89,29,70),(3,85,30,66),(4,81,16,62),(5,77,17,73),(6,88,18,69),(7,84,19,65),(8,80,20,61),(9,76,21,72),(10,87,22,68),(11,83,23,64),(12,79,24,75),(13,90,25,71),(14,86,26,67),(15,82,27,63),(31,106,57,91),(32,117,58,102),(33,113,59,98),(34,109,60,94),(35,120,46,105),(36,116,47,101),(37,112,48,97),(38,108,49,93),(39,119,50,104),(40,115,51,100),(41,111,52,96),(42,107,53,92),(43,118,54,103),(44,114,55,99),(45,110,56,95),(121,203,136,185),(122,199,137,181),(123,210,138,192),(124,206,139,188),(125,202,140,184),(126,198,141,195),(127,209,142,191),(128,205,143,187),(129,201,144,183),(130,197,145,194),(131,208,146,190),(132,204,147,186),(133,200,148,182),(134,196,149,193),(135,207,150,189),(151,229,166,223),(152,240,167,219),(153,236,168,215),(154,232,169,211),(155,228,170,222),(156,239,171,218),(157,235,172,214),(158,231,173,225),(159,227,174,221),(160,238,175,217),(161,234,176,213),(162,230,177,224),(163,226,178,220),(164,237,179,216),(165,233,180,212)])
Matrix representation ►G ⊆ 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] >;
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 |
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
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