direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C15×2- 1+4, C60.300C23, C30.103C24, C4○D4⋊6C30, (C2×Q8)⋊7C30, (Q8×C30)⋊26C2, (C6×Q8)⋊12C10, (Q8×C10)⋊16C6, D4.4(C2×C30), Q8.7(C2×C30), C2.5(C23×C30), C4.10(C22×C30), C20.53(C22×C6), C10.20(C23×C6), C6.20(C23×C10), C12.52(C22×C10), (C2×C60).442C22, (C2×C30).264C23, (D4×C15).53C22, C22.3(C22×C30), (Q8×C15).58C22, (C5×C4○D4)⋊13C6, (C3×C4○D4)⋊9C10, (C2×C4).9(C2×C30), (C15×C4○D4)⋊19C2, (C2×C20).71(C2×C6), (C5×D4).14(C2×C6), (C5×Q8).23(C2×C6), (C2×C12).70(C2×C10), (C3×D4).14(C2×C10), (C2×C10).8(C22×C6), (C2×C6).8(C22×C10), (C3×Q8).15(C2×C10), SmallGroup(480,1185)
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, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >
Subgroups: 312 in 292 conjugacy classes, 272 normal (12 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, C2×C4, D4, Q8, C10, C10, C12, C2×C6, C15, C2×Q8, C4○D4, C20, C2×C10, C2×C12, C3×D4, C3×Q8, C30, C30, 2- 1+4, C2×C20, C5×D4, C5×Q8, C6×Q8, C3×C4○D4, C60, C2×C30, Q8×C10, C5×C4○D4, C3×2- 1+4, C2×C60, D4×C15, Q8×C15, C5×2- 1+4, Q8×C30, C15×C4○D4, C15×2- 1+4
Quotients: C1, C2, C3, C22, C5, C6, C23, C10, C2×C6, C15, C24, C2×C10, C22×C6, C30, 2- 1+4, C22×C10, C23×C6, C2×C30, C23×C10, C3×2- 1+4, C22×C30, C5×2- 1+4, C23×C30, 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 74 161 131)(2 75 162 132)(3 61 163 133)(4 62 164 134)(5 63 165 135)(6 64 151 121)(7 65 152 122)(8 66 153 123)(9 67 154 124)(10 68 155 125)(11 69 156 126)(12 70 157 127)(13 71 158 128)(14 72 159 129)(15 73 160 130)(16 102 56 228)(17 103 57 229)(18 104 58 230)(19 105 59 231)(20 91 60 232)(21 92 46 233)(22 93 47 234)(23 94 48 235)(24 95 49 236)(25 96 50 237)(26 97 51 238)(27 98 52 239)(28 99 53 240)(29 100 54 226)(30 101 55 227)(31 207 109 195)(32 208 110 181)(33 209 111 182)(34 210 112 183)(35 196 113 184)(36 197 114 185)(37 198 115 186)(38 199 116 187)(39 200 117 188)(40 201 118 189)(41 202 119 190)(42 203 120 191)(43 204 106 192)(44 205 107 193)(45 206 108 194)(76 175 149 211)(77 176 150 212)(78 177 136 213)(79 178 137 214)(80 179 138 215)(81 180 139 216)(82 166 140 217)(83 167 141 218)(84 168 142 219)(85 169 143 220)(86 170 144 221)(87 171 145 222)(88 172 146 223)(89 173 147 224)(90 174 148 225)
(31 109)(32 110)(33 111)(34 112)(35 113)(36 114)(37 115)(38 116)(39 117)(40 118)(41 119)(42 120)(43 106)(44 107)(45 108)(61 133)(62 134)(63 135)(64 121)(65 122)(66 123)(67 124)(68 125)(69 126)(70 127)(71 128)(72 129)(73 130)(74 131)(75 132)(91 232)(92 233)(93 234)(94 235)(95 236)(96 237)(97 238)(98 239)(99 240)(100 226)(101 227)(102 228)(103 229)(104 230)(105 231)(166 217)(167 218)(168 219)(169 220)(170 221)(171 222)(172 223)(173 224)(174 225)(175 211)(176 212)(177 213)(178 214)(179 215)(180 216)
(1 51 161 26)(2 52 162 27)(3 53 163 28)(4 54 164 29)(5 55 165 30)(6 56 151 16)(7 57 152 17)(8 58 153 18)(9 59 154 19)(10 60 155 20)(11 46 156 21)(12 47 157 22)(13 48 158 23)(14 49 159 24)(15 50 160 25)(31 224 109 173)(32 225 110 174)(33 211 111 175)(34 212 112 176)(35 213 113 177)(36 214 114 178)(37 215 115 179)(38 216 116 180)(39 217 117 166)(40 218 118 167)(41 219 119 168)(42 220 120 169)(43 221 106 170)(44 222 107 171)(45 223 108 172)(61 240 133 99)(62 226 134 100)(63 227 135 101)(64 228 121 102)(65 229 122 103)(66 230 123 104)(67 231 124 105)(68 232 125 91)(69 233 126 92)(70 234 127 93)(71 235 128 94)(72 236 129 95)(73 237 130 96)(74 238 131 97)(75 239 132 98)(76 182 149 209)(77 183 150 210)(78 184 136 196)(79 185 137 197)(80 186 138 198)(81 187 139 199)(82 188 140 200)(83 189 141 201)(84 190 142 202)(85 191 143 203)(86 192 144 204)(87 193 145 205)(88 194 146 206)(89 195 147 207)(90 181 148 208)
(1 147 161 89)(2 148 162 90)(3 149 163 76)(4 150 164 77)(5 136 165 78)(6 137 151 79)(7 138 152 80)(8 139 153 81)(9 140 154 82)(10 141 155 83)(11 142 156 84)(12 143 157 85)(13 144 158 86)(14 145 159 87)(15 146 160 88)(16 197 56 185)(17 198 57 186)(18 199 58 187)(19 200 59 188)(20 201 60 189)(21 202 46 190)(22 203 47 191)(23 204 48 192)(24 205 49 193)(25 206 50 194)(26 207 51 195)(27 208 52 181)(28 209 53 182)(29 210 54 183)(30 196 55 184)(31 97 109 238)(32 98 110 239)(33 99 111 240)(34 100 112 226)(35 101 113 227)(36 102 114 228)(37 103 115 229)(38 104 116 230)(39 105 117 231)(40 91 118 232)(41 92 119 233)(42 93 120 234)(43 94 106 235)(44 95 107 236)(45 96 108 237)(61 211 133 175)(62 212 134 176)(63 213 135 177)(64 214 121 178)(65 215 122 179)(66 216 123 180)(67 217 124 166)(68 218 125 167)(69 219 126 168)(70 220 127 169)(71 221 128 170)(72 222 129 171)(73 223 130 172)(74 224 131 173)(75 225 132 174)
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), (1,74,161,131)(2,75,162,132)(3,61,163,133)(4,62,164,134)(5,63,165,135)(6,64,151,121)(7,65,152,122)(8,66,153,123)(9,67,154,124)(10,68,155,125)(11,69,156,126)(12,70,157,127)(13,71,158,128)(14,72,159,129)(15,73,160,130)(16,102,56,228)(17,103,57,229)(18,104,58,230)(19,105,59,231)(20,91,60,232)(21,92,46,233)(22,93,47,234)(23,94,48,235)(24,95,49,236)(25,96,50,237)(26,97,51,238)(27,98,52,239)(28,99,53,240)(29,100,54,226)(30,101,55,227)(31,207,109,195)(32,208,110,181)(33,209,111,182)(34,210,112,183)(35,196,113,184)(36,197,114,185)(37,198,115,186)(38,199,116,187)(39,200,117,188)(40,201,118,189)(41,202,119,190)(42,203,120,191)(43,204,106,192)(44,205,107,193)(45,206,108,194)(76,175,149,211)(77,176,150,212)(78,177,136,213)(79,178,137,214)(80,179,138,215)(81,180,139,216)(82,166,140,217)(83,167,141,218)(84,168,142,219)(85,169,143,220)(86,170,144,221)(87,171,145,222)(88,172,146,223)(89,173,147,224)(90,174,148,225), (31,109)(32,110)(33,111)(34,112)(35,113)(36,114)(37,115)(38,116)(39,117)(40,118)(41,119)(42,120)(43,106)(44,107)(45,108)(61,133)(62,134)(63,135)(64,121)(65,122)(66,123)(67,124)(68,125)(69,126)(70,127)(71,128)(72,129)(73,130)(74,131)(75,132)(91,232)(92,233)(93,234)(94,235)(95,236)(96,237)(97,238)(98,239)(99,240)(100,226)(101,227)(102,228)(103,229)(104,230)(105,231)(166,217)(167,218)(168,219)(169,220)(170,221)(171,222)(172,223)(173,224)(174,225)(175,211)(176,212)(177,213)(178,214)(179,215)(180,216), (1,51,161,26)(2,52,162,27)(3,53,163,28)(4,54,164,29)(5,55,165,30)(6,56,151,16)(7,57,152,17)(8,58,153,18)(9,59,154,19)(10,60,155,20)(11,46,156,21)(12,47,157,22)(13,48,158,23)(14,49,159,24)(15,50,160,25)(31,224,109,173)(32,225,110,174)(33,211,111,175)(34,212,112,176)(35,213,113,177)(36,214,114,178)(37,215,115,179)(38,216,116,180)(39,217,117,166)(40,218,118,167)(41,219,119,168)(42,220,120,169)(43,221,106,170)(44,222,107,171)(45,223,108,172)(61,240,133,99)(62,226,134,100)(63,227,135,101)(64,228,121,102)(65,229,122,103)(66,230,123,104)(67,231,124,105)(68,232,125,91)(69,233,126,92)(70,234,127,93)(71,235,128,94)(72,236,129,95)(73,237,130,96)(74,238,131,97)(75,239,132,98)(76,182,149,209)(77,183,150,210)(78,184,136,196)(79,185,137,197)(80,186,138,198)(81,187,139,199)(82,188,140,200)(83,189,141,201)(84,190,142,202)(85,191,143,203)(86,192,144,204)(87,193,145,205)(88,194,146,206)(89,195,147,207)(90,181,148,208), (1,147,161,89)(2,148,162,90)(3,149,163,76)(4,150,164,77)(5,136,165,78)(6,137,151,79)(7,138,152,80)(8,139,153,81)(9,140,154,82)(10,141,155,83)(11,142,156,84)(12,143,157,85)(13,144,158,86)(14,145,159,87)(15,146,160,88)(16,197,56,185)(17,198,57,186)(18,199,58,187)(19,200,59,188)(20,201,60,189)(21,202,46,190)(22,203,47,191)(23,204,48,192)(24,205,49,193)(25,206,50,194)(26,207,51,195)(27,208,52,181)(28,209,53,182)(29,210,54,183)(30,196,55,184)(31,97,109,238)(32,98,110,239)(33,99,111,240)(34,100,112,226)(35,101,113,227)(36,102,114,228)(37,103,115,229)(38,104,116,230)(39,105,117,231)(40,91,118,232)(41,92,119,233)(42,93,120,234)(43,94,106,235)(44,95,107,236)(45,96,108,237)(61,211,133,175)(62,212,134,176)(63,213,135,177)(64,214,121,178)(65,215,122,179)(66,216,123,180)(67,217,124,166)(68,218,125,167)(69,219,126,168)(70,220,127,169)(71,221,128,170)(72,222,129,171)(73,223,130,172)(74,224,131,173)(75,225,132,174)>;
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), (1,74,161,131)(2,75,162,132)(3,61,163,133)(4,62,164,134)(5,63,165,135)(6,64,151,121)(7,65,152,122)(8,66,153,123)(9,67,154,124)(10,68,155,125)(11,69,156,126)(12,70,157,127)(13,71,158,128)(14,72,159,129)(15,73,160,130)(16,102,56,228)(17,103,57,229)(18,104,58,230)(19,105,59,231)(20,91,60,232)(21,92,46,233)(22,93,47,234)(23,94,48,235)(24,95,49,236)(25,96,50,237)(26,97,51,238)(27,98,52,239)(28,99,53,240)(29,100,54,226)(30,101,55,227)(31,207,109,195)(32,208,110,181)(33,209,111,182)(34,210,112,183)(35,196,113,184)(36,197,114,185)(37,198,115,186)(38,199,116,187)(39,200,117,188)(40,201,118,189)(41,202,119,190)(42,203,120,191)(43,204,106,192)(44,205,107,193)(45,206,108,194)(76,175,149,211)(77,176,150,212)(78,177,136,213)(79,178,137,214)(80,179,138,215)(81,180,139,216)(82,166,140,217)(83,167,141,218)(84,168,142,219)(85,169,143,220)(86,170,144,221)(87,171,145,222)(88,172,146,223)(89,173,147,224)(90,174,148,225), (31,109)(32,110)(33,111)(34,112)(35,113)(36,114)(37,115)(38,116)(39,117)(40,118)(41,119)(42,120)(43,106)(44,107)(45,108)(61,133)(62,134)(63,135)(64,121)(65,122)(66,123)(67,124)(68,125)(69,126)(70,127)(71,128)(72,129)(73,130)(74,131)(75,132)(91,232)(92,233)(93,234)(94,235)(95,236)(96,237)(97,238)(98,239)(99,240)(100,226)(101,227)(102,228)(103,229)(104,230)(105,231)(166,217)(167,218)(168,219)(169,220)(170,221)(171,222)(172,223)(173,224)(174,225)(175,211)(176,212)(177,213)(178,214)(179,215)(180,216), (1,51,161,26)(2,52,162,27)(3,53,163,28)(4,54,164,29)(5,55,165,30)(6,56,151,16)(7,57,152,17)(8,58,153,18)(9,59,154,19)(10,60,155,20)(11,46,156,21)(12,47,157,22)(13,48,158,23)(14,49,159,24)(15,50,160,25)(31,224,109,173)(32,225,110,174)(33,211,111,175)(34,212,112,176)(35,213,113,177)(36,214,114,178)(37,215,115,179)(38,216,116,180)(39,217,117,166)(40,218,118,167)(41,219,119,168)(42,220,120,169)(43,221,106,170)(44,222,107,171)(45,223,108,172)(61,240,133,99)(62,226,134,100)(63,227,135,101)(64,228,121,102)(65,229,122,103)(66,230,123,104)(67,231,124,105)(68,232,125,91)(69,233,126,92)(70,234,127,93)(71,235,128,94)(72,236,129,95)(73,237,130,96)(74,238,131,97)(75,239,132,98)(76,182,149,209)(77,183,150,210)(78,184,136,196)(79,185,137,197)(80,186,138,198)(81,187,139,199)(82,188,140,200)(83,189,141,201)(84,190,142,202)(85,191,143,203)(86,192,144,204)(87,193,145,205)(88,194,146,206)(89,195,147,207)(90,181,148,208), (1,147,161,89)(2,148,162,90)(3,149,163,76)(4,150,164,77)(5,136,165,78)(6,137,151,79)(7,138,152,80)(8,139,153,81)(9,140,154,82)(10,141,155,83)(11,142,156,84)(12,143,157,85)(13,144,158,86)(14,145,159,87)(15,146,160,88)(16,197,56,185)(17,198,57,186)(18,199,58,187)(19,200,59,188)(20,201,60,189)(21,202,46,190)(22,203,47,191)(23,204,48,192)(24,205,49,193)(25,206,50,194)(26,207,51,195)(27,208,52,181)(28,209,53,182)(29,210,54,183)(30,196,55,184)(31,97,109,238)(32,98,110,239)(33,99,111,240)(34,100,112,226)(35,101,113,227)(36,102,114,228)(37,103,115,229)(38,104,116,230)(39,105,117,231)(40,91,118,232)(41,92,119,233)(42,93,120,234)(43,94,106,235)(44,95,107,236)(45,96,108,237)(61,211,133,175)(62,212,134,176)(63,213,135,177)(64,214,121,178)(65,215,122,179)(66,216,123,180)(67,217,124,166)(68,218,125,167)(69,219,126,168)(70,220,127,169)(71,221,128,170)(72,222,129,171)(73,223,130,172)(74,224,131,173)(75,225,132,174) );
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)], [(1,74,161,131),(2,75,162,132),(3,61,163,133),(4,62,164,134),(5,63,165,135),(6,64,151,121),(7,65,152,122),(8,66,153,123),(9,67,154,124),(10,68,155,125),(11,69,156,126),(12,70,157,127),(13,71,158,128),(14,72,159,129),(15,73,160,130),(16,102,56,228),(17,103,57,229),(18,104,58,230),(19,105,59,231),(20,91,60,232),(21,92,46,233),(22,93,47,234),(23,94,48,235),(24,95,49,236),(25,96,50,237),(26,97,51,238),(27,98,52,239),(28,99,53,240),(29,100,54,226),(30,101,55,227),(31,207,109,195),(32,208,110,181),(33,209,111,182),(34,210,112,183),(35,196,113,184),(36,197,114,185),(37,198,115,186),(38,199,116,187),(39,200,117,188),(40,201,118,189),(41,202,119,190),(42,203,120,191),(43,204,106,192),(44,205,107,193),(45,206,108,194),(76,175,149,211),(77,176,150,212),(78,177,136,213),(79,178,137,214),(80,179,138,215),(81,180,139,216),(82,166,140,217),(83,167,141,218),(84,168,142,219),(85,169,143,220),(86,170,144,221),(87,171,145,222),(88,172,146,223),(89,173,147,224),(90,174,148,225)], [(31,109),(32,110),(33,111),(34,112),(35,113),(36,114),(37,115),(38,116),(39,117),(40,118),(41,119),(42,120),(43,106),(44,107),(45,108),(61,133),(62,134),(63,135),(64,121),(65,122),(66,123),(67,124),(68,125),(69,126),(70,127),(71,128),(72,129),(73,130),(74,131),(75,132),(91,232),(92,233),(93,234),(94,235),(95,236),(96,237),(97,238),(98,239),(99,240),(100,226),(101,227),(102,228),(103,229),(104,230),(105,231),(166,217),(167,218),(168,219),(169,220),(170,221),(171,222),(172,223),(173,224),(174,225),(175,211),(176,212),(177,213),(178,214),(179,215),(180,216)], [(1,51,161,26),(2,52,162,27),(3,53,163,28),(4,54,164,29),(5,55,165,30),(6,56,151,16),(7,57,152,17),(8,58,153,18),(9,59,154,19),(10,60,155,20),(11,46,156,21),(12,47,157,22),(13,48,158,23),(14,49,159,24),(15,50,160,25),(31,224,109,173),(32,225,110,174),(33,211,111,175),(34,212,112,176),(35,213,113,177),(36,214,114,178),(37,215,115,179),(38,216,116,180),(39,217,117,166),(40,218,118,167),(41,219,119,168),(42,220,120,169),(43,221,106,170),(44,222,107,171),(45,223,108,172),(61,240,133,99),(62,226,134,100),(63,227,135,101),(64,228,121,102),(65,229,122,103),(66,230,123,104),(67,231,124,105),(68,232,125,91),(69,233,126,92),(70,234,127,93),(71,235,128,94),(72,236,129,95),(73,237,130,96),(74,238,131,97),(75,239,132,98),(76,182,149,209),(77,183,150,210),(78,184,136,196),(79,185,137,197),(80,186,138,198),(81,187,139,199),(82,188,140,200),(83,189,141,201),(84,190,142,202),(85,191,143,203),(86,192,144,204),(87,193,145,205),(88,194,146,206),(89,195,147,207),(90,181,148,208)], [(1,147,161,89),(2,148,162,90),(3,149,163,76),(4,150,164,77),(5,136,165,78),(6,137,151,79),(7,138,152,80),(8,139,153,81),(9,140,154,82),(10,141,155,83),(11,142,156,84),(12,143,157,85),(13,144,158,86),(14,145,159,87),(15,146,160,88),(16,197,56,185),(17,198,57,186),(18,199,58,187),(19,200,59,188),(20,201,60,189),(21,202,46,190),(22,203,47,191),(23,204,48,192),(24,205,49,193),(25,206,50,194),(26,207,51,195),(27,208,52,181),(28,209,53,182),(29,210,54,183),(30,196,55,184),(31,97,109,238),(32,98,110,239),(33,99,111,240),(34,100,112,226),(35,101,113,227),(36,102,114,228),(37,103,115,229),(38,104,116,230),(39,105,117,231),(40,91,118,232),(41,92,119,233),(42,93,120,234),(43,94,106,235),(44,95,107,236),(45,96,108,237),(61,211,133,175),(62,212,134,176),(63,213,135,177),(64,214,121,178),(65,215,122,179),(66,216,123,180),(67,217,124,166),(68,218,125,167),(69,219,126,168),(70,220,127,169),(71,221,128,170),(72,222,129,171),(73,223,130,172),(74,224,131,173),(75,225,132,174)]])
255 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2F | 3A | 3B | 4A | ··· | 4J | 5A | 5B | 5C | 5D | 6A | 6B | 6C | ··· | 6L | 10A | 10B | 10C | 10D | 10E | ··· | 10X | 12A | ··· | 12T | 15A | ··· | 15H | 20A | ··· | 20AN | 30A | ··· | 30H | 30I | ··· | 30AV | 60A | ··· | 60CB |
| order | 1 | 2 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | ··· | 6 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
255 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C3 | C5 | C6 | C6 | C10 | C10 | C15 | C30 | C30 | 2- 1+4 | C3×2- 1+4 | C5×2- 1+4 | C15×2- 1+4 |
| kernel | C15×2- 1+4 | Q8×C30 | C15×C4○D4 | C5×2- 1+4 | C3×2- 1+4 | Q8×C10 | C5×C4○D4 | C6×Q8 | C3×C4○D4 | 2- 1+4 | C2×Q8 | C4○D4 | C15 | C5 | C3 | C1 |
| # reps | 1 | 5 | 10 | 2 | 4 | 10 | 20 | 20 | 40 | 8 | 40 | 80 | 1 | 2 | 4 | 8 |
Matrix representation of C15×2- 1+4 ►in GL4(𝔽61) generated by
| 25 | 0 | 0 | 0 |
| 0 | 25 | 0 | 0 |
| 0 | 0 | 25 | 0 |
| 0 | 0 | 0 | 25 |
| 0 | 0 | 59 | 19 |
| 0 | 0 | 19 | 2 |
| 59 | 19 | 0 | 0 |
| 19 | 2 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 1 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 1 | 0 |
| 42 | 59 | 0 | 0 |
| 59 | 19 | 0 | 0 |
| 0 | 0 | 19 | 2 |
| 0 | 0 | 2 | 42 |
G:=sub<GL(4,GF(61))| [25,0,0,0,0,25,0,0,0,0,25,0,0,0,0,25],[0,0,59,19,0,0,19,2,59,19,0,0,19,2,0,0],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[0,60,0,0,1,0,0,0,0,0,0,1,0,0,60,0],[42,59,0,0,59,19,0,0,0,0,19,2,0,0,2,42] >;
C15×2- 1+4 in GAP, Magma, Sage, TeX
C_{15}\times 2_-^{1+4} % in TeX
G:=Group("C15xES-(2,2)"); // GroupNames label
G:=SmallGroup(480,1185);
// by ID
G=gap.SmallGroup(480,1185);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-5,-2,3389,1688,2571,1276,6947]);
// Polycyclic
G:=Group<a,b,c,d,e|a^15=b^4=c^2=1,d^2=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,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