direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D4.8D10, C60.235D4, C60.207C23, D4⋊D5⋊7C6, Q8⋊D5⋊7C6, C4○D20⋊4C6, D4.D5⋊7C6, C5⋊Q16⋊7C6, D4.8(C6×D5), C15⋊38(C4○D8), C10.60(C6×D4), C20.57(C3×D4), (C2×C30).86D4, Q8.13(C6×D5), D20.12(C2×C6), (C3×D4).37D10, C30.417(C2×D4), (C3×Q8).40D10, (C2×C12).369D10, C20.18(C22×C6), C12.125(C5⋊D4), (C2×C60).303C22, Dic10.11(C2×C6), (D4×C15).42C22, (C3×D20).51C22, C12.207(C22×D5), (Q8×C15).45C22, (C3×Dic10).53C22, C5⋊5(C3×C4○D8), C4.18(D5×C2×C6), (C2×C5⋊2C8)⋊8C6, C4○D4⋊4(C3×D5), (C5×C4○D4)⋊6C6, (C3×C4○D4)⋊7D5, (C6×C5⋊2C8)⋊22C2, (C3×D4⋊D5)⋊15C2, (C15×C4○D4)⋊7C2, (C3×Q8⋊D5)⋊15C2, (C5×D4).8(C2×C6), (C2×C10).9(C3×D4), (C2×C4).59(C6×D5), C4.32(C3×C5⋊D4), C2.24(C6×C5⋊D4), (C3×C4○D20)⋊14C2, (C2×C20).40(C2×C6), C5⋊2C8.10(C2×C6), (C3×D4.D5)⋊15C2, (C3×C5⋊Q16)⋊15C2, C6.145(C2×C5⋊D4), (C5×Q8).16(C2×C6), C22.1(C3×C5⋊D4), (C2×C6).23(C5⋊D4), (C3×C5⋊2C8).50C22, SmallGroup(480,743)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D4.8D10
G = < a,b,c,d,e | a3=b4=c2=1, d10=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d9 >
Subgroups: 368 in 124 conjugacy classes, 58 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C2×C8, D8, SD16, Q16, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C3×D5, C30, C30, C4○D8, C5⋊2C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C2×C24, C3×D8, C3×SD16, C3×Q16, C3×C4○D4, C3×C4○D4, C3×Dic5, C60, C60, C6×D5, C2×C30, C2×C30, C2×C5⋊2C8, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C4○D20, C5×C4○D4, C3×C4○D8, C3×C5⋊2C8, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, D4.8D10, C6×C5⋊2C8, C3×D4⋊D5, C3×D4.D5, C3×Q8⋊D5, C3×C5⋊Q16, C3×C4○D20, C15×C4○D4, C3×D4.8D10
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, D10, C3×D4, C22×C6, C3×D5, C4○D8, C5⋊D4, C22×D5, C6×D4, C6×D5, C2×C5⋊D4, C3×C4○D8, C3×C5⋊D4, D5×C2×C6, D4.8D10, C6×C5⋊D4, C3×D4.8D10
►On 240 points
Generators in S240
(1 57 28)(2 58 29)(3 59 30)(4 60 31)(5 41 32)(6 42 33)(7 43 34)(8 44 35)(9 45 36)(10 46 37)(11 47 38)(12 48 39)(13 49 40)(14 50 21)(15 51 22)(16 52 23)(17 53 24)(18 54 25)(19 55 26)(20 56 27)(61 103 81)(62 104 82)(63 105 83)(64 106 84)(65 107 85)(66 108 86)(67 109 87)(68 110 88)(69 111 89)(70 112 90)(71 113 91)(72 114 92)(73 115 93)(74 116 94)(75 117 95)(76 118 96)(77 119 97)(78 120 98)(79 101 99)(80 102 100)(121 174 143)(122 175 144)(123 176 145)(124 177 146)(125 178 147)(126 179 148)(127 180 149)(128 161 150)(129 162 151)(130 163 152)(131 164 153)(132 165 154)(133 166 155)(134 167 156)(135 168 157)(136 169 158)(137 170 159)(138 171 160)(139 172 141)(140 173 142)(181 224 212)(182 225 213)(183 226 214)(184 227 215)(185 228 216)(186 229 217)(187 230 218)(188 231 219)(189 232 220)(190 233 201)(191 234 202)(192 235 203)(193 236 204)(194 237 205)(195 238 206)(196 239 207)(197 240 208)(198 221 209)(199 222 210)(200 223 211)
(1 16 11 6)(2 17 12 7)(3 18 13 8)(4 19 14 9)(5 20 15 10)(21 36 31 26)(22 37 32 27)(23 38 33 28)(24 39 34 29)(25 40 35 30)(41 56 51 46)(42 57 52 47)(43 58 53 48)(44 59 54 49)(45 60 55 50)(61 66 71 76)(62 67 72 77)(63 68 73 78)(64 69 74 79)(65 70 75 80)(81 86 91 96)(82 87 92 97)(83 88 93 98)(84 89 94 99)(85 90 95 100)(101 106 111 116)(102 107 112 117)(103 108 113 118)(104 109 114 119)(105 110 115 120)(121 126 131 136)(122 127 132 137)(123 128 133 138)(124 129 134 139)(125 130 135 140)(141 146 151 156)(142 147 152 157)(143 148 153 158)(144 149 154 159)(145 150 155 160)(161 166 171 176)(162 167 172 177)(163 168 173 178)(164 169 174 179)(165 170 175 180)(181 196 191 186)(182 197 192 187)(183 198 193 188)(184 199 194 189)(185 200 195 190)(201 216 211 206)(202 217 212 207)(203 218 213 208)(204 219 214 209)(205 220 215 210)(221 236 231 226)(222 237 232 227)(223 238 233 228)(224 239 234 229)(225 240 235 230)
(1 72)(2 73)(3 74)(4 75)(5 76)(6 77)(7 78)(8 79)(9 80)(10 61)(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 71)(21 85)(22 86)(23 87)(24 88)(25 89)(26 90)(27 91)(28 92)(29 93)(30 94)(31 95)(32 96)(33 97)(34 98)(35 99)(36 100)(37 81)(38 82)(39 83)(40 84)(41 118)(42 119)(43 120)(44 101)(45 102)(46 103)(47 104)(48 105)(49 106)(50 107)(51 108)(52 109)(53 110)(54 111)(55 112)(56 113)(57 114)(58 115)(59 116)(60 117)(121 188)(122 189)(123 190)(124 191)(125 192)(126 193)(127 194)(128 195)(129 196)(130 197)(131 198)(132 199)(133 200)(134 181)(135 182)(136 183)(137 184)(138 185)(139 186)(140 187)(141 217)(142 218)(143 219)(144 220)(145 201)(146 202)(147 203)(148 204)(149 205)(150 206)(151 207)(152 208)(153 209)(154 210)(155 211)(156 212)(157 213)(158 214)(159 215)(160 216)(161 238)(162 239)(163 240)(164 221)(165 222)(166 223)(167 224)(168 225)(169 226)(170 227)(171 228)(172 229)(173 230)(174 231)(175 232)(176 233)(177 234)(178 235)(179 236)(180 237)
(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)
(1 121 11 131)(2 130 12 140)(3 139 13 129)(4 128 14 138)(5 137 15 127)(6 126 16 136)(7 135 17 125)(8 124 18 134)(9 133 19 123)(10 122 20 132)(21 160 31 150)(22 149 32 159)(23 158 33 148)(24 147 34 157)(25 156 35 146)(26 145 36 155)(27 154 37 144)(28 143 38 153)(29 152 39 142)(30 141 40 151)(41 170 51 180)(42 179 52 169)(43 168 53 178)(44 177 54 167)(45 166 55 176)(46 175 56 165)(47 164 57 174)(48 173 58 163)(49 162 59 172)(50 171 60 161)(61 194 71 184)(62 183 72 193)(63 192 73 182)(64 181 74 191)(65 190 75 200)(66 199 76 189)(67 188 77 198)(68 197 78 187)(69 186 79 196)(70 195 80 185)(81 205 91 215)(82 214 92 204)(83 203 93 213)(84 212 94 202)(85 201 95 211)(86 210 96 220)(87 219 97 209)(88 208 98 218)(89 217 99 207)(90 206 100 216)(101 239 111 229)(102 228 112 238)(103 237 113 227)(104 226 114 236)(105 235 115 225)(106 224 116 234)(107 233 117 223)(108 222 118 232)(109 231 119 221)(110 240 120 230)
G:=sub<Sym(240)| (1,57,28)(2,58,29)(3,59,30)(4,60,31)(5,41,32)(6,42,33)(7,43,34)(8,44,35)(9,45,36)(10,46,37)(11,47,38)(12,48,39)(13,49,40)(14,50,21)(15,51,22)(16,52,23)(17,53,24)(18,54,25)(19,55,26)(20,56,27)(61,103,81)(62,104,82)(63,105,83)(64,106,84)(65,107,85)(66,108,86)(67,109,87)(68,110,88)(69,111,89)(70,112,90)(71,113,91)(72,114,92)(73,115,93)(74,116,94)(75,117,95)(76,118,96)(77,119,97)(78,120,98)(79,101,99)(80,102,100)(121,174,143)(122,175,144)(123,176,145)(124,177,146)(125,178,147)(126,179,148)(127,180,149)(128,161,150)(129,162,151)(130,163,152)(131,164,153)(132,165,154)(133,166,155)(134,167,156)(135,168,157)(136,169,158)(137,170,159)(138,171,160)(139,172,141)(140,173,142)(181,224,212)(182,225,213)(183,226,214)(184,227,215)(185,228,216)(186,229,217)(187,230,218)(188,231,219)(189,232,220)(190,233,201)(191,234,202)(192,235,203)(193,236,204)(194,237,205)(195,238,206)(196,239,207)(197,240,208)(198,221,209)(199,222,210)(200,223,211), (1,16,11,6)(2,17,12,7)(3,18,13,8)(4,19,14,9)(5,20,15,10)(21,36,31,26)(22,37,32,27)(23,38,33,28)(24,39,34,29)(25,40,35,30)(41,56,51,46)(42,57,52,47)(43,58,53,48)(44,59,54,49)(45,60,55,50)(61,66,71,76)(62,67,72,77)(63,68,73,78)(64,69,74,79)(65,70,75,80)(81,86,91,96)(82,87,92,97)(83,88,93,98)(84,89,94,99)(85,90,95,100)(101,106,111,116)(102,107,112,117)(103,108,113,118)(104,109,114,119)(105,110,115,120)(121,126,131,136)(122,127,132,137)(123,128,133,138)(124,129,134,139)(125,130,135,140)(141,146,151,156)(142,147,152,157)(143,148,153,158)(144,149,154,159)(145,150,155,160)(161,166,171,176)(162,167,172,177)(163,168,173,178)(164,169,174,179)(165,170,175,180)(181,196,191,186)(182,197,192,187)(183,198,193,188)(184,199,194,189)(185,200,195,190)(201,216,211,206)(202,217,212,207)(203,218,213,208)(204,219,214,209)(205,220,215,210)(221,236,231,226)(222,237,232,227)(223,238,233,228)(224,239,234,229)(225,240,235,230), (1,72)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,85)(22,86)(23,87)(24,88)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(33,97)(34,98)(35,99)(36,100)(37,81)(38,82)(39,83)(40,84)(41,118)(42,119)(43,120)(44,101)(45,102)(46,103)(47,104)(48,105)(49,106)(50,107)(51,108)(52,109)(53,110)(54,111)(55,112)(56,113)(57,114)(58,115)(59,116)(60,117)(121,188)(122,189)(123,190)(124,191)(125,192)(126,193)(127,194)(128,195)(129,196)(130,197)(131,198)(132,199)(133,200)(134,181)(135,182)(136,183)(137,184)(138,185)(139,186)(140,187)(141,217)(142,218)(143,219)(144,220)(145,201)(146,202)(147,203)(148,204)(149,205)(150,206)(151,207)(152,208)(153,209)(154,210)(155,211)(156,212)(157,213)(158,214)(159,215)(160,216)(161,238)(162,239)(163,240)(164,221)(165,222)(166,223)(167,224)(168,225)(169,226)(170,227)(171,228)(172,229)(173,230)(174,231)(175,232)(176,233)(177,234)(178,235)(179,236)(180,237), (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), (1,121,11,131)(2,130,12,140)(3,139,13,129)(4,128,14,138)(5,137,15,127)(6,126,16,136)(7,135,17,125)(8,124,18,134)(9,133,19,123)(10,122,20,132)(21,160,31,150)(22,149,32,159)(23,158,33,148)(24,147,34,157)(25,156,35,146)(26,145,36,155)(27,154,37,144)(28,143,38,153)(29,152,39,142)(30,141,40,151)(41,170,51,180)(42,179,52,169)(43,168,53,178)(44,177,54,167)(45,166,55,176)(46,175,56,165)(47,164,57,174)(48,173,58,163)(49,162,59,172)(50,171,60,161)(61,194,71,184)(62,183,72,193)(63,192,73,182)(64,181,74,191)(65,190,75,200)(66,199,76,189)(67,188,77,198)(68,197,78,187)(69,186,79,196)(70,195,80,185)(81,205,91,215)(82,214,92,204)(83,203,93,213)(84,212,94,202)(85,201,95,211)(86,210,96,220)(87,219,97,209)(88,208,98,218)(89,217,99,207)(90,206,100,216)(101,239,111,229)(102,228,112,238)(103,237,113,227)(104,226,114,236)(105,235,115,225)(106,224,116,234)(107,233,117,223)(108,222,118,232)(109,231,119,221)(110,240,120,230)>;
G:=Group( (1,57,28)(2,58,29)(3,59,30)(4,60,31)(5,41,32)(6,42,33)(7,43,34)(8,44,35)(9,45,36)(10,46,37)(11,47,38)(12,48,39)(13,49,40)(14,50,21)(15,51,22)(16,52,23)(17,53,24)(18,54,25)(19,55,26)(20,56,27)(61,103,81)(62,104,82)(63,105,83)(64,106,84)(65,107,85)(66,108,86)(67,109,87)(68,110,88)(69,111,89)(70,112,90)(71,113,91)(72,114,92)(73,115,93)(74,116,94)(75,117,95)(76,118,96)(77,119,97)(78,120,98)(79,101,99)(80,102,100)(121,174,143)(122,175,144)(123,176,145)(124,177,146)(125,178,147)(126,179,148)(127,180,149)(128,161,150)(129,162,151)(130,163,152)(131,164,153)(132,165,154)(133,166,155)(134,167,156)(135,168,157)(136,169,158)(137,170,159)(138,171,160)(139,172,141)(140,173,142)(181,224,212)(182,225,213)(183,226,214)(184,227,215)(185,228,216)(186,229,217)(187,230,218)(188,231,219)(189,232,220)(190,233,201)(191,234,202)(192,235,203)(193,236,204)(194,237,205)(195,238,206)(196,239,207)(197,240,208)(198,221,209)(199,222,210)(200,223,211), (1,16,11,6)(2,17,12,7)(3,18,13,8)(4,19,14,9)(5,20,15,10)(21,36,31,26)(22,37,32,27)(23,38,33,28)(24,39,34,29)(25,40,35,30)(41,56,51,46)(42,57,52,47)(43,58,53,48)(44,59,54,49)(45,60,55,50)(61,66,71,76)(62,67,72,77)(63,68,73,78)(64,69,74,79)(65,70,75,80)(81,86,91,96)(82,87,92,97)(83,88,93,98)(84,89,94,99)(85,90,95,100)(101,106,111,116)(102,107,112,117)(103,108,113,118)(104,109,114,119)(105,110,115,120)(121,126,131,136)(122,127,132,137)(123,128,133,138)(124,129,134,139)(125,130,135,140)(141,146,151,156)(142,147,152,157)(143,148,153,158)(144,149,154,159)(145,150,155,160)(161,166,171,176)(162,167,172,177)(163,168,173,178)(164,169,174,179)(165,170,175,180)(181,196,191,186)(182,197,192,187)(183,198,193,188)(184,199,194,189)(185,200,195,190)(201,216,211,206)(202,217,212,207)(203,218,213,208)(204,219,214,209)(205,220,215,210)(221,236,231,226)(222,237,232,227)(223,238,233,228)(224,239,234,229)(225,240,235,230), (1,72)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,85)(22,86)(23,87)(24,88)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(33,97)(34,98)(35,99)(36,100)(37,81)(38,82)(39,83)(40,84)(41,118)(42,119)(43,120)(44,101)(45,102)(46,103)(47,104)(48,105)(49,106)(50,107)(51,108)(52,109)(53,110)(54,111)(55,112)(56,113)(57,114)(58,115)(59,116)(60,117)(121,188)(122,189)(123,190)(124,191)(125,192)(126,193)(127,194)(128,195)(129,196)(130,197)(131,198)(132,199)(133,200)(134,181)(135,182)(136,183)(137,184)(138,185)(139,186)(140,187)(141,217)(142,218)(143,219)(144,220)(145,201)(146,202)(147,203)(148,204)(149,205)(150,206)(151,207)(152,208)(153,209)(154,210)(155,211)(156,212)(157,213)(158,214)(159,215)(160,216)(161,238)(162,239)(163,240)(164,221)(165,222)(166,223)(167,224)(168,225)(169,226)(170,227)(171,228)(172,229)(173,230)(174,231)(175,232)(176,233)(177,234)(178,235)(179,236)(180,237), (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), (1,121,11,131)(2,130,12,140)(3,139,13,129)(4,128,14,138)(5,137,15,127)(6,126,16,136)(7,135,17,125)(8,124,18,134)(9,133,19,123)(10,122,20,132)(21,160,31,150)(22,149,32,159)(23,158,33,148)(24,147,34,157)(25,156,35,146)(26,145,36,155)(27,154,37,144)(28,143,38,153)(29,152,39,142)(30,141,40,151)(41,170,51,180)(42,179,52,169)(43,168,53,178)(44,177,54,167)(45,166,55,176)(46,175,56,165)(47,164,57,174)(48,173,58,163)(49,162,59,172)(50,171,60,161)(61,194,71,184)(62,183,72,193)(63,192,73,182)(64,181,74,191)(65,190,75,200)(66,199,76,189)(67,188,77,198)(68,197,78,187)(69,186,79,196)(70,195,80,185)(81,205,91,215)(82,214,92,204)(83,203,93,213)(84,212,94,202)(85,201,95,211)(86,210,96,220)(87,219,97,209)(88,208,98,218)(89,217,99,207)(90,206,100,216)(101,239,111,229)(102,228,112,238)(103,237,113,227)(104,226,114,236)(105,235,115,225)(106,224,116,234)(107,233,117,223)(108,222,118,232)(109,231,119,221)(110,240,120,230) );
G=PermutationGroup([[(1,57,28),(2,58,29),(3,59,30),(4,60,31),(5,41,32),(6,42,33),(7,43,34),(8,44,35),(9,45,36),(10,46,37),(11,47,38),(12,48,39),(13,49,40),(14,50,21),(15,51,22),(16,52,23),(17,53,24),(18,54,25),(19,55,26),(20,56,27),(61,103,81),(62,104,82),(63,105,83),(64,106,84),(65,107,85),(66,108,86),(67,109,87),(68,110,88),(69,111,89),(70,112,90),(71,113,91),(72,114,92),(73,115,93),(74,116,94),(75,117,95),(76,118,96),(77,119,97),(78,120,98),(79,101,99),(80,102,100),(121,174,143),(122,175,144),(123,176,145),(124,177,146),(125,178,147),(126,179,148),(127,180,149),(128,161,150),(129,162,151),(130,163,152),(131,164,153),(132,165,154),(133,166,155),(134,167,156),(135,168,157),(136,169,158),(137,170,159),(138,171,160),(139,172,141),(140,173,142),(181,224,212),(182,225,213),(183,226,214),(184,227,215),(185,228,216),(186,229,217),(187,230,218),(188,231,219),(189,232,220),(190,233,201),(191,234,202),(192,235,203),(193,236,204),(194,237,205),(195,238,206),(196,239,207),(197,240,208),(198,221,209),(199,222,210),(200,223,211)], [(1,16,11,6),(2,17,12,7),(3,18,13,8),(4,19,14,9),(5,20,15,10),(21,36,31,26),(22,37,32,27),(23,38,33,28),(24,39,34,29),(25,40,35,30),(41,56,51,46),(42,57,52,47),(43,58,53,48),(44,59,54,49),(45,60,55,50),(61,66,71,76),(62,67,72,77),(63,68,73,78),(64,69,74,79),(65,70,75,80),(81,86,91,96),(82,87,92,97),(83,88,93,98),(84,89,94,99),(85,90,95,100),(101,106,111,116),(102,107,112,117),(103,108,113,118),(104,109,114,119),(105,110,115,120),(121,126,131,136),(122,127,132,137),(123,128,133,138),(124,129,134,139),(125,130,135,140),(141,146,151,156),(142,147,152,157),(143,148,153,158),(144,149,154,159),(145,150,155,160),(161,166,171,176),(162,167,172,177),(163,168,173,178),(164,169,174,179),(165,170,175,180),(181,196,191,186),(182,197,192,187),(183,198,193,188),(184,199,194,189),(185,200,195,190),(201,216,211,206),(202,217,212,207),(203,218,213,208),(204,219,214,209),(205,220,215,210),(221,236,231,226),(222,237,232,227),(223,238,233,228),(224,239,234,229),(225,240,235,230)], [(1,72),(2,73),(3,74),(4,75),(5,76),(6,77),(7,78),(8,79),(9,80),(10,61),(11,62),(12,63),(13,64),(14,65),(15,66),(16,67),(17,68),(18,69),(19,70),(20,71),(21,85),(22,86),(23,87),(24,88),(25,89),(26,90),(27,91),(28,92),(29,93),(30,94),(31,95),(32,96),(33,97),(34,98),(35,99),(36,100),(37,81),(38,82),(39,83),(40,84),(41,118),(42,119),(43,120),(44,101),(45,102),(46,103),(47,104),(48,105),(49,106),(50,107),(51,108),(52,109),(53,110),(54,111),(55,112),(56,113),(57,114),(58,115),(59,116),(60,117),(121,188),(122,189),(123,190),(124,191),(125,192),(126,193),(127,194),(128,195),(129,196),(130,197),(131,198),(132,199),(133,200),(134,181),(135,182),(136,183),(137,184),(138,185),(139,186),(140,187),(141,217),(142,218),(143,219),(144,220),(145,201),(146,202),(147,203),(148,204),(149,205),(150,206),(151,207),(152,208),(153,209),(154,210),(155,211),(156,212),(157,213),(158,214),(159,215),(160,216),(161,238),(162,239),(163,240),(164,221),(165,222),(166,223),(167,224),(168,225),(169,226),(170,227),(171,228),(172,229),(173,230),(174,231),(175,232),(176,233),(177,234),(178,235),(179,236),(180,237)], [(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)], [(1,121,11,131),(2,130,12,140),(3,139,13,129),(4,128,14,138),(5,137,15,127),(6,126,16,136),(7,135,17,125),(8,124,18,134),(9,133,19,123),(10,122,20,132),(21,160,31,150),(22,149,32,159),(23,158,33,148),(24,147,34,157),(25,156,35,146),(26,145,36,155),(27,154,37,144),(28,143,38,153),(29,152,39,142),(30,141,40,151),(41,170,51,180),(42,179,52,169),(43,168,53,178),(44,177,54,167),(45,166,55,176),(46,175,56,165),(47,164,57,174),(48,173,58,163),(49,162,59,172),(50,171,60,161),(61,194,71,184),(62,183,72,193),(63,192,73,182),(64,181,74,191),(65,190,75,200),(66,199,76,189),(67,188,77,198),(68,197,78,187),(69,186,79,196),(70,195,80,185),(81,205,91,215),(82,214,92,204),(83,203,93,213),(84,212,94,202),(85,201,95,211),(86,210,96,220),(87,219,97,209),(88,208,98,218),(89,217,99,207),(90,206,100,216),(101,239,111,229),(102,228,112,238),(103,237,113,227),(104,226,114,236),(105,235,115,225),(106,224,116,234),(107,233,117,223),(108,222,118,232),(109,231,119,221),(110,240,120,230)]])
102 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | 8B | 8C | 8D | 10A | 10B | 10C | ··· | 10H | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | ··· | 20J | 24A | ··· | 24H | 30A | 30B | 30C | 30D | 30E | ··· | 30P | 60A | ··· | 60H | 60I | ··· | 60T |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 4 | 20 | 1 | 1 | 1 | 1 | 2 | 4 | 20 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 20 | 20 | 10 | 10 | 10 | 10 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
102 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 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | C4○D8 | C5⋊D4 | C5⋊D4 | C6×D5 | C6×D5 | C6×D5 | C3×C4○D8 | C3×C5⋊D4 | C3×C5⋊D4 | D4.8D10 | C3×D4.8D10 |
| kernel | C3×D4.8D10 | C6×C5⋊2C8 | C3×D4⋊D5 | C3×D4.D5 | C3×Q8⋊D5 | C3×C5⋊Q16 | C3×C4○D20 | C15×C4○D4 | D4.8D10 | C2×C5⋊2C8 | D4⋊D5 | D4.D5 | Q8⋊D5 | C5⋊Q16 | C4○D20 | C5×C4○D4 | C60 | C2×C30 | C3×C4○D4 | C2×C12 | C3×D4 | C3×Q8 | C20 | C2×C10 | C4○D4 | C15 | C12 | C2×C6 | C2×C4 | D4 | Q8 | C5 | C4 | C22 | C3 | 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 | 4 | 4 | 4 | 8 | 8 | 8 | 4 | 8 |
Matrix representation of C3×D4.8D10 ►in GL6(𝔽241)
| 225 | 0 | 0 | 0 | 0 | 0 |
| 0 | 225 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 64 | 177 |
| 17 | 2 | 0 | 0 | 0 | 0 |
| 97 | 224 | 0 | 0 | 0 | 0 |
| 0 | 0 | 214 | 192 | 0 | 0 |
| 0 | 0 | 187 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 177 | 128 |
| 0 | 0 | 0 | 0 | 177 | 64 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 52 | 190 | 0 | 0 |
| 0 | 0 | 52 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 177 | 0 |
| 0 | 0 | 0 | 0 | 0 | 177 |
| 206 | 209 | 0 | 0 | 0 | 0 |
| 219 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 199 | 9 | 0 | 0 |
| 0 | 0 | 179 | 42 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 181 |
| 0 | 0 | 0 | 0 | 11 | 211 |
G:=sub<GL(6,GF(241))| [225,0,0,0,0,0,0,225,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,64,64,0,0,0,0,0,177],[17,97,0,0,0,0,2,224,0,0,0,0,0,0,214,187,0,0,0,0,192,27,0,0,0,0,0,0,177,177,0,0,0,0,128,64],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,52,52,0,0,0,0,190,0,0,0,0,0,0,0,177,0,0,0,0,0,0,177],[206,219,0,0,0,0,209,35,0,0,0,0,0,0,199,179,0,0,0,0,9,42,0,0,0,0,0,0,30,11,0,0,0,0,181,211] >;
C3×D4.8D10 in GAP, Magma, Sage, TeX
C_3\times D_4._8D_{10} % in TeX
G:=Group("C3xD4.8D10"); // GroupNames label
G:=SmallGroup(480,743);
// by ID
G=gap.SmallGroup(480,743);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,344,590,2524,648,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^2=1,d^10=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^9>;
// generators/relations