direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C4⋊C4⋊D5, C4⋊Dic5⋊7C6, (C4×Dic5)⋊14C6, (C12×Dic5)⋊32C2, D10⋊C4.4C6, C10.D4⋊13C6, (C2×C12).282D10, C15⋊22(C42⋊2C2), C30.191(C4○D4), C6.121(C4○D20), (C2×C60).270C22, (C2×C30).356C23, C6.52(Q8⋊2D5), C6.118(D4⋊2D5), (C6×Dic5).244C22, (C5×C4⋊C4)⋊9C6, C4⋊C4⋊6(C3×D5), (C3×C4⋊C4)⋊15D5, (C15×C4⋊C4)⋊18C2, C5⋊3(C3×C42⋊2C2), (C2×C4).26(C6×D5), C22.53(D5×C2×C6), (C2×C20).58(C2×C6), (C3×C4⋊Dic5)⋊25C2, C10.26(C3×C4○D4), C2.16(C3×C4○D20), C2.7(C3×Q8⋊2D5), (D5×C2×C6).80C22, C2.14(C3×D4⋊2D5), (C3×D10⋊C4).9C2, (C3×C10.D4)⋊35C2, (C2×C10).39(C22×C6), (C2×Dic5).34(C2×C6), (C22×D5).10(C2×C6), (C2×C6).352(C22×D5), SmallGroup(480,691)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4⋊C4⋊D5
G = < a,b,c,d,e | a3=b4=c4=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, ebe=bc2, cd=dc, ece=b2c, ede=d-1 >
Subgroups: 384 in 120 conjugacy classes, 58 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, D5, C10, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C4⋊C4, Dic5, C20, D10, C2×C10, C2×C12, C2×C12, C22×C6, C3×D5, C30, C42⋊2C2, C2×Dic5, C2×C20, C22×D5, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C3×Dic5, C60, C6×D5, C2×C30, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C3×C42⋊2C2, C6×Dic5, C2×C60, D5×C2×C6, C4⋊C4⋊D5, C12×Dic5, C3×C10.D4, C3×C4⋊Dic5, C3×D10⋊C4, C15×C4⋊C4, C3×C4⋊C4⋊D5
Quotients: C1, C2, C3, C22, C6, C23, D5, C2×C6, C4○D4, D10, C22×C6, C3×D5, C42⋊2C2, C22×D5, C3×C4○D4, C6×D5, C4○D20, D4⋊2D5, Q8⋊2D5, C3×C42⋊2C2, D5×C2×C6, C4⋊C4⋊D5, C3×C4○D20, C3×D4⋊2D5, C3×Q8⋊2D5, C3×C4⋊C4⋊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 76 6 71)(2 77 7 72)(3 78 8 73)(4 79 9 74)(5 80 10 75)(11 66 16 61)(12 67 17 62)(13 68 18 63)(14 69 19 64)(15 70 20 65)(21 96 26 91)(22 97 27 92)(23 98 28 93)(24 99 29 94)(25 100 30 95)(31 86 36 81)(32 87 37 82)(33 88 38 83)(34 89 39 84)(35 90 40 85)(41 116 46 111)(42 117 47 112)(43 118 48 113)(44 119 49 114)(45 120 50 115)(51 106 56 101)(52 107 57 102)(53 108 58 103)(54 109 59 104)(55 110 60 105)(121 191 126 196)(122 192 127 197)(123 193 128 198)(124 194 129 199)(125 195 130 200)(131 181 136 186)(132 182 137 187)(133 183 138 188)(134 184 139 189)(135 185 140 190)(141 211 146 216)(142 212 147 217)(143 213 148 218)(144 214 149 219)(145 215 150 220)(151 201 156 206)(152 202 157 207)(153 203 158 208)(154 204 159 209)(155 205 160 210)(161 231 166 236)(162 232 167 237)(163 233 168 238)(164 234 169 239)(165 235 170 240)(171 221 176 226)(172 222 177 227)(173 223 178 228)(174 224 179 229)(175 225 180 230)
(1 131 11 121)(2 132 12 122)(3 133 13 123)(4 134 14 124)(5 135 15 125)(6 136 16 126)(7 137 17 127)(8 138 18 128)(9 139 19 129)(10 140 20 130)(21 151 31 141)(22 152 32 142)(23 153 33 143)(24 154 34 144)(25 155 35 145)(26 156 36 146)(27 157 37 147)(28 158 38 148)(29 159 39 149)(30 160 40 150)(41 171 51 161)(42 172 52 162)(43 173 53 163)(44 174 54 164)(45 175 55 165)(46 176 56 166)(47 177 57 167)(48 178 58 168)(49 179 59 169)(50 180 60 170)(61 191 71 181)(62 192 72 182)(63 193 73 183)(64 194 74 184)(65 195 75 185)(66 196 76 186)(67 197 77 187)(68 198 78 188)(69 199 79 189)(70 200 80 190)(81 211 91 201)(82 212 92 202)(83 213 93 203)(84 214 94 204)(85 215 95 205)(86 216 96 206)(87 217 97 207)(88 218 98 208)(89 219 99 209)(90 220 100 210)(101 231 111 221)(102 232 112 222)(103 233 113 223)(104 234 114 224)(105 235 115 225)(106 236 116 226)(107 237 117 227)(108 238 118 228)(109 239 119 229)(110 240 120 230)
(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 5)(2 4)(6 10)(7 9)(11 15)(12 14)(16 20)(17 19)(21 25)(22 24)(26 30)(27 29)(31 35)(32 34)(36 40)(37 39)(41 45)(42 44)(46 50)(47 49)(51 55)(52 54)(56 60)(57 59)(61 75)(62 74)(63 73)(64 72)(65 71)(66 80)(67 79)(68 78)(69 77)(70 76)(81 95)(82 94)(83 93)(84 92)(85 91)(86 100)(87 99)(88 98)(89 97)(90 96)(101 115)(102 114)(103 113)(104 112)(105 111)(106 120)(107 119)(108 118)(109 117)(110 116)(121 130)(122 129)(123 128)(124 127)(125 126)(131 140)(132 139)(133 138)(134 137)(135 136)(141 150)(142 149)(143 148)(144 147)(145 146)(151 160)(152 159)(153 158)(154 157)(155 156)(161 170)(162 169)(163 168)(164 167)(165 166)(171 180)(172 179)(173 178)(174 177)(175 176)(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), (1,76,6,71)(2,77,7,72)(3,78,8,73)(4,79,9,74)(5,80,10,75)(11,66,16,61)(12,67,17,62)(13,68,18,63)(14,69,19,64)(15,70,20,65)(21,96,26,91)(22,97,27,92)(23,98,28,93)(24,99,29,94)(25,100,30,95)(31,86,36,81)(32,87,37,82)(33,88,38,83)(34,89,39,84)(35,90,40,85)(41,116,46,111)(42,117,47,112)(43,118,48,113)(44,119,49,114)(45,120,50,115)(51,106,56,101)(52,107,57,102)(53,108,58,103)(54,109,59,104)(55,110,60,105)(121,191,126,196)(122,192,127,197)(123,193,128,198)(124,194,129,199)(125,195,130,200)(131,181,136,186)(132,182,137,187)(133,183,138,188)(134,184,139,189)(135,185,140,190)(141,211,146,216)(142,212,147,217)(143,213,148,218)(144,214,149,219)(145,215,150,220)(151,201,156,206)(152,202,157,207)(153,203,158,208)(154,204,159,209)(155,205,160,210)(161,231,166,236)(162,232,167,237)(163,233,168,238)(164,234,169,239)(165,235,170,240)(171,221,176,226)(172,222,177,227)(173,223,178,228)(174,224,179,229)(175,225,180,230), (1,131,11,121)(2,132,12,122)(3,133,13,123)(4,134,14,124)(5,135,15,125)(6,136,16,126)(7,137,17,127)(8,138,18,128)(9,139,19,129)(10,140,20,130)(21,151,31,141)(22,152,32,142)(23,153,33,143)(24,154,34,144)(25,155,35,145)(26,156,36,146)(27,157,37,147)(28,158,38,148)(29,159,39,149)(30,160,40,150)(41,171,51,161)(42,172,52,162)(43,173,53,163)(44,174,54,164)(45,175,55,165)(46,176,56,166)(47,177,57,167)(48,178,58,168)(49,179,59,169)(50,180,60,170)(61,191,71,181)(62,192,72,182)(63,193,73,183)(64,194,74,184)(65,195,75,185)(66,196,76,186)(67,197,77,187)(68,198,78,188)(69,199,79,189)(70,200,80,190)(81,211,91,201)(82,212,92,202)(83,213,93,203)(84,214,94,204)(85,215,95,205)(86,216,96,206)(87,217,97,207)(88,218,98,208)(89,219,99,209)(90,220,100,210)(101,231,111,221)(102,232,112,222)(103,233,113,223)(104,234,114,224)(105,235,115,225)(106,236,116,226)(107,237,117,227)(108,238,118,228)(109,239,119,229)(110,240,120,230), (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), (1,5)(2,4)(6,10)(7,9)(11,15)(12,14)(16,20)(17,19)(21,25)(22,24)(26,30)(27,29)(31,35)(32,34)(36,40)(37,39)(41,45)(42,44)(46,50)(47,49)(51,55)(52,54)(56,60)(57,59)(61,75)(62,74)(63,73)(64,72)(65,71)(66,80)(67,79)(68,78)(69,77)(70,76)(81,95)(82,94)(83,93)(84,92)(85,91)(86,100)(87,99)(88,98)(89,97)(90,96)(101,115)(102,114)(103,113)(104,112)(105,111)(106,120)(107,119)(108,118)(109,117)(110,116)(121,130)(122,129)(123,128)(124,127)(125,126)(131,140)(132,139)(133,138)(134,137)(135,136)(141,150)(142,149)(143,148)(144,147)(145,146)(151,160)(152,159)(153,158)(154,157)(155,156)(161,170)(162,169)(163,168)(164,167)(165,166)(171,180)(172,179)(173,178)(174,177)(175,176)(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), (1,76,6,71)(2,77,7,72)(3,78,8,73)(4,79,9,74)(5,80,10,75)(11,66,16,61)(12,67,17,62)(13,68,18,63)(14,69,19,64)(15,70,20,65)(21,96,26,91)(22,97,27,92)(23,98,28,93)(24,99,29,94)(25,100,30,95)(31,86,36,81)(32,87,37,82)(33,88,38,83)(34,89,39,84)(35,90,40,85)(41,116,46,111)(42,117,47,112)(43,118,48,113)(44,119,49,114)(45,120,50,115)(51,106,56,101)(52,107,57,102)(53,108,58,103)(54,109,59,104)(55,110,60,105)(121,191,126,196)(122,192,127,197)(123,193,128,198)(124,194,129,199)(125,195,130,200)(131,181,136,186)(132,182,137,187)(133,183,138,188)(134,184,139,189)(135,185,140,190)(141,211,146,216)(142,212,147,217)(143,213,148,218)(144,214,149,219)(145,215,150,220)(151,201,156,206)(152,202,157,207)(153,203,158,208)(154,204,159,209)(155,205,160,210)(161,231,166,236)(162,232,167,237)(163,233,168,238)(164,234,169,239)(165,235,170,240)(171,221,176,226)(172,222,177,227)(173,223,178,228)(174,224,179,229)(175,225,180,230), (1,131,11,121)(2,132,12,122)(3,133,13,123)(4,134,14,124)(5,135,15,125)(6,136,16,126)(7,137,17,127)(8,138,18,128)(9,139,19,129)(10,140,20,130)(21,151,31,141)(22,152,32,142)(23,153,33,143)(24,154,34,144)(25,155,35,145)(26,156,36,146)(27,157,37,147)(28,158,38,148)(29,159,39,149)(30,160,40,150)(41,171,51,161)(42,172,52,162)(43,173,53,163)(44,174,54,164)(45,175,55,165)(46,176,56,166)(47,177,57,167)(48,178,58,168)(49,179,59,169)(50,180,60,170)(61,191,71,181)(62,192,72,182)(63,193,73,183)(64,194,74,184)(65,195,75,185)(66,196,76,186)(67,197,77,187)(68,198,78,188)(69,199,79,189)(70,200,80,190)(81,211,91,201)(82,212,92,202)(83,213,93,203)(84,214,94,204)(85,215,95,205)(86,216,96,206)(87,217,97,207)(88,218,98,208)(89,219,99,209)(90,220,100,210)(101,231,111,221)(102,232,112,222)(103,233,113,223)(104,234,114,224)(105,235,115,225)(106,236,116,226)(107,237,117,227)(108,238,118,228)(109,239,119,229)(110,240,120,230), (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), (1,5)(2,4)(6,10)(7,9)(11,15)(12,14)(16,20)(17,19)(21,25)(22,24)(26,30)(27,29)(31,35)(32,34)(36,40)(37,39)(41,45)(42,44)(46,50)(47,49)(51,55)(52,54)(56,60)(57,59)(61,75)(62,74)(63,73)(64,72)(65,71)(66,80)(67,79)(68,78)(69,77)(70,76)(81,95)(82,94)(83,93)(84,92)(85,91)(86,100)(87,99)(88,98)(89,97)(90,96)(101,115)(102,114)(103,113)(104,112)(105,111)(106,120)(107,119)(108,118)(109,117)(110,116)(121,130)(122,129)(123,128)(124,127)(125,126)(131,140)(132,139)(133,138)(134,137)(135,136)(141,150)(142,149)(143,148)(144,147)(145,146)(151,160)(152,159)(153,158)(154,157)(155,156)(161,170)(162,169)(163,168)(164,167)(165,166)(171,180)(172,179)(173,178)(174,177)(175,176)(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)], [(1,76,6,71),(2,77,7,72),(3,78,8,73),(4,79,9,74),(5,80,10,75),(11,66,16,61),(12,67,17,62),(13,68,18,63),(14,69,19,64),(15,70,20,65),(21,96,26,91),(22,97,27,92),(23,98,28,93),(24,99,29,94),(25,100,30,95),(31,86,36,81),(32,87,37,82),(33,88,38,83),(34,89,39,84),(35,90,40,85),(41,116,46,111),(42,117,47,112),(43,118,48,113),(44,119,49,114),(45,120,50,115),(51,106,56,101),(52,107,57,102),(53,108,58,103),(54,109,59,104),(55,110,60,105),(121,191,126,196),(122,192,127,197),(123,193,128,198),(124,194,129,199),(125,195,130,200),(131,181,136,186),(132,182,137,187),(133,183,138,188),(134,184,139,189),(135,185,140,190),(141,211,146,216),(142,212,147,217),(143,213,148,218),(144,214,149,219),(145,215,150,220),(151,201,156,206),(152,202,157,207),(153,203,158,208),(154,204,159,209),(155,205,160,210),(161,231,166,236),(162,232,167,237),(163,233,168,238),(164,234,169,239),(165,235,170,240),(171,221,176,226),(172,222,177,227),(173,223,178,228),(174,224,179,229),(175,225,180,230)], [(1,131,11,121),(2,132,12,122),(3,133,13,123),(4,134,14,124),(5,135,15,125),(6,136,16,126),(7,137,17,127),(8,138,18,128),(9,139,19,129),(10,140,20,130),(21,151,31,141),(22,152,32,142),(23,153,33,143),(24,154,34,144),(25,155,35,145),(26,156,36,146),(27,157,37,147),(28,158,38,148),(29,159,39,149),(30,160,40,150),(41,171,51,161),(42,172,52,162),(43,173,53,163),(44,174,54,164),(45,175,55,165),(46,176,56,166),(47,177,57,167),(48,178,58,168),(49,179,59,169),(50,180,60,170),(61,191,71,181),(62,192,72,182),(63,193,73,183),(64,194,74,184),(65,195,75,185),(66,196,76,186),(67,197,77,187),(68,198,78,188),(69,199,79,189),(70,200,80,190),(81,211,91,201),(82,212,92,202),(83,213,93,203),(84,214,94,204),(85,215,95,205),(86,216,96,206),(87,217,97,207),(88,218,98,208),(89,219,99,209),(90,220,100,210),(101,231,111,221),(102,232,112,222),(103,233,113,223),(104,234,114,224),(105,235,115,225),(106,236,116,226),(107,237,117,227),(108,238,118,228),(109,239,119,229),(110,240,120,230)], [(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)], [(1,5),(2,4),(6,10),(7,9),(11,15),(12,14),(16,20),(17,19),(21,25),(22,24),(26,30),(27,29),(31,35),(32,34),(36,40),(37,39),(41,45),(42,44),(46,50),(47,49),(51,55),(52,54),(56,60),(57,59),(61,75),(62,74),(63,73),(64,72),(65,71),(66,80),(67,79),(68,78),(69,77),(70,76),(81,95),(82,94),(83,93),(84,92),(85,91),(86,100),(87,99),(88,98),(89,97),(90,96),(101,115),(102,114),(103,113),(104,112),(105,111),(106,120),(107,119),(108,118),(109,117),(110,116),(121,130),(122,129),(123,128),(124,127),(125,126),(131,140),(132,139),(133,138),(134,137),(135,136),(141,150),(142,149),(143,148),(144,147),(145,146),(151,160),(152,159),(153,158),(154,157),(155,156),(161,170),(162,169),(163,168),(164,167),(165,166),(171,180),(172,179),(173,178),(174,177),(175,176),(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)]])
102 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 6A | ··· | 6F | 6G | 6H | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | ··· | 12P | 12Q | 12R | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 20 | 1 | 1 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 2 | 2 | 1 | ··· | 1 | 20 | 20 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D5 | C4○D4 | D10 | C3×D5 | C3×C4○D4 | C6×D5 | C4○D20 | C3×C4○D20 | D4⋊2D5 | Q8⋊2D5 | C3×D4⋊2D5 | C3×Q8⋊2D5 |
| kernel | C3×C4⋊C4⋊D5 | C12×Dic5 | C3×C10.D4 | C3×C4⋊Dic5 | C3×D10⋊C4 | C15×C4⋊C4 | C4⋊C4⋊D5 | C4×Dic5 | C10.D4 | C4⋊Dic5 | D10⋊C4 | C5×C4⋊C4 | C3×C4⋊C4 | C30 | C2×C12 | C4⋊C4 | C10 | C2×C4 | C6 | C2 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 3 | 1 | 2 | 2 | 2 | 2 | 6 | 2 | 2 | 6 | 6 | 4 | 12 | 12 | 8 | 16 | 2 | 2 | 4 | 4 |
Matrix representation of C3×C4⋊C4⋊D5 ►in GL4(𝔽61) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 47 | 0 |
| 0 | 0 | 0 | 47 |
| 30 | 17 | 0 | 0 |
| 44 | 31 | 0 | 0 |
| 0 | 0 | 27 | 2 |
| 0 | 0 | 1 | 34 |
| 50 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 |
| 0 | 0 | 53 | 22 |
| 0 | 0 | 22 | 8 |
| 0 | 60 | 0 | 0 |
| 1 | 43 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 18 | 60 | 0 | 0 |
| 18 | 43 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 34 | 60 |
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,47,0,0,0,0,47],[30,44,0,0,17,31,0,0,0,0,27,1,0,0,2,34],[50,0,0,0,0,50,0,0,0,0,53,22,0,0,22,8],[0,1,0,0,60,43,0,0,0,0,1,0,0,0,0,1],[18,18,0,0,60,43,0,0,0,0,1,34,0,0,0,60] >;
C3×C4⋊C4⋊D5 in GAP, Magma, Sage, TeX
C_3\times C_4\rtimes C_4\rtimes D_5
% in TeX
G:=Group("C3xC4:C4:D5"); // GroupNames label
G:=SmallGroup(480,691);
// by ID
G=gap.SmallGroup(480,691);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,701,176,590,555,268,18822]);
// 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,c*b*c^-1=b^-1,b*d=d*b,e*b*e=b*c^2,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations