direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D10.12D4, C4⋊Dic5⋊4C6, (C6×D5).69D4, C6.172(D4×D5), C10.18(C6×D4), C23.D5⋊4C6, D10⋊C4⋊5C6, C23.4(C6×D5), D10.11(C3×D4), C30.331(C2×D4), (C22×C6).4D10, C10.D4⋊10C6, (C2×C12).273D10, C30.187(C4○D4), C6.115(C4○D20), (C2×C60).263C22, (C2×C30).341C23, C6.110(D4⋊2D5), C15⋊27(C22.D4), (C22×C30).99C22, (C6×Dic5).155C22, C2.8(C3×D4×D5), (C2×C4×D5)⋊10C6, (D5×C2×C12)⋊26C2, (C5×C22⋊C4)⋊5C6, C22⋊C4⋊3(C3×D5), C10.7(C3×C4○D4), (C2×C4).24(C6×D5), (C2×C5⋊D4).3C6, C22.42(D5×C2×C6), (C2×C20).50(C2×C6), (C3×C4⋊Dic5)⋊22C2, (C3×C22⋊C4)⋊11D5, C2.10(C3×C4○D20), C2.8(C3×D4⋊2D5), (C6×C5⋊D4).10C2, (C15×C22⋊C4)⋊14C2, C5⋊1(C3×C22.D4), (C3×C23.D5)⋊20C2, (C2×Dic5).7(C2×C6), (C3×D10⋊C4)⋊16C2, (D5×C2×C6).126C22, (C3×C10.D4)⋊26C2, (C22×C10).18(C2×C6), (C2×C10).24(C22×C6), (C22×D5).23(C2×C6), (C2×C6).337(C22×D5), SmallGroup(480,676)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D10.12D4
G = < a,b,c,d,e | a3=b10=c2=d4=1, e2=b5, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=b5c, ce=ec, ede-1=b5d-1 >
Subgroups: 512 in 156 conjugacy classes, 62 normal (58 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C12, C2×C6, C2×C6, C15, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C3×D5, C30, C30, C22.D4, C4×D5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, C2×C30, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C2×C5⋊D4, C3×C22.D4, D5×C12, C6×Dic5, C3×C5⋊D4, C2×C60, D5×C2×C6, C22×C30, D10.12D4, C3×C10.D4, C3×C4⋊Dic5, C3×D10⋊C4, C3×C23.D5, C15×C22⋊C4, D5×C2×C12, C6×C5⋊D4, C3×D10.12D4
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, C4○D4, D10, C3×D4, C22×C6, C3×D5, C22.D4, C22×D5, C6×D4, C3×C4○D4, C6×D5, C4○D20, D4×D5, D4⋊2D5, C3×C22.D4, D5×C2×C6, D10.12D4, C3×C4○D20, C3×D4×D5, C3×D4⋊2D5, C3×D10.12D4
►On 240 points
Generators in S240
(1 56 36)(2 57 37)(3 58 38)(4 59 39)(5 60 40)(6 51 31)(7 52 32)(8 53 33)(9 54 34)(10 55 35)(11 225 205)(12 226 206)(13 227 207)(14 228 208)(15 229 209)(16 230 210)(17 221 201)(18 222 202)(19 223 203)(20 224 204)(21 61 41)(22 62 42)(23 63 43)(24 64 44)(25 65 45)(26 66 46)(27 67 47)(28 68 48)(29 69 49)(30 70 50)(71 111 91)(72 112 92)(73 113 93)(74 114 94)(75 115 95)(76 116 96)(77 117 97)(78 118 98)(79 119 99)(80 120 100)(81 121 101)(82 122 102)(83 123 103)(84 124 104)(85 125 105)(86 126 106)(87 127 107)(88 128 108)(89 129 109)(90 130 110)(131 171 151)(132 172 152)(133 173 153)(134 174 154)(135 175 155)(136 176 156)(137 177 157)(138 178 158)(139 179 159)(140 180 160)(141 181 161)(142 182 162)(143 183 163)(144 184 164)(145 185 165)(146 186 166)(147 187 167)(148 188 168)(149 189 169)(150 190 170)(191 231 211)(192 232 212)(193 233 213)(194 234 214)(195 235 215)(196 236 216)(197 237 217)(198 238 218)(199 239 219)(200 240 220)
(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 24)(2 23)(3 22)(4 21)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 240)(12 239)(13 238)(14 237)(15 236)(16 235)(17 234)(18 233)(19 232)(20 231)(31 49)(32 48)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(40 50)(51 69)(52 68)(53 67)(54 66)(55 65)(56 64)(57 63)(58 62)(59 61)(60 70)(71 84)(72 83)(73 82)(74 81)(75 90)(76 89)(77 88)(78 87)(79 86)(80 85)(91 104)(92 103)(93 102)(94 101)(95 110)(96 109)(97 108)(98 107)(99 106)(100 105)(111 124)(112 123)(113 122)(114 121)(115 130)(116 129)(117 128)(118 127)(119 126)(120 125)(131 149)(132 148)(133 147)(134 146)(135 145)(136 144)(137 143)(138 142)(139 141)(140 150)(151 169)(152 168)(153 167)(154 166)(155 165)(156 164)(157 163)(158 162)(159 161)(160 170)(171 189)(172 188)(173 187)(174 186)(175 185)(176 184)(177 183)(178 182)(179 181)(180 190)(191 204)(192 203)(193 202)(194 201)(195 210)(196 209)(197 208)(198 207)(199 206)(200 205)(211 224)(212 223)(213 222)(214 221)(215 230)(216 229)(217 228)(218 227)(219 226)(220 225)
(1 210 30 196)(2 201 21 197)(3 202 22 198)(4 203 23 199)(5 204 24 200)(6 205 25 191)(7 206 26 192)(8 207 27 193)(9 208 28 194)(10 209 29 195)(11 65 231 51)(12 66 232 52)(13 67 233 53)(14 68 234 54)(15 69 235 55)(16 70 236 56)(17 61 237 57)(18 62 238 58)(19 63 239 59)(20 64 240 60)(31 225 45 211)(32 226 46 212)(33 227 47 213)(34 228 48 214)(35 229 49 215)(36 230 50 216)(37 221 41 217)(38 222 42 218)(39 223 43 219)(40 224 44 220)(71 145 85 131)(72 146 86 132)(73 147 87 133)(74 148 88 134)(75 149 89 135)(76 150 90 136)(77 141 81 137)(78 142 82 138)(79 143 83 139)(80 144 84 140)(91 165 105 151)(92 166 106 152)(93 167 107 153)(94 168 108 154)(95 169 109 155)(96 170 110 156)(97 161 101 157)(98 162 102 158)(99 163 103 159)(100 164 104 160)(111 185 125 171)(112 186 126 172)(113 187 127 173)(114 188 128 174)(115 189 129 175)(116 190 130 176)(117 181 121 177)(118 182 122 178)(119 183 123 179)(120 184 124 180)
(1 131 6 136)(2 132 7 137)(3 133 8 138)(4 134 9 139)(5 135 10 140)(11 125 16 130)(12 126 17 121)(13 127 18 122)(14 128 19 123)(15 129 20 124)(21 146 26 141)(22 147 27 142)(23 148 28 143)(24 149 29 144)(25 150 30 145)(31 156 36 151)(32 157 37 152)(33 158 38 153)(34 159 39 154)(35 160 40 155)(41 166 46 161)(42 167 47 162)(43 168 48 163)(44 169 49 164)(45 170 50 165)(51 176 56 171)(52 177 57 172)(53 178 58 173)(54 179 59 174)(55 180 60 175)(61 186 66 181)(62 187 67 182)(63 188 68 183)(64 189 69 184)(65 190 70 185)(71 196 76 191)(72 197 77 192)(73 198 78 193)(74 199 79 194)(75 200 80 195)(81 206 86 201)(82 207 87 202)(83 208 88 203)(84 209 89 204)(85 210 90 205)(91 216 96 211)(92 217 97 212)(93 218 98 213)(94 219 99 214)(95 220 100 215)(101 226 106 221)(102 227 107 222)(103 228 108 223)(104 229 109 224)(105 230 110 225)(111 236 116 231)(112 237 117 232)(113 238 118 233)(114 239 119 234)(115 240 120 235)
G:=sub<Sym(240)| (1,56,36)(2,57,37)(3,58,38)(4,59,39)(5,60,40)(6,51,31)(7,52,32)(8,53,33)(9,54,34)(10,55,35)(11,225,205)(12,226,206)(13,227,207)(14,228,208)(15,229,209)(16,230,210)(17,221,201)(18,222,202)(19,223,203)(20,224,204)(21,61,41)(22,62,42)(23,63,43)(24,64,44)(25,65,45)(26,66,46)(27,67,47)(28,68,48)(29,69,49)(30,70,50)(71,111,91)(72,112,92)(73,113,93)(74,114,94)(75,115,95)(76,116,96)(77,117,97)(78,118,98)(79,119,99)(80,120,100)(81,121,101)(82,122,102)(83,123,103)(84,124,104)(85,125,105)(86,126,106)(87,127,107)(88,128,108)(89,129,109)(90,130,110)(131,171,151)(132,172,152)(133,173,153)(134,174,154)(135,175,155)(136,176,156)(137,177,157)(138,178,158)(139,179,159)(140,180,160)(141,181,161)(142,182,162)(143,183,163)(144,184,164)(145,185,165)(146,186,166)(147,187,167)(148,188,168)(149,189,169)(150,190,170)(191,231,211)(192,232,212)(193,233,213)(194,234,214)(195,235,215)(196,236,216)(197,237,217)(198,238,218)(199,239,219)(200,240,220), (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), (1,24)(2,23)(3,22)(4,21)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,240)(12,239)(13,238)(14,237)(15,236)(16,235)(17,234)(18,233)(19,232)(20,231)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(40,50)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)(60,70)(71,84)(72,83)(73,82)(74,81)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(91,104)(92,103)(93,102)(94,101)(95,110)(96,109)(97,108)(98,107)(99,106)(100,105)(111,124)(112,123)(113,122)(114,121)(115,130)(116,129)(117,128)(118,127)(119,126)(120,125)(131,149)(132,148)(133,147)(134,146)(135,145)(136,144)(137,143)(138,142)(139,141)(140,150)(151,169)(152,168)(153,167)(154,166)(155,165)(156,164)(157,163)(158,162)(159,161)(160,170)(171,189)(172,188)(173,187)(174,186)(175,185)(176,184)(177,183)(178,182)(179,181)(180,190)(191,204)(192,203)(193,202)(194,201)(195,210)(196,209)(197,208)(198,207)(199,206)(200,205)(211,224)(212,223)(213,222)(214,221)(215,230)(216,229)(217,228)(218,227)(219,226)(220,225), (1,210,30,196)(2,201,21,197)(3,202,22,198)(4,203,23,199)(5,204,24,200)(6,205,25,191)(7,206,26,192)(8,207,27,193)(9,208,28,194)(10,209,29,195)(11,65,231,51)(12,66,232,52)(13,67,233,53)(14,68,234,54)(15,69,235,55)(16,70,236,56)(17,61,237,57)(18,62,238,58)(19,63,239,59)(20,64,240,60)(31,225,45,211)(32,226,46,212)(33,227,47,213)(34,228,48,214)(35,229,49,215)(36,230,50,216)(37,221,41,217)(38,222,42,218)(39,223,43,219)(40,224,44,220)(71,145,85,131)(72,146,86,132)(73,147,87,133)(74,148,88,134)(75,149,89,135)(76,150,90,136)(77,141,81,137)(78,142,82,138)(79,143,83,139)(80,144,84,140)(91,165,105,151)(92,166,106,152)(93,167,107,153)(94,168,108,154)(95,169,109,155)(96,170,110,156)(97,161,101,157)(98,162,102,158)(99,163,103,159)(100,164,104,160)(111,185,125,171)(112,186,126,172)(113,187,127,173)(114,188,128,174)(115,189,129,175)(116,190,130,176)(117,181,121,177)(118,182,122,178)(119,183,123,179)(120,184,124,180), (1,131,6,136)(2,132,7,137)(3,133,8,138)(4,134,9,139)(5,135,10,140)(11,125,16,130)(12,126,17,121)(13,127,18,122)(14,128,19,123)(15,129,20,124)(21,146,26,141)(22,147,27,142)(23,148,28,143)(24,149,29,144)(25,150,30,145)(31,156,36,151)(32,157,37,152)(33,158,38,153)(34,159,39,154)(35,160,40,155)(41,166,46,161)(42,167,47,162)(43,168,48,163)(44,169,49,164)(45,170,50,165)(51,176,56,171)(52,177,57,172)(53,178,58,173)(54,179,59,174)(55,180,60,175)(61,186,66,181)(62,187,67,182)(63,188,68,183)(64,189,69,184)(65,190,70,185)(71,196,76,191)(72,197,77,192)(73,198,78,193)(74,199,79,194)(75,200,80,195)(81,206,86,201)(82,207,87,202)(83,208,88,203)(84,209,89,204)(85,210,90,205)(91,216,96,211)(92,217,97,212)(93,218,98,213)(94,219,99,214)(95,220,100,215)(101,226,106,221)(102,227,107,222)(103,228,108,223)(104,229,109,224)(105,230,110,225)(111,236,116,231)(112,237,117,232)(113,238,118,233)(114,239,119,234)(115,240,120,235)>;
G:=Group( (1,56,36)(2,57,37)(3,58,38)(4,59,39)(5,60,40)(6,51,31)(7,52,32)(8,53,33)(9,54,34)(10,55,35)(11,225,205)(12,226,206)(13,227,207)(14,228,208)(15,229,209)(16,230,210)(17,221,201)(18,222,202)(19,223,203)(20,224,204)(21,61,41)(22,62,42)(23,63,43)(24,64,44)(25,65,45)(26,66,46)(27,67,47)(28,68,48)(29,69,49)(30,70,50)(71,111,91)(72,112,92)(73,113,93)(74,114,94)(75,115,95)(76,116,96)(77,117,97)(78,118,98)(79,119,99)(80,120,100)(81,121,101)(82,122,102)(83,123,103)(84,124,104)(85,125,105)(86,126,106)(87,127,107)(88,128,108)(89,129,109)(90,130,110)(131,171,151)(132,172,152)(133,173,153)(134,174,154)(135,175,155)(136,176,156)(137,177,157)(138,178,158)(139,179,159)(140,180,160)(141,181,161)(142,182,162)(143,183,163)(144,184,164)(145,185,165)(146,186,166)(147,187,167)(148,188,168)(149,189,169)(150,190,170)(191,231,211)(192,232,212)(193,233,213)(194,234,214)(195,235,215)(196,236,216)(197,237,217)(198,238,218)(199,239,219)(200,240,220), (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), (1,24)(2,23)(3,22)(4,21)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,240)(12,239)(13,238)(14,237)(15,236)(16,235)(17,234)(18,233)(19,232)(20,231)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(40,50)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)(60,70)(71,84)(72,83)(73,82)(74,81)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(91,104)(92,103)(93,102)(94,101)(95,110)(96,109)(97,108)(98,107)(99,106)(100,105)(111,124)(112,123)(113,122)(114,121)(115,130)(116,129)(117,128)(118,127)(119,126)(120,125)(131,149)(132,148)(133,147)(134,146)(135,145)(136,144)(137,143)(138,142)(139,141)(140,150)(151,169)(152,168)(153,167)(154,166)(155,165)(156,164)(157,163)(158,162)(159,161)(160,170)(171,189)(172,188)(173,187)(174,186)(175,185)(176,184)(177,183)(178,182)(179,181)(180,190)(191,204)(192,203)(193,202)(194,201)(195,210)(196,209)(197,208)(198,207)(199,206)(200,205)(211,224)(212,223)(213,222)(214,221)(215,230)(216,229)(217,228)(218,227)(219,226)(220,225), (1,210,30,196)(2,201,21,197)(3,202,22,198)(4,203,23,199)(5,204,24,200)(6,205,25,191)(7,206,26,192)(8,207,27,193)(9,208,28,194)(10,209,29,195)(11,65,231,51)(12,66,232,52)(13,67,233,53)(14,68,234,54)(15,69,235,55)(16,70,236,56)(17,61,237,57)(18,62,238,58)(19,63,239,59)(20,64,240,60)(31,225,45,211)(32,226,46,212)(33,227,47,213)(34,228,48,214)(35,229,49,215)(36,230,50,216)(37,221,41,217)(38,222,42,218)(39,223,43,219)(40,224,44,220)(71,145,85,131)(72,146,86,132)(73,147,87,133)(74,148,88,134)(75,149,89,135)(76,150,90,136)(77,141,81,137)(78,142,82,138)(79,143,83,139)(80,144,84,140)(91,165,105,151)(92,166,106,152)(93,167,107,153)(94,168,108,154)(95,169,109,155)(96,170,110,156)(97,161,101,157)(98,162,102,158)(99,163,103,159)(100,164,104,160)(111,185,125,171)(112,186,126,172)(113,187,127,173)(114,188,128,174)(115,189,129,175)(116,190,130,176)(117,181,121,177)(118,182,122,178)(119,183,123,179)(120,184,124,180), (1,131,6,136)(2,132,7,137)(3,133,8,138)(4,134,9,139)(5,135,10,140)(11,125,16,130)(12,126,17,121)(13,127,18,122)(14,128,19,123)(15,129,20,124)(21,146,26,141)(22,147,27,142)(23,148,28,143)(24,149,29,144)(25,150,30,145)(31,156,36,151)(32,157,37,152)(33,158,38,153)(34,159,39,154)(35,160,40,155)(41,166,46,161)(42,167,47,162)(43,168,48,163)(44,169,49,164)(45,170,50,165)(51,176,56,171)(52,177,57,172)(53,178,58,173)(54,179,59,174)(55,180,60,175)(61,186,66,181)(62,187,67,182)(63,188,68,183)(64,189,69,184)(65,190,70,185)(71,196,76,191)(72,197,77,192)(73,198,78,193)(74,199,79,194)(75,200,80,195)(81,206,86,201)(82,207,87,202)(83,208,88,203)(84,209,89,204)(85,210,90,205)(91,216,96,211)(92,217,97,212)(93,218,98,213)(94,219,99,214)(95,220,100,215)(101,226,106,221)(102,227,107,222)(103,228,108,223)(104,229,109,224)(105,230,110,225)(111,236,116,231)(112,237,117,232)(113,238,118,233)(114,239,119,234)(115,240,120,235) );
G=PermutationGroup([[(1,56,36),(2,57,37),(3,58,38),(4,59,39),(5,60,40),(6,51,31),(7,52,32),(8,53,33),(9,54,34),(10,55,35),(11,225,205),(12,226,206),(13,227,207),(14,228,208),(15,229,209),(16,230,210),(17,221,201),(18,222,202),(19,223,203),(20,224,204),(21,61,41),(22,62,42),(23,63,43),(24,64,44),(25,65,45),(26,66,46),(27,67,47),(28,68,48),(29,69,49),(30,70,50),(71,111,91),(72,112,92),(73,113,93),(74,114,94),(75,115,95),(76,116,96),(77,117,97),(78,118,98),(79,119,99),(80,120,100),(81,121,101),(82,122,102),(83,123,103),(84,124,104),(85,125,105),(86,126,106),(87,127,107),(88,128,108),(89,129,109),(90,130,110),(131,171,151),(132,172,152),(133,173,153),(134,174,154),(135,175,155),(136,176,156),(137,177,157),(138,178,158),(139,179,159),(140,180,160),(141,181,161),(142,182,162),(143,183,163),(144,184,164),(145,185,165),(146,186,166),(147,187,167),(148,188,168),(149,189,169),(150,190,170),(191,231,211),(192,232,212),(193,233,213),(194,234,214),(195,235,215),(196,236,216),(197,237,217),(198,238,218),(199,239,219),(200,240,220)], [(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)], [(1,24),(2,23),(3,22),(4,21),(5,30),(6,29),(7,28),(8,27),(9,26),(10,25),(11,240),(12,239),(13,238),(14,237),(15,236),(16,235),(17,234),(18,233),(19,232),(20,231),(31,49),(32,48),(33,47),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41),(40,50),(51,69),(52,68),(53,67),(54,66),(55,65),(56,64),(57,63),(58,62),(59,61),(60,70),(71,84),(72,83),(73,82),(74,81),(75,90),(76,89),(77,88),(78,87),(79,86),(80,85),(91,104),(92,103),(93,102),(94,101),(95,110),(96,109),(97,108),(98,107),(99,106),(100,105),(111,124),(112,123),(113,122),(114,121),(115,130),(116,129),(117,128),(118,127),(119,126),(120,125),(131,149),(132,148),(133,147),(134,146),(135,145),(136,144),(137,143),(138,142),(139,141),(140,150),(151,169),(152,168),(153,167),(154,166),(155,165),(156,164),(157,163),(158,162),(159,161),(160,170),(171,189),(172,188),(173,187),(174,186),(175,185),(176,184),(177,183),(178,182),(179,181),(180,190),(191,204),(192,203),(193,202),(194,201),(195,210),(196,209),(197,208),(198,207),(199,206),(200,205),(211,224),(212,223),(213,222),(214,221),(215,230),(216,229),(217,228),(218,227),(219,226),(220,225)], [(1,210,30,196),(2,201,21,197),(3,202,22,198),(4,203,23,199),(5,204,24,200),(6,205,25,191),(7,206,26,192),(8,207,27,193),(9,208,28,194),(10,209,29,195),(11,65,231,51),(12,66,232,52),(13,67,233,53),(14,68,234,54),(15,69,235,55),(16,70,236,56),(17,61,237,57),(18,62,238,58),(19,63,239,59),(20,64,240,60),(31,225,45,211),(32,226,46,212),(33,227,47,213),(34,228,48,214),(35,229,49,215),(36,230,50,216),(37,221,41,217),(38,222,42,218),(39,223,43,219),(40,224,44,220),(71,145,85,131),(72,146,86,132),(73,147,87,133),(74,148,88,134),(75,149,89,135),(76,150,90,136),(77,141,81,137),(78,142,82,138),(79,143,83,139),(80,144,84,140),(91,165,105,151),(92,166,106,152),(93,167,107,153),(94,168,108,154),(95,169,109,155),(96,170,110,156),(97,161,101,157),(98,162,102,158),(99,163,103,159),(100,164,104,160),(111,185,125,171),(112,186,126,172),(113,187,127,173),(114,188,128,174),(115,189,129,175),(116,190,130,176),(117,181,121,177),(118,182,122,178),(119,183,123,179),(120,184,124,180)], [(1,131,6,136),(2,132,7,137),(3,133,8,138),(4,134,9,139),(5,135,10,140),(11,125,16,130),(12,126,17,121),(13,127,18,122),(14,128,19,123),(15,129,20,124),(21,146,26,141),(22,147,27,142),(23,148,28,143),(24,149,29,144),(25,150,30,145),(31,156,36,151),(32,157,37,152),(33,158,38,153),(34,159,39,154),(35,160,40,155),(41,166,46,161),(42,167,47,162),(43,168,48,163),(44,169,49,164),(45,170,50,165),(51,176,56,171),(52,177,57,172),(53,178,58,173),(54,179,59,174),(55,180,60,175),(61,186,66,181),(62,187,67,182),(63,188,68,183),(64,189,69,184),(65,190,70,185),(71,196,76,191),(72,197,77,192),(73,198,78,193),(74,199,79,194),(75,200,80,195),(81,206,86,201),(82,207,87,202),(83,208,88,203),(84,209,89,204),(85,210,90,205),(91,216,96,211),(92,217,97,212),(93,218,98,213),(94,219,99,214),(95,220,100,215),(101,226,106,221),(102,227,107,222),(103,228,108,223),(104,229,109,224),(105,230,110,225),(111,236,116,231),(112,237,117,232),(113,238,118,233),(114,239,119,234),(115,240,120,235)]])
102 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 12K | 12L | 12M | 12N | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 30M | ··· | 30T | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 4 | 10 | 10 | 1 | 1 | 2 | 2 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 1 | ··· | 1 | 4 | 4 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 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 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | C6 | D4 | D5 | C4○D4 | D10 | D10 | C3×D4 | C3×D5 | C3×C4○D4 | C6×D5 | C6×D5 | C4○D20 | C3×C4○D20 | D4×D5 | D4⋊2D5 | C3×D4×D5 | C3×D4⋊2D5 |
| kernel | C3×D10.12D4 | C3×C10.D4 | C3×C4⋊Dic5 | C3×D10⋊C4 | C3×C23.D5 | C15×C22⋊C4 | D5×C2×C12 | C6×C5⋊D4 | D10.12D4 | C10.D4 | C4⋊Dic5 | D10⋊C4 | C23.D5 | C5×C22⋊C4 | C2×C4×D5 | C2×C5⋊D4 | C6×D5 | C3×C22⋊C4 | C30 | C2×C12 | C22×C6 | D10 | C22⋊C4 | C10 | C2×C4 | C23 | C6 | C2 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 4 | 4 | 8 | 8 | 4 | 8 | 16 | 2 | 2 | 4 | 4 |
Matrix representation of C3×D10.12D4 ►in GL4(𝔽61) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 43 | 0 | 0 |
| 17 | 18 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 60 |
| 25 | 50 | 0 | 0 |
| 7 | 36 | 0 | 0 |
| 0 | 0 | 0 | 11 |
| 0 | 0 | 50 | 0 |
| 11 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 11 |
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[0,17,0,0,43,18,0,0,0,0,60,0,0,0,0,60],[60,1,0,0,0,1,0,0,0,0,1,0,0,0,0,60],[25,7,0,0,50,36,0,0,0,0,0,50,0,0,11,0],[11,0,0,0,0,11,0,0,0,0,50,0,0,0,0,11] >;
C3×D10.12D4 in GAP, Magma, Sage, TeX
C_3\times D_{10}._{12}D_4 % in TeX
G:=Group("C3xD10.12D4"); // GroupNames label
G:=SmallGroup(480,676);
// by ID
G=gap.SmallGroup(480,676);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,176,590,555,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^10=c^2=d^4=1,e^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^5*c,c*e=e*c,e*d*e^-1=b^5*d^-1>;
// generators/relations