direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q8×D15, C30.61C24, C60.85C23, D30.45C23, Dic30⋊24C22, Dic15.29C23, C6⋊3(Q8×D5), (C6×Q8)⋊5D5, C10⋊3(S3×Q8), C30⋊8(C2×Q8), (C5×Q8)⋊21D6, (Q8×C10)⋊9S3, (Q8×C30)⋊5C2, C15⋊9(C22×Q8), (C3×Q8)⋊18D10, (C2×C4).61D30, (C2×C20).171D6, C2.9(C23×D15), C6.61(C23×D5), (C2×Dic30)⋊16C2, (C2×C12).169D10, (C2×C60).87C22, C10.61(S3×C23), (Q8×C15)⋊20C22, C4.22(C22×D15), C20.135(C22×S3), (C2×C30).322C23, (C4×D15).48C22, C12.133(C22×D5), C22.31(C22×D15), (C2×Dic15).178C22, (C22×D15).128C22, C5⋊4(C2×S3×Q8), C3⋊4(C2×Q8×D5), (C2×C4×D15).5C2, (C2×C6).318(C22×D5), (C2×C10).318(C22×S3), SmallGroup(480,1172)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1492 in 312 conjugacy classes, 135 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×6], C4 [×6], C22, C22 [×6], C5, S3 [×4], C6, C6 [×2], C2×C4 [×3], C2×C4 [×15], Q8 [×4], Q8 [×12], C23, D5 [×4], C10, C10 [×2], Dic3 [×6], C12 [×6], D6 [×6], C2×C6, C15, C22×C4 [×3], C2×Q8, C2×Q8 [×11], Dic5 [×6], C20 [×6], D10 [×6], C2×C10, Dic6 [×12], C4×S3 [×12], C2×Dic3 [×3], C2×C12 [×3], C3×Q8 [×4], C22×S3, D15 [×4], C30, C30 [×2], C22×Q8, Dic10 [×12], C4×D5 [×12], C2×Dic5 [×3], C2×C20 [×3], C5×Q8 [×4], C22×D5, C2×Dic6 [×3], S3×C2×C4 [×3], S3×Q8 [×8], C6×Q8, Dic15 [×6], C60 [×6], D30 [×6], C2×C30, C2×Dic10 [×3], C2×C4×D5 [×3], Q8×D5 [×8], Q8×C10, C2×S3×Q8, Dic30 [×12], C4×D15 [×12], C2×Dic15 [×3], C2×C60 [×3], Q8×C15 [×4], C22×D15, C2×Q8×D5, C2×Dic30 [×3], C2×C4×D15 [×3], Q8×D15 [×8], Q8×C30, C2×Q8×D15
Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D5, D6 [×7], C2×Q8 [×6], C24, D10 [×7], C22×S3 [×7], D15, C22×Q8, C22×D5 [×7], S3×Q8 [×2], S3×C23, D30 [×7], Q8×D5 [×2], C23×D5, C2×S3×Q8, C22×D15 [×7], C2×Q8×D5, Q8×D15 [×2], C23×D15, C2×Q8×D15
Generators and relations
G = < a,b,c,d,e | a2=b4=d15=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
►On 240 points
Generators in S240
(1 129)(2 130)(3 131)(4 132)(5 133)(6 134)(7 135)(8 121)(9 122)(10 123)(11 124)(12 125)(13 126)(14 127)(15 128)(16 147)(17 148)(18 149)(19 150)(20 136)(21 137)(22 138)(23 139)(24 140)(25 141)(26 142)(27 143)(28 144)(29 145)(30 146)(31 159)(32 160)(33 161)(34 162)(35 163)(36 164)(37 165)(38 151)(39 152)(40 153)(41 154)(42 155)(43 156)(44 157)(45 158)(46 170)(47 171)(48 172)(49 173)(50 174)(51 175)(52 176)(53 177)(54 178)(55 179)(56 180)(57 166)(58 167)(59 168)(60 169)(61 188)(62 189)(63 190)(64 191)(65 192)(66 193)(67 194)(68 195)(69 181)(70 182)(71 183)(72 184)(73 185)(74 186)(75 187)(76 204)(77 205)(78 206)(79 207)(80 208)(81 209)(82 210)(83 196)(84 197)(85 198)(86 199)(87 200)(88 201)(89 202)(90 203)(91 218)(92 219)(93 220)(94 221)(95 222)(96 223)(97 224)(98 225)(99 211)(100 212)(101 213)(102 214)(103 215)(104 216)(105 217)(106 226)(107 227)(108 228)(109 229)(110 230)(111 231)(112 232)(113 233)(114 234)(115 235)(116 236)(117 237)(118 238)(119 239)(120 240)
(1 54 28 39)(2 55 29 40)(3 56 30 41)(4 57 16 42)(5 58 17 43)(6 59 18 44)(7 60 19 45)(8 46 20 31)(9 47 21 32)(10 48 22 33)(11 49 23 34)(12 50 24 35)(13 51 25 36)(14 52 26 37)(15 53 27 38)(61 95 87 113)(62 96 88 114)(63 97 89 115)(64 98 90 116)(65 99 76 117)(66 100 77 118)(67 101 78 119)(68 102 79 120)(69 103 80 106)(70 104 81 107)(71 105 82 108)(72 91 83 109)(73 92 84 110)(74 93 85 111)(75 94 86 112)(121 170 136 159)(122 171 137 160)(123 172 138 161)(124 173 139 162)(125 174 140 163)(126 175 141 164)(127 176 142 165)(128 177 143 151)(129 178 144 152)(130 179 145 153)(131 180 146 154)(132 166 147 155)(133 167 148 156)(134 168 149 157)(135 169 150 158)(181 215 208 226)(182 216 209 227)(183 217 210 228)(184 218 196 229)(185 219 197 230)(186 220 198 231)(187 221 199 232)(188 222 200 233)(189 223 201 234)(190 224 202 235)(191 225 203 236)(192 211 204 237)(193 212 205 238)(194 213 206 239)(195 214 207 240)
(1 197 28 185)(2 198 29 186)(3 199 30 187)(4 200 16 188)(5 201 17 189)(6 202 18 190)(7 203 19 191)(8 204 20 192)(9 205 21 193)(10 206 22 194)(11 207 23 195)(12 208 24 181)(13 209 25 182)(14 210 26 183)(15 196 27 184)(31 237 46 211)(32 238 47 212)(33 239 48 213)(34 240 49 214)(35 226 50 215)(36 227 51 216)(37 228 52 217)(38 229 53 218)(39 230 54 219)(40 231 55 220)(41 232 56 221)(42 233 57 222)(43 234 58 223)(44 235 59 224)(45 236 60 225)(61 132 87 147)(62 133 88 148)(63 134 89 149)(64 135 90 150)(65 121 76 136)(66 122 77 137)(67 123 78 138)(68 124 79 139)(69 125 80 140)(70 126 81 141)(71 127 82 142)(72 128 83 143)(73 129 84 144)(74 130 85 145)(75 131 86 146)(91 151 109 177)(92 152 110 178)(93 153 111 179)(94 154 112 180)(95 155 113 166)(96 156 114 167)(97 157 115 168)(98 158 116 169)(99 159 117 170)(100 160 118 171)(101 161 119 172)(102 162 120 173)(103 163 106 174)(104 164 107 175)(105 165 108 176)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 24)(17 23)(18 22)(19 21)(25 30)(26 29)(27 28)(32 45)(33 44)(34 43)(35 42)(36 41)(37 40)(38 39)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)(61 69)(62 68)(63 67)(64 66)(70 75)(71 74)(72 73)(77 90)(78 89)(79 88)(80 87)(81 86)(82 85)(83 84)(91 92)(93 105)(94 104)(95 103)(96 102)(97 101)(98 100)(106 113)(107 112)(108 111)(109 110)(114 120)(115 119)(116 118)(122 135)(123 134)(124 133)(125 132)(126 131)(127 130)(128 129)(137 150)(138 149)(139 148)(140 147)(141 146)(142 145)(143 144)(151 152)(153 165)(154 164)(155 163)(156 162)(157 161)(158 160)(166 174)(167 173)(168 172)(169 171)(175 180)(176 179)(177 178)(181 188)(182 187)(183 186)(184 185)(189 195)(190 194)(191 193)(196 197)(198 210)(199 209)(200 208)(201 207)(202 206)(203 205)(212 225)(213 224)(214 223)(215 222)(216 221)(217 220)(218 219)(226 233)(227 232)(228 231)(229 230)(234 240)(235 239)(236 238)
G:=sub<Sym(240)| (1,129)(2,130)(3,131)(4,132)(5,133)(6,134)(7,135)(8,121)(9,122)(10,123)(11,124)(12,125)(13,126)(14,127)(15,128)(16,147)(17,148)(18,149)(19,150)(20,136)(21,137)(22,138)(23,139)(24,140)(25,141)(26,142)(27,143)(28,144)(29,145)(30,146)(31,159)(32,160)(33,161)(34,162)(35,163)(36,164)(37,165)(38,151)(39,152)(40,153)(41,154)(42,155)(43,156)(44,157)(45,158)(46,170)(47,171)(48,172)(49,173)(50,174)(51,175)(52,176)(53,177)(54,178)(55,179)(56,180)(57,166)(58,167)(59,168)(60,169)(61,188)(62,189)(63,190)(64,191)(65,192)(66,193)(67,194)(68,195)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,204)(77,205)(78,206)(79,207)(80,208)(81,209)(82,210)(83,196)(84,197)(85,198)(86,199)(87,200)(88,201)(89,202)(90,203)(91,218)(92,219)(93,220)(94,221)(95,222)(96,223)(97,224)(98,225)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,226)(107,227)(108,228)(109,229)(110,230)(111,231)(112,232)(113,233)(114,234)(115,235)(116,236)(117,237)(118,238)(119,239)(120,240), (1,54,28,39)(2,55,29,40)(3,56,30,41)(4,57,16,42)(5,58,17,43)(6,59,18,44)(7,60,19,45)(8,46,20,31)(9,47,21,32)(10,48,22,33)(11,49,23,34)(12,50,24,35)(13,51,25,36)(14,52,26,37)(15,53,27,38)(61,95,87,113)(62,96,88,114)(63,97,89,115)(64,98,90,116)(65,99,76,117)(66,100,77,118)(67,101,78,119)(68,102,79,120)(69,103,80,106)(70,104,81,107)(71,105,82,108)(72,91,83,109)(73,92,84,110)(74,93,85,111)(75,94,86,112)(121,170,136,159)(122,171,137,160)(123,172,138,161)(124,173,139,162)(125,174,140,163)(126,175,141,164)(127,176,142,165)(128,177,143,151)(129,178,144,152)(130,179,145,153)(131,180,146,154)(132,166,147,155)(133,167,148,156)(134,168,149,157)(135,169,150,158)(181,215,208,226)(182,216,209,227)(183,217,210,228)(184,218,196,229)(185,219,197,230)(186,220,198,231)(187,221,199,232)(188,222,200,233)(189,223,201,234)(190,224,202,235)(191,225,203,236)(192,211,204,237)(193,212,205,238)(194,213,206,239)(195,214,207,240), (1,197,28,185)(2,198,29,186)(3,199,30,187)(4,200,16,188)(5,201,17,189)(6,202,18,190)(7,203,19,191)(8,204,20,192)(9,205,21,193)(10,206,22,194)(11,207,23,195)(12,208,24,181)(13,209,25,182)(14,210,26,183)(15,196,27,184)(31,237,46,211)(32,238,47,212)(33,239,48,213)(34,240,49,214)(35,226,50,215)(36,227,51,216)(37,228,52,217)(38,229,53,218)(39,230,54,219)(40,231,55,220)(41,232,56,221)(42,233,57,222)(43,234,58,223)(44,235,59,224)(45,236,60,225)(61,132,87,147)(62,133,88,148)(63,134,89,149)(64,135,90,150)(65,121,76,136)(66,122,77,137)(67,123,78,138)(68,124,79,139)(69,125,80,140)(70,126,81,141)(71,127,82,142)(72,128,83,143)(73,129,84,144)(74,130,85,145)(75,131,86,146)(91,151,109,177)(92,152,110,178)(93,153,111,179)(94,154,112,180)(95,155,113,166)(96,156,114,167)(97,157,115,168)(98,158,116,169)(99,159,117,170)(100,160,118,171)(101,161,119,172)(102,162,120,173)(103,163,106,174)(104,164,107,175)(105,165,108,176), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,24)(17,23)(18,22)(19,21)(25,30)(26,29)(27,28)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(61,69)(62,68)(63,67)(64,66)(70,75)(71,74)(72,73)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,84)(91,92)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100)(106,113)(107,112)(108,111)(109,110)(114,120)(115,119)(116,118)(122,135)(123,134)(124,133)(125,132)(126,131)(127,130)(128,129)(137,150)(138,149)(139,148)(140,147)(141,146)(142,145)(143,144)(151,152)(153,165)(154,164)(155,163)(156,162)(157,161)(158,160)(166,174)(167,173)(168,172)(169,171)(175,180)(176,179)(177,178)(181,188)(182,187)(183,186)(184,185)(189,195)(190,194)(191,193)(196,197)(198,210)(199,209)(200,208)(201,207)(202,206)(203,205)(212,225)(213,224)(214,223)(215,222)(216,221)(217,220)(218,219)(226,233)(227,232)(228,231)(229,230)(234,240)(235,239)(236,238)>;
G:=Group( (1,129)(2,130)(3,131)(4,132)(5,133)(6,134)(7,135)(8,121)(9,122)(10,123)(11,124)(12,125)(13,126)(14,127)(15,128)(16,147)(17,148)(18,149)(19,150)(20,136)(21,137)(22,138)(23,139)(24,140)(25,141)(26,142)(27,143)(28,144)(29,145)(30,146)(31,159)(32,160)(33,161)(34,162)(35,163)(36,164)(37,165)(38,151)(39,152)(40,153)(41,154)(42,155)(43,156)(44,157)(45,158)(46,170)(47,171)(48,172)(49,173)(50,174)(51,175)(52,176)(53,177)(54,178)(55,179)(56,180)(57,166)(58,167)(59,168)(60,169)(61,188)(62,189)(63,190)(64,191)(65,192)(66,193)(67,194)(68,195)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,204)(77,205)(78,206)(79,207)(80,208)(81,209)(82,210)(83,196)(84,197)(85,198)(86,199)(87,200)(88,201)(89,202)(90,203)(91,218)(92,219)(93,220)(94,221)(95,222)(96,223)(97,224)(98,225)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,226)(107,227)(108,228)(109,229)(110,230)(111,231)(112,232)(113,233)(114,234)(115,235)(116,236)(117,237)(118,238)(119,239)(120,240), (1,54,28,39)(2,55,29,40)(3,56,30,41)(4,57,16,42)(5,58,17,43)(6,59,18,44)(7,60,19,45)(8,46,20,31)(9,47,21,32)(10,48,22,33)(11,49,23,34)(12,50,24,35)(13,51,25,36)(14,52,26,37)(15,53,27,38)(61,95,87,113)(62,96,88,114)(63,97,89,115)(64,98,90,116)(65,99,76,117)(66,100,77,118)(67,101,78,119)(68,102,79,120)(69,103,80,106)(70,104,81,107)(71,105,82,108)(72,91,83,109)(73,92,84,110)(74,93,85,111)(75,94,86,112)(121,170,136,159)(122,171,137,160)(123,172,138,161)(124,173,139,162)(125,174,140,163)(126,175,141,164)(127,176,142,165)(128,177,143,151)(129,178,144,152)(130,179,145,153)(131,180,146,154)(132,166,147,155)(133,167,148,156)(134,168,149,157)(135,169,150,158)(181,215,208,226)(182,216,209,227)(183,217,210,228)(184,218,196,229)(185,219,197,230)(186,220,198,231)(187,221,199,232)(188,222,200,233)(189,223,201,234)(190,224,202,235)(191,225,203,236)(192,211,204,237)(193,212,205,238)(194,213,206,239)(195,214,207,240), (1,197,28,185)(2,198,29,186)(3,199,30,187)(4,200,16,188)(5,201,17,189)(6,202,18,190)(7,203,19,191)(8,204,20,192)(9,205,21,193)(10,206,22,194)(11,207,23,195)(12,208,24,181)(13,209,25,182)(14,210,26,183)(15,196,27,184)(31,237,46,211)(32,238,47,212)(33,239,48,213)(34,240,49,214)(35,226,50,215)(36,227,51,216)(37,228,52,217)(38,229,53,218)(39,230,54,219)(40,231,55,220)(41,232,56,221)(42,233,57,222)(43,234,58,223)(44,235,59,224)(45,236,60,225)(61,132,87,147)(62,133,88,148)(63,134,89,149)(64,135,90,150)(65,121,76,136)(66,122,77,137)(67,123,78,138)(68,124,79,139)(69,125,80,140)(70,126,81,141)(71,127,82,142)(72,128,83,143)(73,129,84,144)(74,130,85,145)(75,131,86,146)(91,151,109,177)(92,152,110,178)(93,153,111,179)(94,154,112,180)(95,155,113,166)(96,156,114,167)(97,157,115,168)(98,158,116,169)(99,159,117,170)(100,160,118,171)(101,161,119,172)(102,162,120,173)(103,163,106,174)(104,164,107,175)(105,165,108,176), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,24)(17,23)(18,22)(19,21)(25,30)(26,29)(27,28)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(61,69)(62,68)(63,67)(64,66)(70,75)(71,74)(72,73)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,84)(91,92)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100)(106,113)(107,112)(108,111)(109,110)(114,120)(115,119)(116,118)(122,135)(123,134)(124,133)(125,132)(126,131)(127,130)(128,129)(137,150)(138,149)(139,148)(140,147)(141,146)(142,145)(143,144)(151,152)(153,165)(154,164)(155,163)(156,162)(157,161)(158,160)(166,174)(167,173)(168,172)(169,171)(175,180)(176,179)(177,178)(181,188)(182,187)(183,186)(184,185)(189,195)(190,194)(191,193)(196,197)(198,210)(199,209)(200,208)(201,207)(202,206)(203,205)(212,225)(213,224)(214,223)(215,222)(216,221)(217,220)(218,219)(226,233)(227,232)(228,231)(229,230)(234,240)(235,239)(236,238) );
G=PermutationGroup([(1,129),(2,130),(3,131),(4,132),(5,133),(6,134),(7,135),(8,121),(9,122),(10,123),(11,124),(12,125),(13,126),(14,127),(15,128),(16,147),(17,148),(18,149),(19,150),(20,136),(21,137),(22,138),(23,139),(24,140),(25,141),(26,142),(27,143),(28,144),(29,145),(30,146),(31,159),(32,160),(33,161),(34,162),(35,163),(36,164),(37,165),(38,151),(39,152),(40,153),(41,154),(42,155),(43,156),(44,157),(45,158),(46,170),(47,171),(48,172),(49,173),(50,174),(51,175),(52,176),(53,177),(54,178),(55,179),(56,180),(57,166),(58,167),(59,168),(60,169),(61,188),(62,189),(63,190),(64,191),(65,192),(66,193),(67,194),(68,195),(69,181),(70,182),(71,183),(72,184),(73,185),(74,186),(75,187),(76,204),(77,205),(78,206),(79,207),(80,208),(81,209),(82,210),(83,196),(84,197),(85,198),(86,199),(87,200),(88,201),(89,202),(90,203),(91,218),(92,219),(93,220),(94,221),(95,222),(96,223),(97,224),(98,225),(99,211),(100,212),(101,213),(102,214),(103,215),(104,216),(105,217),(106,226),(107,227),(108,228),(109,229),(110,230),(111,231),(112,232),(113,233),(114,234),(115,235),(116,236),(117,237),(118,238),(119,239),(120,240)], [(1,54,28,39),(2,55,29,40),(3,56,30,41),(4,57,16,42),(5,58,17,43),(6,59,18,44),(7,60,19,45),(8,46,20,31),(9,47,21,32),(10,48,22,33),(11,49,23,34),(12,50,24,35),(13,51,25,36),(14,52,26,37),(15,53,27,38),(61,95,87,113),(62,96,88,114),(63,97,89,115),(64,98,90,116),(65,99,76,117),(66,100,77,118),(67,101,78,119),(68,102,79,120),(69,103,80,106),(70,104,81,107),(71,105,82,108),(72,91,83,109),(73,92,84,110),(74,93,85,111),(75,94,86,112),(121,170,136,159),(122,171,137,160),(123,172,138,161),(124,173,139,162),(125,174,140,163),(126,175,141,164),(127,176,142,165),(128,177,143,151),(129,178,144,152),(130,179,145,153),(131,180,146,154),(132,166,147,155),(133,167,148,156),(134,168,149,157),(135,169,150,158),(181,215,208,226),(182,216,209,227),(183,217,210,228),(184,218,196,229),(185,219,197,230),(186,220,198,231),(187,221,199,232),(188,222,200,233),(189,223,201,234),(190,224,202,235),(191,225,203,236),(192,211,204,237),(193,212,205,238),(194,213,206,239),(195,214,207,240)], [(1,197,28,185),(2,198,29,186),(3,199,30,187),(4,200,16,188),(5,201,17,189),(6,202,18,190),(7,203,19,191),(8,204,20,192),(9,205,21,193),(10,206,22,194),(11,207,23,195),(12,208,24,181),(13,209,25,182),(14,210,26,183),(15,196,27,184),(31,237,46,211),(32,238,47,212),(33,239,48,213),(34,240,49,214),(35,226,50,215),(36,227,51,216),(37,228,52,217),(38,229,53,218),(39,230,54,219),(40,231,55,220),(41,232,56,221),(42,233,57,222),(43,234,58,223),(44,235,59,224),(45,236,60,225),(61,132,87,147),(62,133,88,148),(63,134,89,149),(64,135,90,150),(65,121,76,136),(66,122,77,137),(67,123,78,138),(68,124,79,139),(69,125,80,140),(70,126,81,141),(71,127,82,142),(72,128,83,143),(73,129,84,144),(74,130,85,145),(75,131,86,146),(91,151,109,177),(92,152,110,178),(93,153,111,179),(94,154,112,180),(95,155,113,166),(96,156,114,167),(97,157,115,168),(98,158,116,169),(99,159,117,170),(100,160,118,171),(101,161,119,172),(102,162,120,173),(103,163,106,174),(104,164,107,175),(105,165,108,176)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,24),(17,23),(18,22),(19,21),(25,30),(26,29),(27,28),(32,45),(33,44),(34,43),(35,42),(36,41),(37,40),(38,39),(47,60),(48,59),(49,58),(50,57),(51,56),(52,55),(53,54),(61,69),(62,68),(63,67),(64,66),(70,75),(71,74),(72,73),(77,90),(78,89),(79,88),(80,87),(81,86),(82,85),(83,84),(91,92),(93,105),(94,104),(95,103),(96,102),(97,101),(98,100),(106,113),(107,112),(108,111),(109,110),(114,120),(115,119),(116,118),(122,135),(123,134),(124,133),(125,132),(126,131),(127,130),(128,129),(137,150),(138,149),(139,148),(140,147),(141,146),(142,145),(143,144),(151,152),(153,165),(154,164),(155,163),(156,162),(157,161),(158,160),(166,174),(167,173),(168,172),(169,171),(175,180),(176,179),(177,178),(181,188),(182,187),(183,186),(184,185),(189,195),(190,194),(191,193),(196,197),(198,210),(199,209),(200,208),(201,207),(202,206),(203,205),(212,225),(213,224),(214,223),(215,222),(216,221),(217,220),(218,219),(226,233),(227,232),(228,231),(229,230),(234,240),(235,239),(236,238)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 59 |
| 0 | 0 | 1 | 1 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 56 | 36 |
| 0 | 0 | 23 | 5 |
| 56 | 30 | 0 | 0 |
| 40 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 60 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,60,1,0,0,59,1],[60,0,0,0,0,60,0,0,0,0,56,23,0,0,36,5],[56,40,0,0,30,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,60,60,0,0,0,0,1,0,0,0,0,1] >;
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4F | 4G | ··· | 4L | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | ··· | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 15 | 15 | 15 | 15 | 2 | 2 | ··· | 2 | 30 | ··· | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | + | + | + | + | - | - | - |
| image | C1 | C2 | C2 | C2 | C2 | S3 | Q8 | D5 | D6 | D6 | D10 | D10 | D15 | D30 | D30 | S3×Q8 | Q8×D5 | Q8×D15 |
| kernel | C2×Q8×D15 | C2×Dic30 | C2×C4×D15 | Q8×D15 | Q8×C30 | Q8×C10 | D30 | C6×Q8 | C2×C20 | C5×Q8 | C2×C12 | C3×Q8 | C2×Q8 | C2×C4 | Q8 | C10 | C6 | C2 |
| # reps | 1 | 3 | 3 | 8 | 1 | 1 | 4 | 2 | 3 | 4 | 6 | 8 | 4 | 12 | 16 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_2\times Q_8\times D_{15} % in TeX
G:=Group("C2xQ8xD15"); // GroupNames label
G:=SmallGroup(480,1172);
// by ID
G=gap.SmallGroup(480,1172);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,100,185,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^15=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations
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