direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D5×C4⋊C4, C4⋊3(D5×C12), C60⋊26(C2×C4), C20⋊4(C2×C12), (C4×D5)⋊1C12, (D5×C12)⋊5C4, C12⋊15(C4×D5), C6.50(Q8×D5), C4⋊Dic5⋊11C6, (C6×D5).84D4, C6.176(D4×D5), C10.22(C6×D4), D10.5(C3×Q8), (C6×D5).15Q8, C10.12(C6×Q8), Dic5⋊4(C2×C12), D10.22(C3×D4), C30.335(C2×D4), C30.102(C2×Q8), D10.18(C2×C12), C10.D4⋊11C6, (C2×C12).278D10, C30.180(C22×C4), C10.22(C22×C12), (C2×C60).396C22, (C2×C30).349C23, (C6×Dic5).241C22, C5⋊2(C6×C4⋊C4), (C5×C4⋊C4)⋊2C6, C15⋊15(C2×C4⋊C4), C2.3(C3×D4×D5), C2.2(C3×Q8×D5), (C15×C4⋊C4)⋊11C2, (C2×C4×D5).10C6, C6.105(C2×C4×D5), C2.11(D5×C2×C12), (D5×C2×C12).29C2, (C2×C4).39(C6×D5), C22.16(D5×C2×C6), (C2×C20).54(C2×C6), (C3×C4⋊Dic5)⋊29C2, (C6×D5).67(C2×C4), (C3×Dic5)⋊23(C2×C4), (D5×C2×C6).155C22, (C3×C10.D4)⋊27C2, (C2×C10).32(C22×C6), (C2×Dic5).31(C2×C6), (C22×D5).44(C2×C6), (C2×C6).345(C22×D5), SmallGroup(480,684)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D5×C4⋊C4
G = < a,b,c,d,e | a3=b5=c2=d4=e4=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 528 in 184 conjugacy classes, 98 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C2×C4, C23, D5, C10, C12, C12, C2×C6, C2×C6, C15, C4⋊C4, C4⋊C4, C22×C4, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C12, C2×C12, C2×C12, C22×C6, C3×D5, C30, C2×C4⋊C4, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C3×C4⋊C4, C3×C4⋊C4, C22×C12, C3×Dic5, C3×Dic5, C60, C60, C6×D5, C2×C30, C10.D4, C4⋊Dic5, C5×C4⋊C4, C2×C4×D5, C2×C4×D5, C6×C4⋊C4, D5×C12, D5×C12, C6×Dic5, C6×Dic5, C2×C60, C2×C60, D5×C2×C6, D5×C4⋊C4, C3×C10.D4, C3×C4⋊Dic5, C15×C4⋊C4, D5×C2×C12, D5×C2×C12, C3×D5×C4⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C23, D5, C12, C2×C6, C4⋊C4, C22×C4, C2×D4, C2×Q8, D10, C2×C12, C3×D4, C3×Q8, C22×C6, C3×D5, C2×C4⋊C4, C4×D5, C22×D5, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, C6×D5, C2×C4×D5, D4×D5, Q8×D5, C6×C4⋊C4, D5×C12, D5×C2×C6, D5×C4⋊C4, D5×C2×C12, C3×D4×D5, C3×Q8×D5, C3×D5×C4⋊C4
►On 240 points
Generators in S240
(1 44 24)(2 45 25)(3 41 21)(4 42 22)(5 43 23)(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 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 18)(2 17)(3 16)(4 20)(5 19)(6 11)(7 15)(8 14)(9 13)(10 12)(21 36)(22 40)(23 39)(24 38)(25 37)(26 31)(27 35)(28 34)(29 33)(30 32)(41 56)(42 60)(43 59)(44 58)(45 57)(46 51)(47 55)(48 54)(49 53)(50 52)(61 76)(62 80)(63 79)(64 78)(65 77)(66 71)(67 75)(68 74)(69 73)(70 72)(81 96)(82 100)(83 99)(84 98)(85 97)(86 91)(87 95)(88 94)(89 93)(90 92)(101 116)(102 120)(103 119)(104 118)(105 117)(106 111)(107 115)(108 114)(109 113)(110 112)(121 136)(122 140)(123 139)(124 138)(125 137)(126 131)(127 135)(128 134)(129 133)(130 132)(141 156)(142 160)(143 159)(144 158)(145 157)(146 151)(147 155)(148 154)(149 153)(150 152)(161 176)(162 180)(163 179)(164 178)(165 177)(166 171)(167 175)(168 174)(169 173)(170 172)(181 196)(182 200)(183 199)(184 198)(185 197)(186 191)(187 195)(188 194)(189 193)(190 192)(201 216)(202 220)(203 219)(204 218)(205 217)(206 211)(207 215)(208 214)(209 213)(210 212)(221 236)(222 240)(223 239)(224 238)(225 237)(226 231)(227 235)(228 234)(229 233)(230 232)
(1 139 9 134)(2 140 10 135)(3 136 6 131)(4 137 7 132)(5 138 8 133)(11 126 16 121)(12 127 17 122)(13 128 18 123)(14 129 19 124)(15 130 20 125)(21 156 26 151)(22 157 27 152)(23 158 28 153)(24 159 29 154)(25 160 30 155)(31 146 36 141)(32 147 37 142)(33 148 38 143)(34 149 39 144)(35 150 40 145)(41 176 46 171)(42 177 47 172)(43 178 48 173)(44 179 49 174)(45 180 50 175)(51 166 56 161)(52 167 57 162)(53 168 58 163)(54 169 59 164)(55 170 60 165)(61 196 66 191)(62 197 67 192)(63 198 68 193)(64 199 69 194)(65 200 70 195)(71 186 76 181)(72 187 77 182)(73 188 78 183)(74 189 79 184)(75 190 80 185)(81 216 86 211)(82 217 87 212)(83 218 88 213)(84 219 89 214)(85 220 90 215)(91 206 96 201)(92 207 97 202)(93 208 98 203)(94 209 99 204)(95 210 100 205)(101 236 106 231)(102 237 107 232)(103 238 108 233)(104 239 109 234)(105 240 110 235)(111 226 116 221)(112 227 117 222)(113 228 118 223)(114 229 119 224)(115 230 120 225)
(1 74 14 64)(2 75 15 65)(3 71 11 61)(4 72 12 62)(5 73 13 63)(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)(121 196 131 186)(122 197 132 187)(123 198 133 188)(124 199 134 189)(125 200 135 190)(126 191 136 181)(127 192 137 182)(128 193 138 183)(129 194 139 184)(130 195 140 185)(141 216 151 206)(142 217 152 207)(143 218 153 208)(144 219 154 209)(145 220 155 210)(146 211 156 201)(147 212 157 202)(148 213 158 203)(149 214 159 204)(150 215 160 205)(161 236 171 226)(162 237 172 227)(163 238 173 228)(164 239 174 229)(165 240 175 230)(166 231 176 221)(167 232 177 222)(168 233 178 223)(169 234 179 224)(170 235 180 225)
G:=sub<Sym(240)| (1,44,24)(2,45,25)(3,41,21)(4,42,22)(5,43,23)(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), (1,2,3,4,5)(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), (1,18)(2,17)(3,16)(4,20)(5,19)(6,11)(7,15)(8,14)(9,13)(10,12)(21,36)(22,40)(23,39)(24,38)(25,37)(26,31)(27,35)(28,34)(29,33)(30,32)(41,56)(42,60)(43,59)(44,58)(45,57)(46,51)(47,55)(48,54)(49,53)(50,52)(61,76)(62,80)(63,79)(64,78)(65,77)(66,71)(67,75)(68,74)(69,73)(70,72)(81,96)(82,100)(83,99)(84,98)(85,97)(86,91)(87,95)(88,94)(89,93)(90,92)(101,116)(102,120)(103,119)(104,118)(105,117)(106,111)(107,115)(108,114)(109,113)(110,112)(121,136)(122,140)(123,139)(124,138)(125,137)(126,131)(127,135)(128,134)(129,133)(130,132)(141,156)(142,160)(143,159)(144,158)(145,157)(146,151)(147,155)(148,154)(149,153)(150,152)(161,176)(162,180)(163,179)(164,178)(165,177)(166,171)(167,175)(168,174)(169,173)(170,172)(181,196)(182,200)(183,199)(184,198)(185,197)(186,191)(187,195)(188,194)(189,193)(190,192)(201,216)(202,220)(203,219)(204,218)(205,217)(206,211)(207,215)(208,214)(209,213)(210,212)(221,236)(222,240)(223,239)(224,238)(225,237)(226,231)(227,235)(228,234)(229,233)(230,232), (1,139,9,134)(2,140,10,135)(3,136,6,131)(4,137,7,132)(5,138,8,133)(11,126,16,121)(12,127,17,122)(13,128,18,123)(14,129,19,124)(15,130,20,125)(21,156,26,151)(22,157,27,152)(23,158,28,153)(24,159,29,154)(25,160,30,155)(31,146,36,141)(32,147,37,142)(33,148,38,143)(34,149,39,144)(35,150,40,145)(41,176,46,171)(42,177,47,172)(43,178,48,173)(44,179,49,174)(45,180,50,175)(51,166,56,161)(52,167,57,162)(53,168,58,163)(54,169,59,164)(55,170,60,165)(61,196,66,191)(62,197,67,192)(63,198,68,193)(64,199,69,194)(65,200,70,195)(71,186,76,181)(72,187,77,182)(73,188,78,183)(74,189,79,184)(75,190,80,185)(81,216,86,211)(82,217,87,212)(83,218,88,213)(84,219,89,214)(85,220,90,215)(91,206,96,201)(92,207,97,202)(93,208,98,203)(94,209,99,204)(95,210,100,205)(101,236,106,231)(102,237,107,232)(103,238,108,233)(104,239,109,234)(105,240,110,235)(111,226,116,221)(112,227,117,222)(113,228,118,223)(114,229,119,224)(115,230,120,225), (1,74,14,64)(2,75,15,65)(3,71,11,61)(4,72,12,62)(5,73,13,63)(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)(121,196,131,186)(122,197,132,187)(123,198,133,188)(124,199,134,189)(125,200,135,190)(126,191,136,181)(127,192,137,182)(128,193,138,183)(129,194,139,184)(130,195,140,185)(141,216,151,206)(142,217,152,207)(143,218,153,208)(144,219,154,209)(145,220,155,210)(146,211,156,201)(147,212,157,202)(148,213,158,203)(149,214,159,204)(150,215,160,205)(161,236,171,226)(162,237,172,227)(163,238,173,228)(164,239,174,229)(165,240,175,230)(166,231,176,221)(167,232,177,222)(168,233,178,223)(169,234,179,224)(170,235,180,225)>;
G:=Group( (1,44,24)(2,45,25)(3,41,21)(4,42,22)(5,43,23)(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), (1,2,3,4,5)(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), (1,18)(2,17)(3,16)(4,20)(5,19)(6,11)(7,15)(8,14)(9,13)(10,12)(21,36)(22,40)(23,39)(24,38)(25,37)(26,31)(27,35)(28,34)(29,33)(30,32)(41,56)(42,60)(43,59)(44,58)(45,57)(46,51)(47,55)(48,54)(49,53)(50,52)(61,76)(62,80)(63,79)(64,78)(65,77)(66,71)(67,75)(68,74)(69,73)(70,72)(81,96)(82,100)(83,99)(84,98)(85,97)(86,91)(87,95)(88,94)(89,93)(90,92)(101,116)(102,120)(103,119)(104,118)(105,117)(106,111)(107,115)(108,114)(109,113)(110,112)(121,136)(122,140)(123,139)(124,138)(125,137)(126,131)(127,135)(128,134)(129,133)(130,132)(141,156)(142,160)(143,159)(144,158)(145,157)(146,151)(147,155)(148,154)(149,153)(150,152)(161,176)(162,180)(163,179)(164,178)(165,177)(166,171)(167,175)(168,174)(169,173)(170,172)(181,196)(182,200)(183,199)(184,198)(185,197)(186,191)(187,195)(188,194)(189,193)(190,192)(201,216)(202,220)(203,219)(204,218)(205,217)(206,211)(207,215)(208,214)(209,213)(210,212)(221,236)(222,240)(223,239)(224,238)(225,237)(226,231)(227,235)(228,234)(229,233)(230,232), (1,139,9,134)(2,140,10,135)(3,136,6,131)(4,137,7,132)(5,138,8,133)(11,126,16,121)(12,127,17,122)(13,128,18,123)(14,129,19,124)(15,130,20,125)(21,156,26,151)(22,157,27,152)(23,158,28,153)(24,159,29,154)(25,160,30,155)(31,146,36,141)(32,147,37,142)(33,148,38,143)(34,149,39,144)(35,150,40,145)(41,176,46,171)(42,177,47,172)(43,178,48,173)(44,179,49,174)(45,180,50,175)(51,166,56,161)(52,167,57,162)(53,168,58,163)(54,169,59,164)(55,170,60,165)(61,196,66,191)(62,197,67,192)(63,198,68,193)(64,199,69,194)(65,200,70,195)(71,186,76,181)(72,187,77,182)(73,188,78,183)(74,189,79,184)(75,190,80,185)(81,216,86,211)(82,217,87,212)(83,218,88,213)(84,219,89,214)(85,220,90,215)(91,206,96,201)(92,207,97,202)(93,208,98,203)(94,209,99,204)(95,210,100,205)(101,236,106,231)(102,237,107,232)(103,238,108,233)(104,239,109,234)(105,240,110,235)(111,226,116,221)(112,227,117,222)(113,228,118,223)(114,229,119,224)(115,230,120,225), (1,74,14,64)(2,75,15,65)(3,71,11,61)(4,72,12,62)(5,73,13,63)(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)(121,196,131,186)(122,197,132,187)(123,198,133,188)(124,199,134,189)(125,200,135,190)(126,191,136,181)(127,192,137,182)(128,193,138,183)(129,194,139,184)(130,195,140,185)(141,216,151,206)(142,217,152,207)(143,218,153,208)(144,219,154,209)(145,220,155,210)(146,211,156,201)(147,212,157,202)(148,213,158,203)(149,214,159,204)(150,215,160,205)(161,236,171,226)(162,237,172,227)(163,238,173,228)(164,239,174,229)(165,240,175,230)(166,231,176,221)(167,232,177,222)(168,233,178,223)(169,234,179,224)(170,235,180,225) );
G=PermutationGroup([[(1,44,24),(2,45,25),(3,41,21),(4,42,22),(5,43,23),(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)], [(1,2,3,4,5),(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)], [(1,18),(2,17),(3,16),(4,20),(5,19),(6,11),(7,15),(8,14),(9,13),(10,12),(21,36),(22,40),(23,39),(24,38),(25,37),(26,31),(27,35),(28,34),(29,33),(30,32),(41,56),(42,60),(43,59),(44,58),(45,57),(46,51),(47,55),(48,54),(49,53),(50,52),(61,76),(62,80),(63,79),(64,78),(65,77),(66,71),(67,75),(68,74),(69,73),(70,72),(81,96),(82,100),(83,99),(84,98),(85,97),(86,91),(87,95),(88,94),(89,93),(90,92),(101,116),(102,120),(103,119),(104,118),(105,117),(106,111),(107,115),(108,114),(109,113),(110,112),(121,136),(122,140),(123,139),(124,138),(125,137),(126,131),(127,135),(128,134),(129,133),(130,132),(141,156),(142,160),(143,159),(144,158),(145,157),(146,151),(147,155),(148,154),(149,153),(150,152),(161,176),(162,180),(163,179),(164,178),(165,177),(166,171),(167,175),(168,174),(169,173),(170,172),(181,196),(182,200),(183,199),(184,198),(185,197),(186,191),(187,195),(188,194),(189,193),(190,192),(201,216),(202,220),(203,219),(204,218),(205,217),(206,211),(207,215),(208,214),(209,213),(210,212),(221,236),(222,240),(223,239),(224,238),(225,237),(226,231),(227,235),(228,234),(229,233),(230,232)], [(1,139,9,134),(2,140,10,135),(3,136,6,131),(4,137,7,132),(5,138,8,133),(11,126,16,121),(12,127,17,122),(13,128,18,123),(14,129,19,124),(15,130,20,125),(21,156,26,151),(22,157,27,152),(23,158,28,153),(24,159,29,154),(25,160,30,155),(31,146,36,141),(32,147,37,142),(33,148,38,143),(34,149,39,144),(35,150,40,145),(41,176,46,171),(42,177,47,172),(43,178,48,173),(44,179,49,174),(45,180,50,175),(51,166,56,161),(52,167,57,162),(53,168,58,163),(54,169,59,164),(55,170,60,165),(61,196,66,191),(62,197,67,192),(63,198,68,193),(64,199,69,194),(65,200,70,195),(71,186,76,181),(72,187,77,182),(73,188,78,183),(74,189,79,184),(75,190,80,185),(81,216,86,211),(82,217,87,212),(83,218,88,213),(84,219,89,214),(85,220,90,215),(91,206,96,201),(92,207,97,202),(93,208,98,203),(94,209,99,204),(95,210,100,205),(101,236,106,231),(102,237,107,232),(103,238,108,233),(104,239,109,234),(105,240,110,235),(111,226,116,221),(112,227,117,222),(113,228,118,223),(114,229,119,224),(115,230,120,225)], [(1,74,14,64),(2,75,15,65),(3,71,11,61),(4,72,12,62),(5,73,13,63),(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),(121,196,131,186),(122,197,132,187),(123,198,133,188),(124,199,134,189),(125,200,135,190),(126,191,136,181),(127,192,137,182),(128,193,138,183),(129,194,139,184),(130,195,140,185),(141,216,151,206),(142,217,152,207),(143,218,153,208),(144,219,154,209),(145,220,155,210),(146,211,156,201),(147,212,157,202),(148,213,158,203),(149,214,159,204),(150,215,160,205),(161,236,171,226),(162,237,172,227),(163,238,173,228),(164,239,174,229),(165,240,175,230),(166,231,176,221),(167,232,177,222),(168,233,178,223),(169,234,179,224),(170,235,180,225)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | ··· | 4F | 4G | ··· | 4L | 5A | 5B | 6A | ··· | 6F | 6G | ··· | 6N | 10A | ··· | 10F | 12A | ··· | 12L | 12M | ··· | 12X | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 1 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C12 | D4 | Q8 | D5 | D10 | C3×D4 | C3×Q8 | C3×D5 | C4×D5 | C6×D5 | D5×C12 | D4×D5 | Q8×D5 | C3×D4×D5 | C3×Q8×D5 |
| kernel | C3×D5×C4⋊C4 | C3×C10.D4 | C3×C4⋊Dic5 | C15×C4⋊C4 | D5×C2×C12 | D5×C4⋊C4 | D5×C12 | C10.D4 | C4⋊Dic5 | C5×C4⋊C4 | C2×C4×D5 | C4×D5 | C6×D5 | C6×D5 | C3×C4⋊C4 | C2×C12 | D10 | D10 | C4⋊C4 | C12 | C2×C4 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 3 | 2 | 8 | 4 | 2 | 2 | 6 | 16 | 2 | 2 | 2 | 6 | 4 | 4 | 4 | 8 | 12 | 16 | 2 | 2 | 4 | 4 |
Matrix representation of C3×D5×C4⋊C4 ►in GL4(𝔽61) generated by
| 47 | 0 | 0 | 0 |
| 0 | 47 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 43 | 1 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 18 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 40 | 30 |
| 0 | 0 | 30 | 21 |
| 50 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 0 |
G:=sub<GL(4,GF(61))| [47,0,0,0,0,47,0,0,0,0,1,0,0,0,0,1],[43,60,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,18,1,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,40,30,0,0,30,21],[50,0,0,0,0,50,0,0,0,0,0,60,0,0,1,0] >;
C3×D5×C4⋊C4 in GAP, Magma, Sage, TeX
C_3\times D_5\times C_4\rtimes C_4
% in TeX
G:=Group("C3xD5xC4:C4"); // GroupNames label
G:=SmallGroup(480,684);
// by ID
G=gap.SmallGroup(480,684);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,344,555,142,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^5=c^2=d^4=e^4=1,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,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations