direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C15×2- (1+4), C30.103C24, C60.300C23, C4○D4⋊6C30, (C2×Q8)⋊7C30, (C6×Q8)⋊12C10, (Q8×C30)⋊26C2, (Q8×C10)⋊16C6, D4.4(C2×C30), Q8.7(C2×C30), C2.5(C23×C30), C6.20(C23×C10), C10.20(C23×C6), C20.53(C22×C6), C4.10(C22×C30), C12.52(C22×C10), (C2×C60).442C22, (C2×C30).264C23, (D4×C15).53C22, C22.3(C22×C30), (Q8×C15).58C22, (C3×C4○D4)⋊9C10, (C5×C4○D4)⋊13C6, (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×C6).8(C22×C10), (C2×C10).8(C22×C6), (C3×Q8).15(C2×C10), SmallGroup(480,1185)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 312 in 292 conjugacy classes, 272 normal (12 characteristic)
C1, C2, C2 [×5], C3, C4 [×10], C22 [×5], C5, C6, C6 [×5], C2×C4 [×15], D4 [×10], Q8 [×10], C10, C10 [×5], C12 [×10], C2×C6 [×5], C15, C2×Q8 [×5], C4○D4 [×10], C20 [×10], C2×C10 [×5], C2×C12 [×15], C3×D4 [×10], C3×Q8 [×10], C30, C30 [×5], 2- (1+4), C2×C20 [×15], C5×D4 [×10], C5×Q8 [×10], C6×Q8 [×5], C3×C4○D4 [×10], C60 [×10], C2×C30 [×5], Q8×C10 [×5], C5×C4○D4 [×10], C3×2- (1+4), C2×C60 [×15], D4×C15 [×10], Q8×C15 [×10], C5×2- (1+4), Q8×C30 [×5], C15×C4○D4 [×10], C15×2- (1+4)
Quotients:
C1, C2 [×15], C3, C22 [×35], C5, C6 [×15], C23 [×15], C10 [×15], C2×C6 [×35], C15, C24, C2×C10 [×35], C22×C6 [×15], C30 [×15], 2- (1+4), C22×C10 [×15], C23×C6, C2×C30 [×35], C23×C10, C3×2- (1+4), C22×C30 [×15], C5×2- (1+4), C23×C30, C15×2- (1+4)
Generators and relations
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 >
►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 29 188 76)(2 30 189 77)(3 16 190 78)(4 17 191 79)(5 18 192 80)(6 19 193 81)(7 20 194 82)(8 21 195 83)(9 22 181 84)(10 23 182 85)(11 24 183 86)(12 25 184 87)(13 26 185 88)(14 27 186 89)(15 28 187 90)(31 91 169 198)(32 92 170 199)(33 93 171 200)(34 94 172 201)(35 95 173 202)(36 96 174 203)(37 97 175 204)(38 98 176 205)(39 99 177 206)(40 100 178 207)(41 101 179 208)(42 102 180 209)(43 103 166 210)(44 104 167 196)(45 105 168 197)(46 75 122 120)(47 61 123 106)(48 62 124 107)(49 63 125 108)(50 64 126 109)(51 65 127 110)(52 66 128 111)(53 67 129 112)(54 68 130 113)(55 69 131 114)(56 70 132 115)(57 71 133 116)(58 72 134 117)(59 73 135 118)(60 74 121 119)(136 228 157 216)(137 229 158 217)(138 230 159 218)(139 231 160 219)(140 232 161 220)(141 233 162 221)(142 234 163 222)(143 235 164 223)(144 236 165 224)(145 237 151 225)(146 238 152 211)(147 239 153 212)(148 240 154 213)(149 226 155 214)(150 227 156 215)
(16 78)(17 79)(18 80)(19 81)(20 82)(21 83)(22 84)(23 85)(24 86)(25 87)(26 88)(27 89)(28 90)(29 76)(30 77)(31 169)(32 170)(33 171)(34 172)(35 173)(36 174)(37 175)(38 176)(39 177)(40 178)(41 179)(42 180)(43 166)(44 167)(45 168)(61 106)(62 107)(63 108)(64 109)(65 110)(66 111)(67 112)(68 113)(69 114)(70 115)(71 116)(72 117)(73 118)(74 119)(75 120)(136 157)(137 158)(138 159)(139 160)(140 161)(141 162)(142 163)(143 164)(144 165)(145 151)(146 152)(147 153)(148 154)(149 155)(150 156)
(1 132 188 56)(2 133 189 57)(3 134 190 58)(4 135 191 59)(5 121 192 60)(6 122 193 46)(7 123 194 47)(8 124 195 48)(9 125 181 49)(10 126 182 50)(11 127 183 51)(12 128 184 52)(13 129 185 53)(14 130 186 54)(15 131 187 55)(16 117 78 72)(17 118 79 73)(18 119 80 74)(19 120 81 75)(20 106 82 61)(21 107 83 62)(22 108 84 63)(23 109 85 64)(24 110 86 65)(25 111 87 66)(26 112 88 67)(27 113 89 68)(28 114 90 69)(29 115 76 70)(30 116 77 71)(31 162 169 141)(32 163 170 142)(33 164 171 143)(34 165 172 144)(35 151 173 145)(36 152 174 146)(37 153 175 147)(38 154 176 148)(39 155 177 149)(40 156 178 150)(41 157 179 136)(42 158 180 137)(43 159 166 138)(44 160 167 139)(45 161 168 140)(91 221 198 233)(92 222 199 234)(93 223 200 235)(94 224 201 236)(95 225 202 237)(96 211 203 238)(97 212 204 239)(98 213 205 240)(99 214 206 226)(100 215 207 227)(101 216 208 228)(102 217 209 229)(103 218 210 230)(104 219 196 231)(105 220 197 232)
(1 215 188 227)(2 216 189 228)(3 217 190 229)(4 218 191 230)(5 219 192 231)(6 220 193 232)(7 221 194 233)(8 222 195 234)(9 223 181 235)(10 224 182 236)(11 225 183 237)(12 211 184 238)(13 212 185 239)(14 213 186 240)(15 214 187 226)(16 137 78 158)(17 138 79 159)(18 139 80 160)(19 140 81 161)(20 141 82 162)(21 142 83 163)(22 143 84 164)(23 144 85 165)(24 145 86 151)(25 146 87 152)(26 147 88 153)(27 148 89 154)(28 149 90 155)(29 150 76 156)(30 136 77 157)(31 106 169 61)(32 107 170 62)(33 108 171 63)(34 109 172 64)(35 110 173 65)(36 111 174 66)(37 112 175 67)(38 113 176 68)(39 114 177 69)(40 115 178 70)(41 116 179 71)(42 117 180 72)(43 118 166 73)(44 119 167 74)(45 120 168 75)(46 197 122 105)(47 198 123 91)(48 199 124 92)(49 200 125 93)(50 201 126 94)(51 202 127 95)(52 203 128 96)(53 204 129 97)(54 205 130 98)(55 206 131 99)(56 207 132 100)(57 208 133 101)(58 209 134 102)(59 210 135 103)(60 196 121 104)
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), (1,29,188,76)(2,30,189,77)(3,16,190,78)(4,17,191,79)(5,18,192,80)(6,19,193,81)(7,20,194,82)(8,21,195,83)(9,22,181,84)(10,23,182,85)(11,24,183,86)(12,25,184,87)(13,26,185,88)(14,27,186,89)(15,28,187,90)(31,91,169,198)(32,92,170,199)(33,93,171,200)(34,94,172,201)(35,95,173,202)(36,96,174,203)(37,97,175,204)(38,98,176,205)(39,99,177,206)(40,100,178,207)(41,101,179,208)(42,102,180,209)(43,103,166,210)(44,104,167,196)(45,105,168,197)(46,75,122,120)(47,61,123,106)(48,62,124,107)(49,63,125,108)(50,64,126,109)(51,65,127,110)(52,66,128,111)(53,67,129,112)(54,68,130,113)(55,69,131,114)(56,70,132,115)(57,71,133,116)(58,72,134,117)(59,73,135,118)(60,74,121,119)(136,228,157,216)(137,229,158,217)(138,230,159,218)(139,231,160,219)(140,232,161,220)(141,233,162,221)(142,234,163,222)(143,235,164,223)(144,236,165,224)(145,237,151,225)(146,238,152,211)(147,239,153,212)(148,240,154,213)(149,226,155,214)(150,227,156,215), (16,78)(17,79)(18,80)(19,81)(20,82)(21,83)(22,84)(23,85)(24,86)(25,87)(26,88)(27,89)(28,90)(29,76)(30,77)(31,169)(32,170)(33,171)(34,172)(35,173)(36,174)(37,175)(38,176)(39,177)(40,178)(41,179)(42,180)(43,166)(44,167)(45,168)(61,106)(62,107)(63,108)(64,109)(65,110)(66,111)(67,112)(68,113)(69,114)(70,115)(71,116)(72,117)(73,118)(74,119)(75,120)(136,157)(137,158)(138,159)(139,160)(140,161)(141,162)(142,163)(143,164)(144,165)(145,151)(146,152)(147,153)(148,154)(149,155)(150,156), (1,132,188,56)(2,133,189,57)(3,134,190,58)(4,135,191,59)(5,121,192,60)(6,122,193,46)(7,123,194,47)(8,124,195,48)(9,125,181,49)(10,126,182,50)(11,127,183,51)(12,128,184,52)(13,129,185,53)(14,130,186,54)(15,131,187,55)(16,117,78,72)(17,118,79,73)(18,119,80,74)(19,120,81,75)(20,106,82,61)(21,107,83,62)(22,108,84,63)(23,109,85,64)(24,110,86,65)(25,111,87,66)(26,112,88,67)(27,113,89,68)(28,114,90,69)(29,115,76,70)(30,116,77,71)(31,162,169,141)(32,163,170,142)(33,164,171,143)(34,165,172,144)(35,151,173,145)(36,152,174,146)(37,153,175,147)(38,154,176,148)(39,155,177,149)(40,156,178,150)(41,157,179,136)(42,158,180,137)(43,159,166,138)(44,160,167,139)(45,161,168,140)(91,221,198,233)(92,222,199,234)(93,223,200,235)(94,224,201,236)(95,225,202,237)(96,211,203,238)(97,212,204,239)(98,213,205,240)(99,214,206,226)(100,215,207,227)(101,216,208,228)(102,217,209,229)(103,218,210,230)(104,219,196,231)(105,220,197,232), (1,215,188,227)(2,216,189,228)(3,217,190,229)(4,218,191,230)(5,219,192,231)(6,220,193,232)(7,221,194,233)(8,222,195,234)(9,223,181,235)(10,224,182,236)(11,225,183,237)(12,211,184,238)(13,212,185,239)(14,213,186,240)(15,214,187,226)(16,137,78,158)(17,138,79,159)(18,139,80,160)(19,140,81,161)(20,141,82,162)(21,142,83,163)(22,143,84,164)(23,144,85,165)(24,145,86,151)(25,146,87,152)(26,147,88,153)(27,148,89,154)(28,149,90,155)(29,150,76,156)(30,136,77,157)(31,106,169,61)(32,107,170,62)(33,108,171,63)(34,109,172,64)(35,110,173,65)(36,111,174,66)(37,112,175,67)(38,113,176,68)(39,114,177,69)(40,115,178,70)(41,116,179,71)(42,117,180,72)(43,118,166,73)(44,119,167,74)(45,120,168,75)(46,197,122,105)(47,198,123,91)(48,199,124,92)(49,200,125,93)(50,201,126,94)(51,202,127,95)(52,203,128,96)(53,204,129,97)(54,205,130,98)(55,206,131,99)(56,207,132,100)(57,208,133,101)(58,209,134,102)(59,210,135,103)(60,196,121,104)>;
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), (1,29,188,76)(2,30,189,77)(3,16,190,78)(4,17,191,79)(5,18,192,80)(6,19,193,81)(7,20,194,82)(8,21,195,83)(9,22,181,84)(10,23,182,85)(11,24,183,86)(12,25,184,87)(13,26,185,88)(14,27,186,89)(15,28,187,90)(31,91,169,198)(32,92,170,199)(33,93,171,200)(34,94,172,201)(35,95,173,202)(36,96,174,203)(37,97,175,204)(38,98,176,205)(39,99,177,206)(40,100,178,207)(41,101,179,208)(42,102,180,209)(43,103,166,210)(44,104,167,196)(45,105,168,197)(46,75,122,120)(47,61,123,106)(48,62,124,107)(49,63,125,108)(50,64,126,109)(51,65,127,110)(52,66,128,111)(53,67,129,112)(54,68,130,113)(55,69,131,114)(56,70,132,115)(57,71,133,116)(58,72,134,117)(59,73,135,118)(60,74,121,119)(136,228,157,216)(137,229,158,217)(138,230,159,218)(139,231,160,219)(140,232,161,220)(141,233,162,221)(142,234,163,222)(143,235,164,223)(144,236,165,224)(145,237,151,225)(146,238,152,211)(147,239,153,212)(148,240,154,213)(149,226,155,214)(150,227,156,215), (16,78)(17,79)(18,80)(19,81)(20,82)(21,83)(22,84)(23,85)(24,86)(25,87)(26,88)(27,89)(28,90)(29,76)(30,77)(31,169)(32,170)(33,171)(34,172)(35,173)(36,174)(37,175)(38,176)(39,177)(40,178)(41,179)(42,180)(43,166)(44,167)(45,168)(61,106)(62,107)(63,108)(64,109)(65,110)(66,111)(67,112)(68,113)(69,114)(70,115)(71,116)(72,117)(73,118)(74,119)(75,120)(136,157)(137,158)(138,159)(139,160)(140,161)(141,162)(142,163)(143,164)(144,165)(145,151)(146,152)(147,153)(148,154)(149,155)(150,156), (1,132,188,56)(2,133,189,57)(3,134,190,58)(4,135,191,59)(5,121,192,60)(6,122,193,46)(7,123,194,47)(8,124,195,48)(9,125,181,49)(10,126,182,50)(11,127,183,51)(12,128,184,52)(13,129,185,53)(14,130,186,54)(15,131,187,55)(16,117,78,72)(17,118,79,73)(18,119,80,74)(19,120,81,75)(20,106,82,61)(21,107,83,62)(22,108,84,63)(23,109,85,64)(24,110,86,65)(25,111,87,66)(26,112,88,67)(27,113,89,68)(28,114,90,69)(29,115,76,70)(30,116,77,71)(31,162,169,141)(32,163,170,142)(33,164,171,143)(34,165,172,144)(35,151,173,145)(36,152,174,146)(37,153,175,147)(38,154,176,148)(39,155,177,149)(40,156,178,150)(41,157,179,136)(42,158,180,137)(43,159,166,138)(44,160,167,139)(45,161,168,140)(91,221,198,233)(92,222,199,234)(93,223,200,235)(94,224,201,236)(95,225,202,237)(96,211,203,238)(97,212,204,239)(98,213,205,240)(99,214,206,226)(100,215,207,227)(101,216,208,228)(102,217,209,229)(103,218,210,230)(104,219,196,231)(105,220,197,232), (1,215,188,227)(2,216,189,228)(3,217,190,229)(4,218,191,230)(5,219,192,231)(6,220,193,232)(7,221,194,233)(8,222,195,234)(9,223,181,235)(10,224,182,236)(11,225,183,237)(12,211,184,238)(13,212,185,239)(14,213,186,240)(15,214,187,226)(16,137,78,158)(17,138,79,159)(18,139,80,160)(19,140,81,161)(20,141,82,162)(21,142,83,163)(22,143,84,164)(23,144,85,165)(24,145,86,151)(25,146,87,152)(26,147,88,153)(27,148,89,154)(28,149,90,155)(29,150,76,156)(30,136,77,157)(31,106,169,61)(32,107,170,62)(33,108,171,63)(34,109,172,64)(35,110,173,65)(36,111,174,66)(37,112,175,67)(38,113,176,68)(39,114,177,69)(40,115,178,70)(41,116,179,71)(42,117,180,72)(43,118,166,73)(44,119,167,74)(45,120,168,75)(46,197,122,105)(47,198,123,91)(48,199,124,92)(49,200,125,93)(50,201,126,94)(51,202,127,95)(52,203,128,96)(53,204,129,97)(54,205,130,98)(55,206,131,99)(56,207,132,100)(57,208,133,101)(58,209,134,102)(59,210,135,103)(60,196,121,104) );
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)], [(1,29,188,76),(2,30,189,77),(3,16,190,78),(4,17,191,79),(5,18,192,80),(6,19,193,81),(7,20,194,82),(8,21,195,83),(9,22,181,84),(10,23,182,85),(11,24,183,86),(12,25,184,87),(13,26,185,88),(14,27,186,89),(15,28,187,90),(31,91,169,198),(32,92,170,199),(33,93,171,200),(34,94,172,201),(35,95,173,202),(36,96,174,203),(37,97,175,204),(38,98,176,205),(39,99,177,206),(40,100,178,207),(41,101,179,208),(42,102,180,209),(43,103,166,210),(44,104,167,196),(45,105,168,197),(46,75,122,120),(47,61,123,106),(48,62,124,107),(49,63,125,108),(50,64,126,109),(51,65,127,110),(52,66,128,111),(53,67,129,112),(54,68,130,113),(55,69,131,114),(56,70,132,115),(57,71,133,116),(58,72,134,117),(59,73,135,118),(60,74,121,119),(136,228,157,216),(137,229,158,217),(138,230,159,218),(139,231,160,219),(140,232,161,220),(141,233,162,221),(142,234,163,222),(143,235,164,223),(144,236,165,224),(145,237,151,225),(146,238,152,211),(147,239,153,212),(148,240,154,213),(149,226,155,214),(150,227,156,215)], [(16,78),(17,79),(18,80),(19,81),(20,82),(21,83),(22,84),(23,85),(24,86),(25,87),(26,88),(27,89),(28,90),(29,76),(30,77),(31,169),(32,170),(33,171),(34,172),(35,173),(36,174),(37,175),(38,176),(39,177),(40,178),(41,179),(42,180),(43,166),(44,167),(45,168),(61,106),(62,107),(63,108),(64,109),(65,110),(66,111),(67,112),(68,113),(69,114),(70,115),(71,116),(72,117),(73,118),(74,119),(75,120),(136,157),(137,158),(138,159),(139,160),(140,161),(141,162),(142,163),(143,164),(144,165),(145,151),(146,152),(147,153),(148,154),(149,155),(150,156)], [(1,132,188,56),(2,133,189,57),(3,134,190,58),(4,135,191,59),(5,121,192,60),(6,122,193,46),(7,123,194,47),(8,124,195,48),(9,125,181,49),(10,126,182,50),(11,127,183,51),(12,128,184,52),(13,129,185,53),(14,130,186,54),(15,131,187,55),(16,117,78,72),(17,118,79,73),(18,119,80,74),(19,120,81,75),(20,106,82,61),(21,107,83,62),(22,108,84,63),(23,109,85,64),(24,110,86,65),(25,111,87,66),(26,112,88,67),(27,113,89,68),(28,114,90,69),(29,115,76,70),(30,116,77,71),(31,162,169,141),(32,163,170,142),(33,164,171,143),(34,165,172,144),(35,151,173,145),(36,152,174,146),(37,153,175,147),(38,154,176,148),(39,155,177,149),(40,156,178,150),(41,157,179,136),(42,158,180,137),(43,159,166,138),(44,160,167,139),(45,161,168,140),(91,221,198,233),(92,222,199,234),(93,223,200,235),(94,224,201,236),(95,225,202,237),(96,211,203,238),(97,212,204,239),(98,213,205,240),(99,214,206,226),(100,215,207,227),(101,216,208,228),(102,217,209,229),(103,218,210,230),(104,219,196,231),(105,220,197,232)], [(1,215,188,227),(2,216,189,228),(3,217,190,229),(4,218,191,230),(5,219,192,231),(6,220,193,232),(7,221,194,233),(8,222,195,234),(9,223,181,235),(10,224,182,236),(11,225,183,237),(12,211,184,238),(13,212,185,239),(14,213,186,240),(15,214,187,226),(16,137,78,158),(17,138,79,159),(18,139,80,160),(19,140,81,161),(20,141,82,162),(21,142,83,163),(22,143,84,164),(23,144,85,165),(24,145,86,151),(25,146,87,152),(26,147,88,153),(27,148,89,154),(28,149,90,155),(29,150,76,156),(30,136,77,157),(31,106,169,61),(32,107,170,62),(33,108,171,63),(34,109,172,64),(35,110,173,65),(36,111,174,66),(37,112,175,67),(38,113,176,68),(39,114,177,69),(40,115,178,70),(41,116,179,71),(42,117,180,72),(43,118,166,73),(44,119,167,74),(45,120,168,75),(46,197,122,105),(47,198,123,91),(48,199,124,92),(49,200,125,93),(50,201,126,94),(51,202,127,95),(52,203,128,96),(53,204,129,97),(54,205,130,98),(55,206,131,99),(56,207,132,100),(57,208,133,101),(58,209,134,102),(59,210,135,103),(60,196,121,104)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀