direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D15⋊Q8, D30⋊11Q8, Dic10⋊21D6, Dic6⋊21D10, C30.10C24, C60.162C23, D30.34C23, Dic15.36C23, C6⋊1(Q8×D5), C30⋊3(C2×Q8), C10⋊1(S3×Q8), D15⋊1(C2×Q8), C15⋊Q8⋊8C22, C15⋊3(C22×Q8), (C2×Dic6)⋊14D5, (C2×C20).167D6, C6.10(C23×D5), (C2×Dic10)⋊14S3, (C6×Dic10)⋊13C2, (C10×Dic6)⋊13C2, (C2×C12).167D10, C10.10(S3×C23), (C2×C60).209C22, (C2×C30).229C23, C20.127(C22×S3), (C2×Dic5).138D6, (C5×Dic6)⋊27C22, (C4×D15).59C22, C12.128(C22×D5), D30.C2.9C22, (C3×Dic5).7C23, (C5×Dic3).6C23, Dic3.6(C22×D5), Dic5.7(C22×S3), (C2×Dic3).130D10, (C3×Dic10)⋊27C22, (C6×Dic5).130C22, (C2×Dic15).233C22, (C10×Dic3).129C22, (C22×D15).118C22, C5⋊1(C2×S3×Q8), C3⋊1(C2×Q8×D5), (C2×C15⋊Q8)⋊21C2, C4.134(C2×S3×D5), (C2×C4×D15).17C2, C22.98(C2×S3×D5), C2.14(C22×S3×D5), (C2×C4).219(S3×D5), (C2×D30.C2).9C2, (C2×C6).239(C22×D5), (C2×C10).239(C22×S3), SmallGroup(480,1082)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1404 in 312 conjugacy classes, 124 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×10], C22, C22 [×6], C5, S3 [×4], C6, C6 [×2], C2×C4, C2×C4 [×17], Q8 [×16], C23, D5 [×4], C10, C10 [×2], Dic3 [×4], Dic3 [×2], C12 [×2], C12 [×4], D6 [×6], C2×C6, C15, C22×C4 [×3], C2×Q8 [×12], Dic5 [×4], Dic5 [×2], C20 [×2], C20 [×4], D10 [×6], C2×C10, Dic6 [×4], Dic6 [×8], C4×S3 [×12], C2×Dic3 [×2], C2×Dic3, C2×C12, C2×C12 [×2], C3×Q8 [×4], C22×S3, D15 [×4], C30, C30 [×2], C22×Q8, Dic10 [×4], Dic10 [×8], C4×D5 [×12], C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×4], C22×D5, C2×Dic6, C2×Dic6 [×2], S3×C2×C4 [×3], S3×Q8 [×8], C6×Q8, C5×Dic3 [×4], C3×Dic5 [×4], Dic15 [×2], C60 [×2], D30 [×6], C2×C30, C2×Dic10, C2×Dic10 [×2], C2×C4×D5 [×3], Q8×D5 [×8], Q8×C10, C2×S3×Q8, D30.C2 [×8], C15⋊Q8 [×8], C3×Dic10 [×4], C6×Dic5 [×2], C5×Dic6 [×4], C10×Dic3 [×2], C4×D15 [×4], C2×Dic15, C2×C60, C22×D15, C2×Q8×D5, D15⋊Q8 [×8], C2×D30.C2 [×2], C2×C15⋊Q8 [×2], C6×Dic10, C10×Dic6, C2×C4×D15, C2×D15⋊Q8
Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D5, D6 [×7], C2×Q8 [×6], C24, D10 [×7], C22×S3 [×7], C22×Q8, C22×D5 [×7], S3×Q8 [×2], S3×C23, S3×D5, Q8×D5 [×2], C23×D5, C2×S3×Q8, C2×S3×D5 [×3], C2×Q8×D5, D15⋊Q8 [×2], C22×S3×D5, C2×D15⋊Q8
Generators and relations
G = < a,b,c,d,e | a2=b15=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b11, cd=dc, ece-1=b10c, ede-1=d-1 >
►On 240 points
Generators in S240
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 73)(14 74)(15 75)(16 86)(17 87)(18 88)(19 89)(20 90)(21 76)(22 77)(23 78)(24 79)(25 80)(26 81)(27 82)(28 83)(29 84)(30 85)(31 102)(32 103)(33 104)(34 105)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(43 99)(44 100)(45 101)(46 118)(47 119)(48 120)(49 106)(50 107)(51 108)(52 109)(53 110)(54 111)(55 112)(56 113)(57 114)(58 115)(59 116)(60 117)(121 185)(122 186)(123 187)(124 188)(125 189)(126 190)(127 191)(128 192)(129 193)(130 194)(131 195)(132 181)(133 182)(134 183)(135 184)(136 198)(137 199)(138 200)(139 201)(140 202)(141 203)(142 204)(143 205)(144 206)(145 207)(146 208)(147 209)(148 210)(149 196)(150 197)(151 211)(152 212)(153 213)(154 214)(155 215)(156 216)(157 217)(158 218)(159 219)(160 220)(161 221)(162 222)(163 223)(164 224)(165 225)(166 228)(167 229)(168 230)(169 231)(170 232)(171 233)(172 234)(173 235)(174 236)(175 237)(176 238)(177 239)(178 240)(179 226)(180 227)
(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 75)(2 74)(3 73)(4 72)(5 71)(6 70)(7 69)(8 68)(9 67)(10 66)(11 65)(12 64)(13 63)(14 62)(15 61)(16 86)(17 85)(18 84)(19 83)(20 82)(21 81)(22 80)(23 79)(24 78)(25 77)(26 76)(27 90)(28 89)(29 88)(30 87)(31 97)(32 96)(33 95)(34 94)(35 93)(36 92)(37 91)(38 105)(39 104)(40 103)(41 102)(42 101)(43 100)(44 99)(45 98)(46 110)(47 109)(48 108)(49 107)(50 106)(51 120)(52 119)(53 118)(54 117)(55 116)(56 115)(57 114)(58 113)(59 112)(60 111)(121 187)(122 186)(123 185)(124 184)(125 183)(126 182)(127 181)(128 195)(129 194)(130 193)(131 192)(132 191)(133 190)(134 189)(135 188)(136 196)(137 210)(138 209)(139 208)(140 207)(141 206)(142 205)(143 204)(144 203)(145 202)(146 201)(147 200)(148 199)(149 198)(150 197)(151 221)(152 220)(153 219)(154 218)(155 217)(156 216)(157 215)(158 214)(159 213)(160 212)(161 211)(162 225)(163 224)(164 223)(165 222)(166 226)(167 240)(168 239)(169 238)(170 237)(171 236)(172 235)(173 234)(174 233)(175 232)(176 231)(177 230)(178 229)(179 228)(180 227)
(1 50 24 44)(2 51 25 45)(3 52 26 31)(4 53 27 32)(5 54 28 33)(6 55 29 34)(7 56 30 35)(8 57 16 36)(9 58 17 37)(10 59 18 38)(11 60 19 39)(12 46 20 40)(13 47 21 41)(14 48 22 42)(15 49 23 43)(61 107 79 100)(62 108 80 101)(63 109 81 102)(64 110 82 103)(65 111 83 104)(66 112 84 105)(67 113 85 91)(68 114 86 92)(69 115 87 93)(70 116 88 94)(71 117 89 95)(72 118 90 96)(73 119 76 97)(74 120 77 98)(75 106 78 99)(121 155 149 179)(122 156 150 180)(123 157 136 166)(124 158 137 167)(125 159 138 168)(126 160 139 169)(127 161 140 170)(128 162 141 171)(129 163 142 172)(130 164 143 173)(131 165 144 174)(132 151 145 175)(133 152 146 176)(134 153 147 177)(135 154 148 178)(181 211 207 237)(182 212 208 238)(183 213 209 239)(184 214 210 240)(185 215 196 226)(186 216 197 227)(187 217 198 228)(188 218 199 229)(189 219 200 230)(190 220 201 231)(191 221 202 232)(192 222 203 233)(193 223 204 234)(194 224 205 235)(195 225 206 236)
(1 143 24 130)(2 139 25 126)(3 150 26 122)(4 146 27 133)(5 142 28 129)(6 138 29 125)(7 149 30 121)(8 145 16 132)(9 141 17 128)(10 137 18 124)(11 148 19 135)(12 144 20 131)(13 140 21 127)(14 136 22 123)(15 147 23 134)(31 180 52 156)(32 176 53 152)(33 172 54 163)(34 168 55 159)(35 179 56 155)(36 175 57 151)(37 171 58 162)(38 167 59 158)(39 178 60 154)(40 174 46 165)(41 170 47 161)(42 166 48 157)(43 177 49 153)(44 173 50 164)(45 169 51 160)(61 205 79 194)(62 201 80 190)(63 197 81 186)(64 208 82 182)(65 204 83 193)(66 200 84 189)(67 196 85 185)(68 207 86 181)(69 203 87 192)(70 199 88 188)(71 210 89 184)(72 206 90 195)(73 202 76 191)(74 198 77 187)(75 209 78 183)(91 226 113 215)(92 237 114 211)(93 233 115 222)(94 229 116 218)(95 240 117 214)(96 236 118 225)(97 232 119 221)(98 228 120 217)(99 239 106 213)(100 235 107 224)(101 231 108 220)(102 227 109 216)(103 238 110 212)(104 234 111 223)(105 230 112 219)
G:=sub<Sym(240)| (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,73)(14,74)(15,75)(16,86)(17,87)(18,88)(19,89)(20,90)(21,76)(22,77)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,84)(30,85)(31,102)(32,103)(33,104)(34,105)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,118)(47,119)(48,120)(49,106)(50,107)(51,108)(52,109)(53,110)(54,111)(55,112)(56,113)(57,114)(58,115)(59,116)(60,117)(121,185)(122,186)(123,187)(124,188)(125,189)(126,190)(127,191)(128,192)(129,193)(130,194)(131,195)(132,181)(133,182)(134,183)(135,184)(136,198)(137,199)(138,200)(139,201)(140,202)(141,203)(142,204)(143,205)(144,206)(145,207)(146,208)(147,209)(148,210)(149,196)(150,197)(151,211)(152,212)(153,213)(154,214)(155,215)(156,216)(157,217)(158,218)(159,219)(160,220)(161,221)(162,222)(163,223)(164,224)(165,225)(166,228)(167,229)(168,230)(169,231)(170,232)(171,233)(172,234)(173,235)(174,236)(175,237)(176,238)(177,239)(178,240)(179,226)(180,227), (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), (1,75)(2,74)(3,73)(4,72)(5,71)(6,70)(7,69)(8,68)(9,67)(10,66)(11,65)(12,64)(13,63)(14,62)(15,61)(16,86)(17,85)(18,84)(19,83)(20,82)(21,81)(22,80)(23,79)(24,78)(25,77)(26,76)(27,90)(28,89)(29,88)(30,87)(31,97)(32,96)(33,95)(34,94)(35,93)(36,92)(37,91)(38,105)(39,104)(40,103)(41,102)(42,101)(43,100)(44,99)(45,98)(46,110)(47,109)(48,108)(49,107)(50,106)(51,120)(52,119)(53,118)(54,117)(55,116)(56,115)(57,114)(58,113)(59,112)(60,111)(121,187)(122,186)(123,185)(124,184)(125,183)(126,182)(127,181)(128,195)(129,194)(130,193)(131,192)(132,191)(133,190)(134,189)(135,188)(136,196)(137,210)(138,209)(139,208)(140,207)(141,206)(142,205)(143,204)(144,203)(145,202)(146,201)(147,200)(148,199)(149,198)(150,197)(151,221)(152,220)(153,219)(154,218)(155,217)(156,216)(157,215)(158,214)(159,213)(160,212)(161,211)(162,225)(163,224)(164,223)(165,222)(166,226)(167,240)(168,239)(169,238)(170,237)(171,236)(172,235)(173,234)(174,233)(175,232)(176,231)(177,230)(178,229)(179,228)(180,227), (1,50,24,44)(2,51,25,45)(3,52,26,31)(4,53,27,32)(5,54,28,33)(6,55,29,34)(7,56,30,35)(8,57,16,36)(9,58,17,37)(10,59,18,38)(11,60,19,39)(12,46,20,40)(13,47,21,41)(14,48,22,42)(15,49,23,43)(61,107,79,100)(62,108,80,101)(63,109,81,102)(64,110,82,103)(65,111,83,104)(66,112,84,105)(67,113,85,91)(68,114,86,92)(69,115,87,93)(70,116,88,94)(71,117,89,95)(72,118,90,96)(73,119,76,97)(74,120,77,98)(75,106,78,99)(121,155,149,179)(122,156,150,180)(123,157,136,166)(124,158,137,167)(125,159,138,168)(126,160,139,169)(127,161,140,170)(128,162,141,171)(129,163,142,172)(130,164,143,173)(131,165,144,174)(132,151,145,175)(133,152,146,176)(134,153,147,177)(135,154,148,178)(181,211,207,237)(182,212,208,238)(183,213,209,239)(184,214,210,240)(185,215,196,226)(186,216,197,227)(187,217,198,228)(188,218,199,229)(189,219,200,230)(190,220,201,231)(191,221,202,232)(192,222,203,233)(193,223,204,234)(194,224,205,235)(195,225,206,236), (1,143,24,130)(2,139,25,126)(3,150,26,122)(4,146,27,133)(5,142,28,129)(6,138,29,125)(7,149,30,121)(8,145,16,132)(9,141,17,128)(10,137,18,124)(11,148,19,135)(12,144,20,131)(13,140,21,127)(14,136,22,123)(15,147,23,134)(31,180,52,156)(32,176,53,152)(33,172,54,163)(34,168,55,159)(35,179,56,155)(36,175,57,151)(37,171,58,162)(38,167,59,158)(39,178,60,154)(40,174,46,165)(41,170,47,161)(42,166,48,157)(43,177,49,153)(44,173,50,164)(45,169,51,160)(61,205,79,194)(62,201,80,190)(63,197,81,186)(64,208,82,182)(65,204,83,193)(66,200,84,189)(67,196,85,185)(68,207,86,181)(69,203,87,192)(70,199,88,188)(71,210,89,184)(72,206,90,195)(73,202,76,191)(74,198,77,187)(75,209,78,183)(91,226,113,215)(92,237,114,211)(93,233,115,222)(94,229,116,218)(95,240,117,214)(96,236,118,225)(97,232,119,221)(98,228,120,217)(99,239,106,213)(100,235,107,224)(101,231,108,220)(102,227,109,216)(103,238,110,212)(104,234,111,223)(105,230,112,219)>;
G:=Group( (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,73)(14,74)(15,75)(16,86)(17,87)(18,88)(19,89)(20,90)(21,76)(22,77)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,84)(30,85)(31,102)(32,103)(33,104)(34,105)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,118)(47,119)(48,120)(49,106)(50,107)(51,108)(52,109)(53,110)(54,111)(55,112)(56,113)(57,114)(58,115)(59,116)(60,117)(121,185)(122,186)(123,187)(124,188)(125,189)(126,190)(127,191)(128,192)(129,193)(130,194)(131,195)(132,181)(133,182)(134,183)(135,184)(136,198)(137,199)(138,200)(139,201)(140,202)(141,203)(142,204)(143,205)(144,206)(145,207)(146,208)(147,209)(148,210)(149,196)(150,197)(151,211)(152,212)(153,213)(154,214)(155,215)(156,216)(157,217)(158,218)(159,219)(160,220)(161,221)(162,222)(163,223)(164,224)(165,225)(166,228)(167,229)(168,230)(169,231)(170,232)(171,233)(172,234)(173,235)(174,236)(175,237)(176,238)(177,239)(178,240)(179,226)(180,227), (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), (1,75)(2,74)(3,73)(4,72)(5,71)(6,70)(7,69)(8,68)(9,67)(10,66)(11,65)(12,64)(13,63)(14,62)(15,61)(16,86)(17,85)(18,84)(19,83)(20,82)(21,81)(22,80)(23,79)(24,78)(25,77)(26,76)(27,90)(28,89)(29,88)(30,87)(31,97)(32,96)(33,95)(34,94)(35,93)(36,92)(37,91)(38,105)(39,104)(40,103)(41,102)(42,101)(43,100)(44,99)(45,98)(46,110)(47,109)(48,108)(49,107)(50,106)(51,120)(52,119)(53,118)(54,117)(55,116)(56,115)(57,114)(58,113)(59,112)(60,111)(121,187)(122,186)(123,185)(124,184)(125,183)(126,182)(127,181)(128,195)(129,194)(130,193)(131,192)(132,191)(133,190)(134,189)(135,188)(136,196)(137,210)(138,209)(139,208)(140,207)(141,206)(142,205)(143,204)(144,203)(145,202)(146,201)(147,200)(148,199)(149,198)(150,197)(151,221)(152,220)(153,219)(154,218)(155,217)(156,216)(157,215)(158,214)(159,213)(160,212)(161,211)(162,225)(163,224)(164,223)(165,222)(166,226)(167,240)(168,239)(169,238)(170,237)(171,236)(172,235)(173,234)(174,233)(175,232)(176,231)(177,230)(178,229)(179,228)(180,227), (1,50,24,44)(2,51,25,45)(3,52,26,31)(4,53,27,32)(5,54,28,33)(6,55,29,34)(7,56,30,35)(8,57,16,36)(9,58,17,37)(10,59,18,38)(11,60,19,39)(12,46,20,40)(13,47,21,41)(14,48,22,42)(15,49,23,43)(61,107,79,100)(62,108,80,101)(63,109,81,102)(64,110,82,103)(65,111,83,104)(66,112,84,105)(67,113,85,91)(68,114,86,92)(69,115,87,93)(70,116,88,94)(71,117,89,95)(72,118,90,96)(73,119,76,97)(74,120,77,98)(75,106,78,99)(121,155,149,179)(122,156,150,180)(123,157,136,166)(124,158,137,167)(125,159,138,168)(126,160,139,169)(127,161,140,170)(128,162,141,171)(129,163,142,172)(130,164,143,173)(131,165,144,174)(132,151,145,175)(133,152,146,176)(134,153,147,177)(135,154,148,178)(181,211,207,237)(182,212,208,238)(183,213,209,239)(184,214,210,240)(185,215,196,226)(186,216,197,227)(187,217,198,228)(188,218,199,229)(189,219,200,230)(190,220,201,231)(191,221,202,232)(192,222,203,233)(193,223,204,234)(194,224,205,235)(195,225,206,236), (1,143,24,130)(2,139,25,126)(3,150,26,122)(4,146,27,133)(5,142,28,129)(6,138,29,125)(7,149,30,121)(8,145,16,132)(9,141,17,128)(10,137,18,124)(11,148,19,135)(12,144,20,131)(13,140,21,127)(14,136,22,123)(15,147,23,134)(31,180,52,156)(32,176,53,152)(33,172,54,163)(34,168,55,159)(35,179,56,155)(36,175,57,151)(37,171,58,162)(38,167,59,158)(39,178,60,154)(40,174,46,165)(41,170,47,161)(42,166,48,157)(43,177,49,153)(44,173,50,164)(45,169,51,160)(61,205,79,194)(62,201,80,190)(63,197,81,186)(64,208,82,182)(65,204,83,193)(66,200,84,189)(67,196,85,185)(68,207,86,181)(69,203,87,192)(70,199,88,188)(71,210,89,184)(72,206,90,195)(73,202,76,191)(74,198,77,187)(75,209,78,183)(91,226,113,215)(92,237,114,211)(93,233,115,222)(94,229,116,218)(95,240,117,214)(96,236,118,225)(97,232,119,221)(98,228,120,217)(99,239,106,213)(100,235,107,224)(101,231,108,220)(102,227,109,216)(103,238,110,212)(104,234,111,223)(105,230,112,219) );
G=PermutationGroup([(1,61),(2,62),(3,63),(4,64),(5,65),(6,66),(7,67),(8,68),(9,69),(10,70),(11,71),(12,72),(13,73),(14,74),(15,75),(16,86),(17,87),(18,88),(19,89),(20,90),(21,76),(22,77),(23,78),(24,79),(25,80),(26,81),(27,82),(28,83),(29,84),(30,85),(31,102),(32,103),(33,104),(34,105),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,97),(42,98),(43,99),(44,100),(45,101),(46,118),(47,119),(48,120),(49,106),(50,107),(51,108),(52,109),(53,110),(54,111),(55,112),(56,113),(57,114),(58,115),(59,116),(60,117),(121,185),(122,186),(123,187),(124,188),(125,189),(126,190),(127,191),(128,192),(129,193),(130,194),(131,195),(132,181),(133,182),(134,183),(135,184),(136,198),(137,199),(138,200),(139,201),(140,202),(141,203),(142,204),(143,205),(144,206),(145,207),(146,208),(147,209),(148,210),(149,196),(150,197),(151,211),(152,212),(153,213),(154,214),(155,215),(156,216),(157,217),(158,218),(159,219),(160,220),(161,221),(162,222),(163,223),(164,224),(165,225),(166,228),(167,229),(168,230),(169,231),(170,232),(171,233),(172,234),(173,235),(174,236),(175,237),(176,238),(177,239),(178,240),(179,226),(180,227)], [(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)], [(1,75),(2,74),(3,73),(4,72),(5,71),(6,70),(7,69),(8,68),(9,67),(10,66),(11,65),(12,64),(13,63),(14,62),(15,61),(16,86),(17,85),(18,84),(19,83),(20,82),(21,81),(22,80),(23,79),(24,78),(25,77),(26,76),(27,90),(28,89),(29,88),(30,87),(31,97),(32,96),(33,95),(34,94),(35,93),(36,92),(37,91),(38,105),(39,104),(40,103),(41,102),(42,101),(43,100),(44,99),(45,98),(46,110),(47,109),(48,108),(49,107),(50,106),(51,120),(52,119),(53,118),(54,117),(55,116),(56,115),(57,114),(58,113),(59,112),(60,111),(121,187),(122,186),(123,185),(124,184),(125,183),(126,182),(127,181),(128,195),(129,194),(130,193),(131,192),(132,191),(133,190),(134,189),(135,188),(136,196),(137,210),(138,209),(139,208),(140,207),(141,206),(142,205),(143,204),(144,203),(145,202),(146,201),(147,200),(148,199),(149,198),(150,197),(151,221),(152,220),(153,219),(154,218),(155,217),(156,216),(157,215),(158,214),(159,213),(160,212),(161,211),(162,225),(163,224),(164,223),(165,222),(166,226),(167,240),(168,239),(169,238),(170,237),(171,236),(172,235),(173,234),(174,233),(175,232),(176,231),(177,230),(178,229),(179,228),(180,227)], [(1,50,24,44),(2,51,25,45),(3,52,26,31),(4,53,27,32),(5,54,28,33),(6,55,29,34),(7,56,30,35),(8,57,16,36),(9,58,17,37),(10,59,18,38),(11,60,19,39),(12,46,20,40),(13,47,21,41),(14,48,22,42),(15,49,23,43),(61,107,79,100),(62,108,80,101),(63,109,81,102),(64,110,82,103),(65,111,83,104),(66,112,84,105),(67,113,85,91),(68,114,86,92),(69,115,87,93),(70,116,88,94),(71,117,89,95),(72,118,90,96),(73,119,76,97),(74,120,77,98),(75,106,78,99),(121,155,149,179),(122,156,150,180),(123,157,136,166),(124,158,137,167),(125,159,138,168),(126,160,139,169),(127,161,140,170),(128,162,141,171),(129,163,142,172),(130,164,143,173),(131,165,144,174),(132,151,145,175),(133,152,146,176),(134,153,147,177),(135,154,148,178),(181,211,207,237),(182,212,208,238),(183,213,209,239),(184,214,210,240),(185,215,196,226),(186,216,197,227),(187,217,198,228),(188,218,199,229),(189,219,200,230),(190,220,201,231),(191,221,202,232),(192,222,203,233),(193,223,204,234),(194,224,205,235),(195,225,206,236)], [(1,143,24,130),(2,139,25,126),(3,150,26,122),(4,146,27,133),(5,142,28,129),(6,138,29,125),(7,149,30,121),(8,145,16,132),(9,141,17,128),(10,137,18,124),(11,148,19,135),(12,144,20,131),(13,140,21,127),(14,136,22,123),(15,147,23,134),(31,180,52,156),(32,176,53,152),(33,172,54,163),(34,168,55,159),(35,179,56,155),(36,175,57,151),(37,171,58,162),(38,167,59,158),(39,178,60,154),(40,174,46,165),(41,170,47,161),(42,166,48,157),(43,177,49,153),(44,173,50,164),(45,169,51,160),(61,205,79,194),(62,201,80,190),(63,197,81,186),(64,208,82,182),(65,204,83,193),(66,200,84,189),(67,196,85,185),(68,207,86,181),(69,203,87,192),(70,199,88,188),(71,210,89,184),(72,206,90,195),(73,202,76,191),(74,198,77,187),(75,209,78,183),(91,226,113,215),(92,237,114,211),(93,233,115,222),(94,229,116,218),(95,240,117,214),(96,236,118,225),(97,232,119,221),(98,228,120,217),(99,239,106,213),(100,235,107,224),(101,231,108,220),(102,227,109,216),(103,238,110,212),(104,234,111,223),(105,230,112,219)])
Matrix representation ►G ⊆ GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 1 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 18 | 0 | 0 |
| 0 | 0 | 44 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 60 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 60 | 0 | 0 |
| 0 | 0 | 44 | 44 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 59 |
| 0 | 0 | 0 | 0 | 1 | 60 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 33 |
| 0 | 0 | 0 | 0 | 48 | 60 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[0,1,0,0,0,0,60,60,0,0,0,0,0,0,17,44,0,0,0,0,18,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,1,1,0,0,0,0,0,0,17,44,0,0,0,0,60,44,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,1,0,0,0,0,59,60],[0,60,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,48,0,0,0,0,33,60] >;
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 15 | 15 | 15 | 15 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | + | - | + | - | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | S3×Q8 | S3×D5 | Q8×D5 | C2×S3×D5 | C2×S3×D5 | D15⋊Q8 |
| kernel | C2×D15⋊Q8 | D15⋊Q8 | C2×D30.C2 | C2×C15⋊Q8 | C6×Dic10 | C10×Dic6 | C2×C4×D15 | C2×Dic10 | D30 | C2×Dic6 | Dic10 | C2×Dic5 | C2×C20 | Dic6 | C2×Dic3 | C2×C12 | C10 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 8 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 2 | 1 | 8 | 4 | 2 | 2 | 2 | 4 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2\times D_{15}\rtimes Q_8 % in TeX
G:=Group("C2xD15:Q8"); // GroupNames label
G:=SmallGroup(480,1082);
// by ID
G=gap.SmallGroup(480,1082);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,675,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^15=c^2=d^4=1,e^2=d^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,e*b*e^-1=b^11,c*d=d*c,e*c*e^-1=b^10*c,e*d*e^-1=d^-1>;
// generators/relations
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