direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×SD16⋊D5, Dic20⋊6C6, C24.35D10, C120.48C22, C60.195C23, (Q8×D5)⋊5C6, C8.2(C6×D5), C8⋊D5⋊2C6, C40.9(C2×C6), D4.D5⋊4C6, C5⋊Q16⋊1C6, D4.4(C6×D5), Q8.6(C6×D5), SD16⋊2(C3×D5), (C3×SD16)⋊6D5, (C5×SD16)⋊2C6, (C6×D5).73D4, C10.32(C6×D4), C6.186(D4×D5), (C15×SD16)⋊6C2, (C3×D4).28D10, D10.15(C3×D4), D4⋊2D5.1C6, C30.345(C2×D4), C20.6(C22×C6), (C3×Q8).27D10, (C3×Dic20)⋊14C2, C15⋊29(C8.C22), Dic10.2(C2×C6), Dic5.18(C3×D4), (C3×Dic5).79D4, (D5×C12).78C22, (D4×C15).28C22, C12.195(C22×D5), (Q8×C15).27C22, (C3×Dic10).33C22, (C3×Q8×D5)⋊9C2, C4.6(D5×C2×C6), C2.20(C3×D4×D5), C5⋊2(C3×C8.C22), (C3×C8⋊D5)⋊6C2, C5⋊2C8.1(C2×C6), (C4×D5).3(C2×C6), (C3×C5⋊Q16)⋊9C2, (C5×D4).4(C2×C6), (C5×Q8).6(C2×C6), (C3×D4.D5)⋊12C2, (C3×D4⋊2D5).4C2, (C3×C5⋊2C8).31C22, SmallGroup(480,708)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×SD16⋊D5
G = < a,b,c,d,e | a3=b8=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b3, bd=db, ebe=b5, cd=dc, ece=b4c, ede=d-1 >
Subgroups: 384 in 120 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, D4, D4, Q8, Q8, D5, C10, C10, C12, C12, C2×C6, C15, M4(2), SD16, SD16, Q16, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, C2×C10, C24, C24, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C3×D5, C30, C30, C8.C22, C5⋊2C8, C40, Dic10, Dic10, C4×D5, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, C3×M4(2), C3×SD16, C3×SD16, C3×Q16, C6×Q8, C3×C4○D4, C3×Dic5, C3×Dic5, C60, C60, C6×D5, C2×C30, C8⋊D5, Dic20, D4.D5, C5⋊Q16, C5×SD16, D4⋊2D5, Q8×D5, C3×C8.C22, C3×C5⋊2C8, C120, C3×Dic10, C3×Dic10, D5×C12, D5×C12, C6×Dic5, C3×C5⋊D4, D4×C15, Q8×C15, SD16⋊D5, C3×C8⋊D5, C3×Dic20, C3×D4.D5, C3×C5⋊Q16, C15×SD16, C3×D4⋊2D5, C3×Q8×D5, C3×SD16⋊D5
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, D10, C3×D4, C22×C6, C3×D5, C8.C22, C22×D5, C6×D4, C6×D5, D4×D5, C3×C8.C22, D5×C2×C6, SD16⋊D5, C3×D4×D5, C3×SD16⋊D5
►On 240 points
Generators in S240
(1 93 180)(2 94 181)(3 95 182)(4 96 183)(5 89 184)(6 90 177)(7 91 178)(8 92 179)(9 76 231)(10 77 232)(11 78 225)(12 79 226)(13 80 227)(14 73 228)(15 74 229)(16 75 230)(17 218 106)(18 219 107)(19 220 108)(20 221 109)(21 222 110)(22 223 111)(23 224 112)(24 217 105)(25 44 99)(26 45 100)(27 46 101)(28 47 102)(29 48 103)(30 41 104)(31 42 97)(32 43 98)(33 149 120)(34 150 113)(35 151 114)(36 152 115)(37 145 116)(38 146 117)(39 147 118)(40 148 119)(49 185 127)(50 186 128)(51 187 121)(52 188 122)(53 189 123)(54 190 124)(55 191 125)(56 192 126)(57 193 132)(58 194 133)(59 195 134)(60 196 135)(61 197 136)(62 198 129)(63 199 130)(64 200 131)(65 141 240)(66 142 233)(67 143 234)(68 144 235)(69 137 236)(70 138 237)(71 139 238)(72 140 239)(81 165 203)(82 166 204)(83 167 205)(84 168 206)(85 161 207)(86 162 208)(87 163 201)(88 164 202)(153 169 210)(154 170 211)(155 171 212)(156 172 213)(157 173 214)(158 174 215)(159 175 216)(160 176 209)
(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)(2 32)(3 27)(4 30)(5 25)(6 28)(7 31)(8 26)(9 116)(10 119)(11 114)(12 117)(13 120)(14 115)(15 118)(16 113)(17 67)(18 70)(19 65)(20 68)(21 71)(22 66)(23 69)(24 72)(33 80)(34 75)(35 78)(36 73)(37 76)(38 79)(39 74)(40 77)(41 96)(42 91)(43 94)(44 89)(45 92)(46 95)(47 90)(48 93)(49 87)(50 82)(51 85)(52 88)(53 83)(54 86)(55 81)(56 84)(57 156)(58 159)(59 154)(60 157)(61 160)(62 155)(63 158)(64 153)(97 178)(98 181)(99 184)(100 179)(101 182)(102 177)(103 180)(104 183)(105 239)(106 234)(107 237)(108 240)(109 235)(110 238)(111 233)(112 236)(121 207)(122 202)(123 205)(124 208)(125 203)(126 206)(127 201)(128 204)(129 212)(130 215)(131 210)(132 213)(133 216)(134 211)(135 214)(136 209)(137 224)(138 219)(139 222)(140 217)(141 220)(142 223)(143 218)(144 221)(145 231)(146 226)(147 229)(148 232)(149 227)(150 230)(151 225)(152 228)(161 187)(162 190)(163 185)(164 188)(165 191)(166 186)(167 189)(168 192)(169 200)(170 195)(171 198)(172 193)(173 196)(174 199)(175 194)(176 197)
(1 225 70 51 59)(2 226 71 52 60)(3 227 72 53 61)(4 228 65 54 62)(5 229 66 55 63)(6 230 67 56 64)(7 231 68 49 57)(8 232 69 50 58)(9 144 185 193 91)(10 137 186 194 92)(11 138 187 195 93)(12 139 188 196 94)(13 140 189 197 95)(14 141 190 198 96)(15 142 191 199 89)(16 143 192 200 90)(17 84 153 28 150)(18 85 154 29 151)(19 86 155 30 152)(20 87 156 31 145)(21 88 157 32 146)(22 81 158 25 147)(23 82 159 26 148)(24 83 160 27 149)(33 105 205 209 101)(34 106 206 210 102)(35 107 207 211 103)(36 108 208 212 104)(37 109 201 213 97)(38 110 202 214 98)(39 111 203 215 99)(40 112 204 216 100)(41 115 220 162 171)(42 116 221 163 172)(43 117 222 164 173)(44 118 223 165 174)(45 119 224 166 175)(46 120 217 167 176)(47 113 218 168 169)(48 114 219 161 170)(73 240 124 129 183)(74 233 125 130 184)(75 234 126 131 177)(76 235 127 132 178)(77 236 128 133 179)(78 237 121 134 180)(79 238 122 135 181)(80 239 123 136 182)
(1 59)(2 64)(3 61)(4 58)(5 63)(6 60)(7 57)(8 62)(9 185)(10 190)(11 187)(12 192)(13 189)(14 186)(15 191)(16 188)(18 22)(20 24)(25 154)(26 159)(27 156)(28 153)(29 158)(30 155)(31 160)(32 157)(33 201)(34 206)(35 203)(36 208)(37 205)(38 202)(39 207)(40 204)(41 171)(42 176)(43 173)(44 170)(45 175)(46 172)(47 169)(48 174)(49 231)(50 228)(51 225)(52 230)(53 227)(54 232)(55 229)(56 226)(65 69)(67 71)(73 128)(74 125)(75 122)(76 127)(77 124)(78 121)(79 126)(80 123)(81 151)(82 148)(83 145)(84 150)(85 147)(86 152)(87 149)(88 146)(89 199)(90 196)(91 193)(92 198)(93 195)(94 200)(95 197)(96 194)(97 209)(98 214)(99 211)(100 216)(101 213)(102 210)(103 215)(104 212)(105 109)(107 111)(113 168)(114 165)(115 162)(116 167)(117 164)(118 161)(119 166)(120 163)(129 179)(130 184)(131 181)(132 178)(133 183)(134 180)(135 177)(136 182)(137 141)(139 143)(217 221)(219 223)(234 238)(236 240)
G:=sub<Sym(240)| (1,93,180)(2,94,181)(3,95,182)(4,96,183)(5,89,184)(6,90,177)(7,91,178)(8,92,179)(9,76,231)(10,77,232)(11,78,225)(12,79,226)(13,80,227)(14,73,228)(15,74,229)(16,75,230)(17,218,106)(18,219,107)(19,220,108)(20,221,109)(21,222,110)(22,223,111)(23,224,112)(24,217,105)(25,44,99)(26,45,100)(27,46,101)(28,47,102)(29,48,103)(30,41,104)(31,42,97)(32,43,98)(33,149,120)(34,150,113)(35,151,114)(36,152,115)(37,145,116)(38,146,117)(39,147,118)(40,148,119)(49,185,127)(50,186,128)(51,187,121)(52,188,122)(53,189,123)(54,190,124)(55,191,125)(56,192,126)(57,193,132)(58,194,133)(59,195,134)(60,196,135)(61,197,136)(62,198,129)(63,199,130)(64,200,131)(65,141,240)(66,142,233)(67,143,234)(68,144,235)(69,137,236)(70,138,237)(71,139,238)(72,140,239)(81,165,203)(82,166,204)(83,167,205)(84,168,206)(85,161,207)(86,162,208)(87,163,201)(88,164,202)(153,169,210)(154,170,211)(155,171,212)(156,172,213)(157,173,214)(158,174,215)(159,175,216)(160,176,209), (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)(2,32)(3,27)(4,30)(5,25)(6,28)(7,31)(8,26)(9,116)(10,119)(11,114)(12,117)(13,120)(14,115)(15,118)(16,113)(17,67)(18,70)(19,65)(20,68)(21,71)(22,66)(23,69)(24,72)(33,80)(34,75)(35,78)(36,73)(37,76)(38,79)(39,74)(40,77)(41,96)(42,91)(43,94)(44,89)(45,92)(46,95)(47,90)(48,93)(49,87)(50,82)(51,85)(52,88)(53,83)(54,86)(55,81)(56,84)(57,156)(58,159)(59,154)(60,157)(61,160)(62,155)(63,158)(64,153)(97,178)(98,181)(99,184)(100,179)(101,182)(102,177)(103,180)(104,183)(105,239)(106,234)(107,237)(108,240)(109,235)(110,238)(111,233)(112,236)(121,207)(122,202)(123,205)(124,208)(125,203)(126,206)(127,201)(128,204)(129,212)(130,215)(131,210)(132,213)(133,216)(134,211)(135,214)(136,209)(137,224)(138,219)(139,222)(140,217)(141,220)(142,223)(143,218)(144,221)(145,231)(146,226)(147,229)(148,232)(149,227)(150,230)(151,225)(152,228)(161,187)(162,190)(163,185)(164,188)(165,191)(166,186)(167,189)(168,192)(169,200)(170,195)(171,198)(172,193)(173,196)(174,199)(175,194)(176,197), (1,225,70,51,59)(2,226,71,52,60)(3,227,72,53,61)(4,228,65,54,62)(5,229,66,55,63)(6,230,67,56,64)(7,231,68,49,57)(8,232,69,50,58)(9,144,185,193,91)(10,137,186,194,92)(11,138,187,195,93)(12,139,188,196,94)(13,140,189,197,95)(14,141,190,198,96)(15,142,191,199,89)(16,143,192,200,90)(17,84,153,28,150)(18,85,154,29,151)(19,86,155,30,152)(20,87,156,31,145)(21,88,157,32,146)(22,81,158,25,147)(23,82,159,26,148)(24,83,160,27,149)(33,105,205,209,101)(34,106,206,210,102)(35,107,207,211,103)(36,108,208,212,104)(37,109,201,213,97)(38,110,202,214,98)(39,111,203,215,99)(40,112,204,216,100)(41,115,220,162,171)(42,116,221,163,172)(43,117,222,164,173)(44,118,223,165,174)(45,119,224,166,175)(46,120,217,167,176)(47,113,218,168,169)(48,114,219,161,170)(73,240,124,129,183)(74,233,125,130,184)(75,234,126,131,177)(76,235,127,132,178)(77,236,128,133,179)(78,237,121,134,180)(79,238,122,135,181)(80,239,123,136,182), (1,59)(2,64)(3,61)(4,58)(5,63)(6,60)(7,57)(8,62)(9,185)(10,190)(11,187)(12,192)(13,189)(14,186)(15,191)(16,188)(18,22)(20,24)(25,154)(26,159)(27,156)(28,153)(29,158)(30,155)(31,160)(32,157)(33,201)(34,206)(35,203)(36,208)(37,205)(38,202)(39,207)(40,204)(41,171)(42,176)(43,173)(44,170)(45,175)(46,172)(47,169)(48,174)(49,231)(50,228)(51,225)(52,230)(53,227)(54,232)(55,229)(56,226)(65,69)(67,71)(73,128)(74,125)(75,122)(76,127)(77,124)(78,121)(79,126)(80,123)(81,151)(82,148)(83,145)(84,150)(85,147)(86,152)(87,149)(88,146)(89,199)(90,196)(91,193)(92,198)(93,195)(94,200)(95,197)(96,194)(97,209)(98,214)(99,211)(100,216)(101,213)(102,210)(103,215)(104,212)(105,109)(107,111)(113,168)(114,165)(115,162)(116,167)(117,164)(118,161)(119,166)(120,163)(129,179)(130,184)(131,181)(132,178)(133,183)(134,180)(135,177)(136,182)(137,141)(139,143)(217,221)(219,223)(234,238)(236,240)>;
G:=Group( (1,93,180)(2,94,181)(3,95,182)(4,96,183)(5,89,184)(6,90,177)(7,91,178)(8,92,179)(9,76,231)(10,77,232)(11,78,225)(12,79,226)(13,80,227)(14,73,228)(15,74,229)(16,75,230)(17,218,106)(18,219,107)(19,220,108)(20,221,109)(21,222,110)(22,223,111)(23,224,112)(24,217,105)(25,44,99)(26,45,100)(27,46,101)(28,47,102)(29,48,103)(30,41,104)(31,42,97)(32,43,98)(33,149,120)(34,150,113)(35,151,114)(36,152,115)(37,145,116)(38,146,117)(39,147,118)(40,148,119)(49,185,127)(50,186,128)(51,187,121)(52,188,122)(53,189,123)(54,190,124)(55,191,125)(56,192,126)(57,193,132)(58,194,133)(59,195,134)(60,196,135)(61,197,136)(62,198,129)(63,199,130)(64,200,131)(65,141,240)(66,142,233)(67,143,234)(68,144,235)(69,137,236)(70,138,237)(71,139,238)(72,140,239)(81,165,203)(82,166,204)(83,167,205)(84,168,206)(85,161,207)(86,162,208)(87,163,201)(88,164,202)(153,169,210)(154,170,211)(155,171,212)(156,172,213)(157,173,214)(158,174,215)(159,175,216)(160,176,209), (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)(2,32)(3,27)(4,30)(5,25)(6,28)(7,31)(8,26)(9,116)(10,119)(11,114)(12,117)(13,120)(14,115)(15,118)(16,113)(17,67)(18,70)(19,65)(20,68)(21,71)(22,66)(23,69)(24,72)(33,80)(34,75)(35,78)(36,73)(37,76)(38,79)(39,74)(40,77)(41,96)(42,91)(43,94)(44,89)(45,92)(46,95)(47,90)(48,93)(49,87)(50,82)(51,85)(52,88)(53,83)(54,86)(55,81)(56,84)(57,156)(58,159)(59,154)(60,157)(61,160)(62,155)(63,158)(64,153)(97,178)(98,181)(99,184)(100,179)(101,182)(102,177)(103,180)(104,183)(105,239)(106,234)(107,237)(108,240)(109,235)(110,238)(111,233)(112,236)(121,207)(122,202)(123,205)(124,208)(125,203)(126,206)(127,201)(128,204)(129,212)(130,215)(131,210)(132,213)(133,216)(134,211)(135,214)(136,209)(137,224)(138,219)(139,222)(140,217)(141,220)(142,223)(143,218)(144,221)(145,231)(146,226)(147,229)(148,232)(149,227)(150,230)(151,225)(152,228)(161,187)(162,190)(163,185)(164,188)(165,191)(166,186)(167,189)(168,192)(169,200)(170,195)(171,198)(172,193)(173,196)(174,199)(175,194)(176,197), (1,225,70,51,59)(2,226,71,52,60)(3,227,72,53,61)(4,228,65,54,62)(5,229,66,55,63)(6,230,67,56,64)(7,231,68,49,57)(8,232,69,50,58)(9,144,185,193,91)(10,137,186,194,92)(11,138,187,195,93)(12,139,188,196,94)(13,140,189,197,95)(14,141,190,198,96)(15,142,191,199,89)(16,143,192,200,90)(17,84,153,28,150)(18,85,154,29,151)(19,86,155,30,152)(20,87,156,31,145)(21,88,157,32,146)(22,81,158,25,147)(23,82,159,26,148)(24,83,160,27,149)(33,105,205,209,101)(34,106,206,210,102)(35,107,207,211,103)(36,108,208,212,104)(37,109,201,213,97)(38,110,202,214,98)(39,111,203,215,99)(40,112,204,216,100)(41,115,220,162,171)(42,116,221,163,172)(43,117,222,164,173)(44,118,223,165,174)(45,119,224,166,175)(46,120,217,167,176)(47,113,218,168,169)(48,114,219,161,170)(73,240,124,129,183)(74,233,125,130,184)(75,234,126,131,177)(76,235,127,132,178)(77,236,128,133,179)(78,237,121,134,180)(79,238,122,135,181)(80,239,123,136,182), (1,59)(2,64)(3,61)(4,58)(5,63)(6,60)(7,57)(8,62)(9,185)(10,190)(11,187)(12,192)(13,189)(14,186)(15,191)(16,188)(18,22)(20,24)(25,154)(26,159)(27,156)(28,153)(29,158)(30,155)(31,160)(32,157)(33,201)(34,206)(35,203)(36,208)(37,205)(38,202)(39,207)(40,204)(41,171)(42,176)(43,173)(44,170)(45,175)(46,172)(47,169)(48,174)(49,231)(50,228)(51,225)(52,230)(53,227)(54,232)(55,229)(56,226)(65,69)(67,71)(73,128)(74,125)(75,122)(76,127)(77,124)(78,121)(79,126)(80,123)(81,151)(82,148)(83,145)(84,150)(85,147)(86,152)(87,149)(88,146)(89,199)(90,196)(91,193)(92,198)(93,195)(94,200)(95,197)(96,194)(97,209)(98,214)(99,211)(100,216)(101,213)(102,210)(103,215)(104,212)(105,109)(107,111)(113,168)(114,165)(115,162)(116,167)(117,164)(118,161)(119,166)(120,163)(129,179)(130,184)(131,181)(132,178)(133,183)(134,180)(135,177)(136,182)(137,141)(139,143)(217,221)(219,223)(234,238)(236,240) );
G=PermutationGroup([[(1,93,180),(2,94,181),(3,95,182),(4,96,183),(5,89,184),(6,90,177),(7,91,178),(8,92,179),(9,76,231),(10,77,232),(11,78,225),(12,79,226),(13,80,227),(14,73,228),(15,74,229),(16,75,230),(17,218,106),(18,219,107),(19,220,108),(20,221,109),(21,222,110),(22,223,111),(23,224,112),(24,217,105),(25,44,99),(26,45,100),(27,46,101),(28,47,102),(29,48,103),(30,41,104),(31,42,97),(32,43,98),(33,149,120),(34,150,113),(35,151,114),(36,152,115),(37,145,116),(38,146,117),(39,147,118),(40,148,119),(49,185,127),(50,186,128),(51,187,121),(52,188,122),(53,189,123),(54,190,124),(55,191,125),(56,192,126),(57,193,132),(58,194,133),(59,195,134),(60,196,135),(61,197,136),(62,198,129),(63,199,130),(64,200,131),(65,141,240),(66,142,233),(67,143,234),(68,144,235),(69,137,236),(70,138,237),(71,139,238),(72,140,239),(81,165,203),(82,166,204),(83,167,205),(84,168,206),(85,161,207),(86,162,208),(87,163,201),(88,164,202),(153,169,210),(154,170,211),(155,171,212),(156,172,213),(157,173,214),(158,174,215),(159,175,216),(160,176,209)], [(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),(2,32),(3,27),(4,30),(5,25),(6,28),(7,31),(8,26),(9,116),(10,119),(11,114),(12,117),(13,120),(14,115),(15,118),(16,113),(17,67),(18,70),(19,65),(20,68),(21,71),(22,66),(23,69),(24,72),(33,80),(34,75),(35,78),(36,73),(37,76),(38,79),(39,74),(40,77),(41,96),(42,91),(43,94),(44,89),(45,92),(46,95),(47,90),(48,93),(49,87),(50,82),(51,85),(52,88),(53,83),(54,86),(55,81),(56,84),(57,156),(58,159),(59,154),(60,157),(61,160),(62,155),(63,158),(64,153),(97,178),(98,181),(99,184),(100,179),(101,182),(102,177),(103,180),(104,183),(105,239),(106,234),(107,237),(108,240),(109,235),(110,238),(111,233),(112,236),(121,207),(122,202),(123,205),(124,208),(125,203),(126,206),(127,201),(128,204),(129,212),(130,215),(131,210),(132,213),(133,216),(134,211),(135,214),(136,209),(137,224),(138,219),(139,222),(140,217),(141,220),(142,223),(143,218),(144,221),(145,231),(146,226),(147,229),(148,232),(149,227),(150,230),(151,225),(152,228),(161,187),(162,190),(163,185),(164,188),(165,191),(166,186),(167,189),(168,192),(169,200),(170,195),(171,198),(172,193),(173,196),(174,199),(175,194),(176,197)], [(1,225,70,51,59),(2,226,71,52,60),(3,227,72,53,61),(4,228,65,54,62),(5,229,66,55,63),(6,230,67,56,64),(7,231,68,49,57),(8,232,69,50,58),(9,144,185,193,91),(10,137,186,194,92),(11,138,187,195,93),(12,139,188,196,94),(13,140,189,197,95),(14,141,190,198,96),(15,142,191,199,89),(16,143,192,200,90),(17,84,153,28,150),(18,85,154,29,151),(19,86,155,30,152),(20,87,156,31,145),(21,88,157,32,146),(22,81,158,25,147),(23,82,159,26,148),(24,83,160,27,149),(33,105,205,209,101),(34,106,206,210,102),(35,107,207,211,103),(36,108,208,212,104),(37,109,201,213,97),(38,110,202,214,98),(39,111,203,215,99),(40,112,204,216,100),(41,115,220,162,171),(42,116,221,163,172),(43,117,222,164,173),(44,118,223,165,174),(45,119,224,166,175),(46,120,217,167,176),(47,113,218,168,169),(48,114,219,161,170),(73,240,124,129,183),(74,233,125,130,184),(75,234,126,131,177),(76,235,127,132,178),(77,236,128,133,179),(78,237,121,134,180),(79,238,122,135,181),(80,239,123,136,182)], [(1,59),(2,64),(3,61),(4,58),(5,63),(6,60),(7,57),(8,62),(9,185),(10,190),(11,187),(12,192),(13,189),(14,186),(15,191),(16,188),(18,22),(20,24),(25,154),(26,159),(27,156),(28,153),(29,158),(30,155),(31,160),(32,157),(33,201),(34,206),(35,203),(36,208),(37,205),(38,202),(39,207),(40,204),(41,171),(42,176),(43,173),(44,170),(45,175),(46,172),(47,169),(48,174),(49,231),(50,228),(51,225),(52,230),(53,227),(54,232),(55,229),(56,226),(65,69),(67,71),(73,128),(74,125),(75,122),(76,127),(77,124),(78,121),(79,126),(80,123),(81,151),(82,148),(83,145),(84,150),(85,147),(86,152),(87,149),(88,146),(89,199),(90,196),(91,193),(92,198),(93,195),(94,200),(95,197),(96,194),(97,209),(98,214),(99,211),(100,216),(101,213),(102,210),(103,215),(104,212),(105,109),(107,111),(113,168),(114,165),(115,162),(116,167),(117,164),(118,161),(119,166),(120,163),(129,179),(130,184),(131,181),(132,178),(133,183),(134,180),(135,177),(136,182),(137,141),(139,143),(217,221),(219,223),(234,238),(236,240)]])
75 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 60E | 60F | 60G | 60H | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 4 | 10 | 1 | 1 | 2 | 4 | 10 | 20 | 20 | 2 | 2 | 1 | 1 | 4 | 4 | 10 | 10 | 4 | 20 | 2 | 2 | 8 | 8 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 20 | 20 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
75 irreducible representations
| dim | 1 | 1 | 1 | 1 | 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | D5 | D10 | D10 | D10 | C3×D4 | C3×D4 | C3×D5 | C6×D5 | C6×D5 | C6×D5 | C8.C22 | D4×D5 | C3×C8.C22 | SD16⋊D5 | C3×D4×D5 | C3×SD16⋊D5 |
| kernel | C3×SD16⋊D5 | C3×C8⋊D5 | C3×Dic20 | C3×D4.D5 | C3×C5⋊Q16 | C15×SD16 | C3×D4⋊2D5 | C3×Q8×D5 | SD16⋊D5 | C8⋊D5 | Dic20 | D4.D5 | C5⋊Q16 | C5×SD16 | D4⋊2D5 | Q8×D5 | C3×Dic5 | C6×D5 | C3×SD16 | C24 | C3×D4 | C3×Q8 | Dic5 | D10 | SD16 | C8 | D4 | Q8 | C15 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C3×SD16⋊D5 ►in GL4(𝔽241) generated by
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 0 | 62 | 208 |
| 0 | 0 | 0 | 33 |
| 35 | 35 | 179 | 66 |
| 0 | 168 | 117 | 62 |
| 64 | 92 | 6 | 59 |
| 149 | 177 | 0 | 182 |
| 206 | 206 | 28 | 149 |
| 0 | 73 | 56 | 213 |
| 51 | 240 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 52 | 240 |
| 0 | 0 | 53 | 240 |
| 51 | 240 | 0 | 0 |
| 190 | 190 | 0 | 0 |
| 0 | 0 | 0 | 51 |
| 0 | 0 | 52 | 0 |
G:=sub<GL(4,GF(241))| [15,0,0,0,0,15,0,0,0,0,15,0,0,0,0,15],[0,0,35,0,0,0,35,168,62,0,179,117,208,33,66,62],[64,149,206,0,92,177,206,73,6,0,28,56,59,182,149,213],[51,1,0,0,240,0,0,0,0,0,52,53,0,0,240,240],[51,190,0,0,240,190,0,0,0,0,0,52,0,0,51,0] >;
C3×SD16⋊D5 in GAP, Magma, Sage, TeX
C_3\times {\rm SD}_{16}\rtimes D_5 % in TeX
G:=Group("C3xSD16:D5"); // GroupNames label
G:=SmallGroup(480,708);
// by ID
G=gap.SmallGroup(480,708);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,344,1094,303,1271,648,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^8=c^2=d^5=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^3,b*d=d*b,e*b*e=b^5,c*d=d*c,e*c*e=b^4*c,e*d*e=d^-1>;
// generators/relations