direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C42⋊D5, (C4×C12)⋊1D5, (C4×C60)⋊21C2, (C4×C20)⋊15C6, (D5×C12)⋊8C4, (C4×D5)⋊3C12, C42⋊4(C3×D5), (C4×Dic5)⋊8C6, C12.92(C4×D5), C4.22(D5×C12), C60.213(C2×C4), C20.47(C2×C12), D10.9(C2×C12), (C12×Dic5)⋊26C2, D10⋊C4.7C6, (C2×C12).374D10, C10.D4⋊17C6, C15⋊22(C42⋊C2), C30.182(C4○D4), C6.110(C4○D20), (C2×C60).444C22, C10.16(C22×C12), C30.174(C22×C4), (C2×C30).330C23, Dic5.11(C2×C12), (C6×Dic5).232C22, (C2×C4×D5).9C6, C6.99(C2×C4×D5), C2.5(D5×C2×C12), C5⋊2(C3×C42⋊C2), (D5×C2×C12).28C2, C2.2(C3×C4○D20), C10.3(C3×C4○D4), (C2×C4).97(C6×D5), C22.10(D5×C2×C6), (C2×C20).74(C2×C6), (C6×D5).53(C2×C4), (D5×C2×C6).122C22, (C3×C10.D4)⋊39C2, (C2×C10).13(C22×C6), (C3×Dic5).61(C2×C4), (C2×Dic5).24(C2×C6), (C22×D5).19(C2×C6), (C2×C6).326(C22×D5), (C3×D10⋊C4).17C2, SmallGroup(480,665)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C42⋊D5
G = < a,b,c,d,e | a3=b4=c4=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >
Subgroups: 432 in 152 conjugacy classes, 82 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, C2×C4, C23, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C42, C42, C22⋊C4, C4⋊C4, C22×C4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C12, C2×C12, C2×C12, C22×C6, C3×D5, C30, C30, C42⋊C2, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×C12, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C3×Dic5, C3×Dic5, C60, C60, C6×D5, C6×D5, C2×C30, C4×Dic5, C10.D4, D10⋊C4, C4×C20, C2×C4×D5, C3×C42⋊C2, D5×C12, C6×Dic5, C6×Dic5, C2×C60, C2×C60, D5×C2×C6, C42⋊D5, C12×Dic5, C3×C10.D4, C3×D10⋊C4, C4×C60, D5×C2×C12, C3×C42⋊D5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, D5, C12, C2×C6, C22×C4, C4○D4, D10, C2×C12, C22×C6, C3×D5, C42⋊C2, C4×D5, C22×D5, C22×C12, C3×C4○D4, C6×D5, C2×C4×D5, C4○D20, C3×C42⋊C2, D5×C12, D5×C2×C6, C42⋊D5, D5×C2×C12, C3×C4○D20, C3×C42⋊D5
►On 240 points
Generators in S240
(1 41 21)(2 42 22)(3 43 23)(4 44 24)(5 45 25)(6 46 26)(7 47 27)(8 48 28)(9 49 29)(10 50 30)(11 51 31)(12 52 32)(13 53 33)(14 54 34)(15 55 35)(16 56 36)(17 57 37)(18 58 38)(19 59 39)(20 60 40)(61 101 81)(62 102 82)(63 103 83)(64 104 84)(65 105 85)(66 106 86)(67 107 87)(68 108 88)(69 109 89)(70 110 90)(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)(121 161 141)(122 162 142)(123 163 143)(124 164 144)(125 165 145)(126 166 146)(127 167 147)(128 168 148)(129 169 149)(130 170 150)(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)(181 221 201)(182 222 202)(183 223 203)(184 224 204)(185 225 205)(186 226 206)(187 227 207)(188 228 208)(189 229 209)(190 230 210)(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)
(1 126 6 121)(2 127 7 122)(3 128 8 123)(4 129 9 124)(5 130 10 125)(11 136 16 131)(12 137 17 132)(13 138 18 133)(14 139 19 134)(15 140 20 135)(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)
(1 71 11 61)(2 72 12 62)(3 73 13 63)(4 74 14 64)(5 75 15 65)(6 76 16 66)(7 77 17 67)(8 78 18 68)(9 79 19 69)(10 80 20 70)(21 91 31 81)(22 92 32 82)(23 93 33 83)(24 94 34 84)(25 95 35 85)(26 96 36 86)(27 97 37 87)(28 98 38 88)(29 99 39 89)(30 100 40 90)(41 111 51 101)(42 112 52 102)(43 113 53 103)(44 114 54 104)(45 115 55 105)(46 116 56 106)(47 117 57 107)(48 118 58 108)(49 119 59 109)(50 120 60 110)(121 191 131 181)(122 192 132 182)(123 193 133 183)(124 194 134 184)(125 195 135 185)(126 196 136 186)(127 197 137 187)(128 198 138 188)(129 199 139 189)(130 200 140 190)(141 211 151 201)(142 212 152 202)(143 213 153 203)(144 214 154 204)(145 215 155 205)(146 216 156 206)(147 217 157 207)(148 218 158 208)(149 219 159 209)(150 220 160 210)(161 231 171 221)(162 232 172 222)(163 233 173 223)(164 234 174 224)(165 235 175 225)(166 236 176 226)(167 237 177 227)(168 238 178 228)(169 239 179 229)(170 240 180 230)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)(31 32 33 34 35)(36 37 38 39 40)(41 42 43 44 45)(46 47 48 49 50)(51 52 53 54 55)(56 57 58 59 60)(61 62 63 64 65)(66 67 68 69 70)(71 72 73 74 75)(76 77 78 79 80)(81 82 83 84 85)(86 87 88 89 90)(91 92 93 94 95)(96 97 98 99 100)(101 102 103 104 105)(106 107 108 109 110)(111 112 113 114 115)(116 117 118 119 120)(121 122 123 124 125)(126 127 128 129 130)(131 132 133 134 135)(136 137 138 139 140)(141 142 143 144 145)(146 147 148 149 150)(151 152 153 154 155)(156 157 158 159 160)(161 162 163 164 165)(166 167 168 169 170)(171 172 173 174 175)(176 177 178 179 180)(181 182 183 184 185)(186 187 188 189 190)(191 192 193 194 195)(196 197 198 199 200)(201 202 203 204 205)(206 207 208 209 210)(211 212 213 214 215)(216 217 218 219 220)(221 222 223 224 225)(226 227 228 229 230)(231 232 233 234 235)(236 237 238 239 240)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 20)(7 19)(8 18)(9 17)(10 16)(21 35)(22 34)(23 33)(24 32)(25 31)(26 40)(27 39)(28 38)(29 37)(30 36)(41 55)(42 54)(43 53)(44 52)(45 51)(46 60)(47 59)(48 58)(49 57)(50 56)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)(81 100)(82 99)(83 98)(84 97)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(101 120)(102 119)(103 118)(104 117)(105 116)(106 115)(107 114)(108 113)(109 112)(110 111)(121 135)(122 134)(123 133)(124 132)(125 131)(126 140)(127 139)(128 138)(129 137)(130 136)(141 155)(142 154)(143 153)(144 152)(145 151)(146 160)(147 159)(148 158)(149 157)(150 156)(161 175)(162 174)(163 173)(164 172)(165 171)(166 180)(167 179)(168 178)(169 177)(170 176)(181 200)(182 199)(183 198)(184 197)(185 196)(186 195)(187 194)(188 193)(189 192)(190 191)(201 220)(202 219)(203 218)(204 217)(205 216)(206 215)(207 214)(208 213)(209 212)(210 211)(221 240)(222 239)(223 238)(224 237)(225 236)(226 235)(227 234)(228 233)(229 232)(230 231)
G:=sub<Sym(240)| (1,41,21)(2,42,22)(3,43,23)(4,44,24)(5,45,25)(6,46,26)(7,47,27)(8,48,28)(9,49,29)(10,50,30)(11,51,31)(12,52,32)(13,53,33)(14,54,34)(15,55,35)(16,56,36)(17,57,37)(18,58,38)(19,59,39)(20,60,40)(61,101,81)(62,102,82)(63,103,83)(64,104,84)(65,105,85)(66,106,86)(67,107,87)(68,108,88)(69,109,89)(70,110,90)(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)(121,161,141)(122,162,142)(123,163,143)(124,164,144)(125,165,145)(126,166,146)(127,167,147)(128,168,148)(129,169,149)(130,170,150)(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)(181,221,201)(182,222,202)(183,223,203)(184,224,204)(185,225,205)(186,226,206)(187,227,207)(188,228,208)(189,229,209)(190,230,210)(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), (1,126,6,121)(2,127,7,122)(3,128,8,123)(4,129,9,124)(5,130,10,125)(11,136,16,131)(12,137,17,132)(13,138,18,133)(14,139,19,134)(15,140,20,135)(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), (1,71,11,61)(2,72,12,62)(3,73,13,63)(4,74,14,64)(5,75,15,65)(6,76,16,66)(7,77,17,67)(8,78,18,68)(9,79,19,69)(10,80,20,70)(21,91,31,81)(22,92,32,82)(23,93,33,83)(24,94,34,84)(25,95,35,85)(26,96,36,86)(27,97,37,87)(28,98,38,88)(29,99,39,89)(30,100,40,90)(41,111,51,101)(42,112,52,102)(43,113,53,103)(44,114,54,104)(45,115,55,105)(46,116,56,106)(47,117,57,107)(48,118,58,108)(49,119,59,109)(50,120,60,110)(121,191,131,181)(122,192,132,182)(123,193,133,183)(124,194,134,184)(125,195,135,185)(126,196,136,186)(127,197,137,187)(128,198,138,188)(129,199,139,189)(130,200,140,190)(141,211,151,201)(142,212,152,202)(143,213,153,203)(144,214,154,204)(145,215,155,205)(146,216,156,206)(147,217,157,207)(148,218,158,208)(149,219,159,209)(150,220,160,210)(161,231,171,221)(162,232,172,222)(163,233,173,223)(164,234,174,224)(165,235,175,225)(166,236,176,226)(167,237,177,227)(168,238,178,228)(169,239,179,229)(170,240,180,230), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100)(101,102,103,104,105)(106,107,108,109,110)(111,112,113,114,115)(116,117,118,119,120)(121,122,123,124,125)(126,127,128,129,130)(131,132,133,134,135)(136,137,138,139,140)(141,142,143,144,145)(146,147,148,149,150)(151,152,153,154,155)(156,157,158,159,160)(161,162,163,164,165)(166,167,168,169,170)(171,172,173,174,175)(176,177,178,179,180)(181,182,183,184,185)(186,187,188,189,190)(191,192,193,194,195)(196,197,198,199,200)(201,202,203,204,205)(206,207,208,209,210)(211,212,213,214,215)(216,217,218,219,220)(221,222,223,224,225)(226,227,228,229,230)(231,232,233,234,235)(236,237,238,239,240), (1,15)(2,14)(3,13)(4,12)(5,11)(6,20)(7,19)(8,18)(9,17)(10,16)(21,35)(22,34)(23,33)(24,32)(25,31)(26,40)(27,39)(28,38)(29,37)(30,36)(41,55)(42,54)(43,53)(44,52)(45,51)(46,60)(47,59)(48,58)(49,57)(50,56)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(101,120)(102,119)(103,118)(104,117)(105,116)(106,115)(107,114)(108,113)(109,112)(110,111)(121,135)(122,134)(123,133)(124,132)(125,131)(126,140)(127,139)(128,138)(129,137)(130,136)(141,155)(142,154)(143,153)(144,152)(145,151)(146,160)(147,159)(148,158)(149,157)(150,156)(161,175)(162,174)(163,173)(164,172)(165,171)(166,180)(167,179)(168,178)(169,177)(170,176)(181,200)(182,199)(183,198)(184,197)(185,196)(186,195)(187,194)(188,193)(189,192)(190,191)(201,220)(202,219)(203,218)(204,217)(205,216)(206,215)(207,214)(208,213)(209,212)(210,211)(221,240)(222,239)(223,238)(224,237)(225,236)(226,235)(227,234)(228,233)(229,232)(230,231)>;
G:=Group( (1,41,21)(2,42,22)(3,43,23)(4,44,24)(5,45,25)(6,46,26)(7,47,27)(8,48,28)(9,49,29)(10,50,30)(11,51,31)(12,52,32)(13,53,33)(14,54,34)(15,55,35)(16,56,36)(17,57,37)(18,58,38)(19,59,39)(20,60,40)(61,101,81)(62,102,82)(63,103,83)(64,104,84)(65,105,85)(66,106,86)(67,107,87)(68,108,88)(69,109,89)(70,110,90)(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)(121,161,141)(122,162,142)(123,163,143)(124,164,144)(125,165,145)(126,166,146)(127,167,147)(128,168,148)(129,169,149)(130,170,150)(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)(181,221,201)(182,222,202)(183,223,203)(184,224,204)(185,225,205)(186,226,206)(187,227,207)(188,228,208)(189,229,209)(190,230,210)(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), (1,126,6,121)(2,127,7,122)(3,128,8,123)(4,129,9,124)(5,130,10,125)(11,136,16,131)(12,137,17,132)(13,138,18,133)(14,139,19,134)(15,140,20,135)(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), (1,71,11,61)(2,72,12,62)(3,73,13,63)(4,74,14,64)(5,75,15,65)(6,76,16,66)(7,77,17,67)(8,78,18,68)(9,79,19,69)(10,80,20,70)(21,91,31,81)(22,92,32,82)(23,93,33,83)(24,94,34,84)(25,95,35,85)(26,96,36,86)(27,97,37,87)(28,98,38,88)(29,99,39,89)(30,100,40,90)(41,111,51,101)(42,112,52,102)(43,113,53,103)(44,114,54,104)(45,115,55,105)(46,116,56,106)(47,117,57,107)(48,118,58,108)(49,119,59,109)(50,120,60,110)(121,191,131,181)(122,192,132,182)(123,193,133,183)(124,194,134,184)(125,195,135,185)(126,196,136,186)(127,197,137,187)(128,198,138,188)(129,199,139,189)(130,200,140,190)(141,211,151,201)(142,212,152,202)(143,213,153,203)(144,214,154,204)(145,215,155,205)(146,216,156,206)(147,217,157,207)(148,218,158,208)(149,219,159,209)(150,220,160,210)(161,231,171,221)(162,232,172,222)(163,233,173,223)(164,234,174,224)(165,235,175,225)(166,236,176,226)(167,237,177,227)(168,238,178,228)(169,239,179,229)(170,240,180,230), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100)(101,102,103,104,105)(106,107,108,109,110)(111,112,113,114,115)(116,117,118,119,120)(121,122,123,124,125)(126,127,128,129,130)(131,132,133,134,135)(136,137,138,139,140)(141,142,143,144,145)(146,147,148,149,150)(151,152,153,154,155)(156,157,158,159,160)(161,162,163,164,165)(166,167,168,169,170)(171,172,173,174,175)(176,177,178,179,180)(181,182,183,184,185)(186,187,188,189,190)(191,192,193,194,195)(196,197,198,199,200)(201,202,203,204,205)(206,207,208,209,210)(211,212,213,214,215)(216,217,218,219,220)(221,222,223,224,225)(226,227,228,229,230)(231,232,233,234,235)(236,237,238,239,240), (1,15)(2,14)(3,13)(4,12)(5,11)(6,20)(7,19)(8,18)(9,17)(10,16)(21,35)(22,34)(23,33)(24,32)(25,31)(26,40)(27,39)(28,38)(29,37)(30,36)(41,55)(42,54)(43,53)(44,52)(45,51)(46,60)(47,59)(48,58)(49,57)(50,56)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(101,120)(102,119)(103,118)(104,117)(105,116)(106,115)(107,114)(108,113)(109,112)(110,111)(121,135)(122,134)(123,133)(124,132)(125,131)(126,140)(127,139)(128,138)(129,137)(130,136)(141,155)(142,154)(143,153)(144,152)(145,151)(146,160)(147,159)(148,158)(149,157)(150,156)(161,175)(162,174)(163,173)(164,172)(165,171)(166,180)(167,179)(168,178)(169,177)(170,176)(181,200)(182,199)(183,198)(184,197)(185,196)(186,195)(187,194)(188,193)(189,192)(190,191)(201,220)(202,219)(203,218)(204,217)(205,216)(206,215)(207,214)(208,213)(209,212)(210,211)(221,240)(222,239)(223,238)(224,237)(225,236)(226,235)(227,234)(228,233)(229,232)(230,231) );
G=PermutationGroup([[(1,41,21),(2,42,22),(3,43,23),(4,44,24),(5,45,25),(6,46,26),(7,47,27),(8,48,28),(9,49,29),(10,50,30),(11,51,31),(12,52,32),(13,53,33),(14,54,34),(15,55,35),(16,56,36),(17,57,37),(18,58,38),(19,59,39),(20,60,40),(61,101,81),(62,102,82),(63,103,83),(64,104,84),(65,105,85),(66,106,86),(67,107,87),(68,108,88),(69,109,89),(70,110,90),(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),(121,161,141),(122,162,142),(123,163,143),(124,164,144),(125,165,145),(126,166,146),(127,167,147),(128,168,148),(129,169,149),(130,170,150),(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),(181,221,201),(182,222,202),(183,223,203),(184,224,204),(185,225,205),(186,226,206),(187,227,207),(188,228,208),(189,229,209),(190,230,210),(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)], [(1,126,6,121),(2,127,7,122),(3,128,8,123),(4,129,9,124),(5,130,10,125),(11,136,16,131),(12,137,17,132),(13,138,18,133),(14,139,19,134),(15,140,20,135),(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)], [(1,71,11,61),(2,72,12,62),(3,73,13,63),(4,74,14,64),(5,75,15,65),(6,76,16,66),(7,77,17,67),(8,78,18,68),(9,79,19,69),(10,80,20,70),(21,91,31,81),(22,92,32,82),(23,93,33,83),(24,94,34,84),(25,95,35,85),(26,96,36,86),(27,97,37,87),(28,98,38,88),(29,99,39,89),(30,100,40,90),(41,111,51,101),(42,112,52,102),(43,113,53,103),(44,114,54,104),(45,115,55,105),(46,116,56,106),(47,117,57,107),(48,118,58,108),(49,119,59,109),(50,120,60,110),(121,191,131,181),(122,192,132,182),(123,193,133,183),(124,194,134,184),(125,195,135,185),(126,196,136,186),(127,197,137,187),(128,198,138,188),(129,199,139,189),(130,200,140,190),(141,211,151,201),(142,212,152,202),(143,213,153,203),(144,214,154,204),(145,215,155,205),(146,216,156,206),(147,217,157,207),(148,218,158,208),(149,219,159,209),(150,220,160,210),(161,231,171,221),(162,232,172,222),(163,233,173,223),(164,234,174,224),(165,235,175,225),(166,236,176,226),(167,237,177,227),(168,238,178,228),(169,239,179,229),(170,240,180,230)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30),(31,32,33,34,35),(36,37,38,39,40),(41,42,43,44,45),(46,47,48,49,50),(51,52,53,54,55),(56,57,58,59,60),(61,62,63,64,65),(66,67,68,69,70),(71,72,73,74,75),(76,77,78,79,80),(81,82,83,84,85),(86,87,88,89,90),(91,92,93,94,95),(96,97,98,99,100),(101,102,103,104,105),(106,107,108,109,110),(111,112,113,114,115),(116,117,118,119,120),(121,122,123,124,125),(126,127,128,129,130),(131,132,133,134,135),(136,137,138,139,140),(141,142,143,144,145),(146,147,148,149,150),(151,152,153,154,155),(156,157,158,159,160),(161,162,163,164,165),(166,167,168,169,170),(171,172,173,174,175),(176,177,178,179,180),(181,182,183,184,185),(186,187,188,189,190),(191,192,193,194,195),(196,197,198,199,200),(201,202,203,204,205),(206,207,208,209,210),(211,212,213,214,215),(216,217,218,219,220),(221,222,223,224,225),(226,227,228,229,230),(231,232,233,234,235),(236,237,238,239,240)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,20),(7,19),(8,18),(9,17),(10,16),(21,35),(22,34),(23,33),(24,32),(25,31),(26,40),(27,39),(28,38),(29,37),(30,36),(41,55),(42,54),(43,53),(44,52),(45,51),(46,60),(47,59),(48,58),(49,57),(50,56),(61,80),(62,79),(63,78),(64,77),(65,76),(66,75),(67,74),(68,73),(69,72),(70,71),(81,100),(82,99),(83,98),(84,97),(85,96),(86,95),(87,94),(88,93),(89,92),(90,91),(101,120),(102,119),(103,118),(104,117),(105,116),(106,115),(107,114),(108,113),(109,112),(110,111),(121,135),(122,134),(123,133),(124,132),(125,131),(126,140),(127,139),(128,138),(129,137),(130,136),(141,155),(142,154),(143,153),(144,152),(145,151),(146,160),(147,159),(148,158),(149,157),(150,156),(161,175),(162,174),(163,173),(164,172),(165,171),(166,180),(167,179),(168,178),(169,177),(170,176),(181,200),(182,199),(183,198),(184,197),(185,196),(186,195),(187,194),(188,193),(189,192),(190,191),(201,220),(202,219),(203,218),(204,217),(205,216),(206,215),(207,214),(208,213),(209,212),(210,211),(221,240),(222,239),(223,238),(224,237),(225,236),(226,235),(227,234),(228,233),(229,232),(230,231)]])
156 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 5A | 5B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 10A | ··· | 10F | 12A | ··· | 12H | 12I | ··· | 12P | 12Q | ··· | 12AB | 15A | 15B | 15C | 15D | 20A | ··· | 20X | 30A | ··· | 30L | 60A | ··· | 60AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 1 | ··· | 1 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
156 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | D5 | C4○D4 | D10 | C3×D5 | C4×D5 | C3×C4○D4 | C6×D5 | C4○D20 | D5×C12 | C3×C4○D20 |
| kernel | C3×C42⋊D5 | C12×Dic5 | C3×C10.D4 | C3×D10⋊C4 | C4×C60 | D5×C2×C12 | C42⋊D5 | D5×C12 | C4×Dic5 | C10.D4 | D10⋊C4 | C4×C20 | C2×C4×D5 | C4×D5 | C4×C12 | C30 | C2×C12 | C42 | C12 | C10 | C2×C4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 8 | 2 | 4 | 4 | 2 | 2 | 16 | 2 | 4 | 6 | 4 | 8 | 8 | 12 | 16 | 16 | 32 |
Matrix representation of C3×C42⋊D5 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 47 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 50 |
| 11 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 29 | 54 |
| 0 | 0 | 7 | 32 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 17 | 60 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 44 | 1 |
| 0 | 0 | 17 | 17 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,47,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,1,0,0,0,0,50,0,0,0,0,50],[11,0,0,0,0,60,0,0,0,0,29,7,0,0,54,32],[1,0,0,0,0,1,0,0,0,0,17,1,0,0,60,0],[1,0,0,0,0,60,0,0,0,0,44,17,0,0,1,17] >;
C3×C42⋊D5 in GAP, Magma, Sage, TeX
C_3\times C_4^2\rtimes D_5
% in TeX
G:=Group("C3xC4^2:D5"); // GroupNames label
G:=SmallGroup(480,665);
// by ID
G=gap.SmallGroup(480,665);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,344,1094,142,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^4=d^5=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations