metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C6×D5).5D4, (C2×D20).5S3, C6.128(D4×D5), (C2×C20).12D6, Dic3⋊C4⋊16D5, (C6×D20).13C2, C30.129(C2×D4), C30.54(C4○D4), C6.52(C4○D20), (C2×C12).221D10, D10⋊Dic3⋊7C2, (C2×C30).97C23, C30.4Q8⋊13C2, (C22×D5).45D6, C3⋊7(D10.13D4), D10.16(C3⋊D4), (C2×C60).252C22, C6.32(Q8⋊2D5), (C2×Dic3).29D10, C5⋊1(C23.23D6), C15⋊6(C22.D4), C2.13(D20⋊5S3), C2.14(D20⋊S3), C10.26(D4⋊2S3), (C10×Dic3).57C22, (C2×Dic15).77C22, (C2×D5×Dic3)⋊3C2, (C2×C4).36(S3×D5), C2.12(D5×C3⋊D4), C10.30(C2×C3⋊D4), (D5×C2×C6).14C22, C22.166(C2×S3×D5), (C5×Dic3⋊C4)⋊16C2, (C2×C6).109(C22×D5), (C2×C10).109(C22×S3), SmallGroup(480,483)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C6×D5).D4
G = < a,b,c,d,e | a6=b5=c2=d4=1, e2=a3, ab=ba, ac=ca, dad-1=eae-1=a-1, cbc=b-1, bd=db, be=eb, dcd-1=a3c, ce=ec, ede-1=d-1 >
Subgroups: 796 in 156 conjugacy classes, 48 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, Dic3, C12, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, D10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C3×D4, C22×C6, C3×D5, C30, C22.D4, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, Dic3⋊C4, Dic3⋊C4, C6.D4, C22×Dic3, C6×D4, C5×Dic3, Dic15, C60, C6×D5, C6×D5, C2×C30, C10.D4, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C23.23D6, D5×Dic3, C3×D20, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, D10.13D4, D10⋊Dic3, C5×Dic3⋊C4, C30.4Q8, C2×D5×Dic3, C6×D20, (C6×D5).D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C3⋊D4, C22×S3, C22.D4, C22×D5, D4⋊2S3, C2×C3⋊D4, S3×D5, C4○D20, D4×D5, Q8⋊2D5, C23.23D6, C2×S3×D5, D10.13D4, D20⋊5S3, D20⋊S3, D5×C3⋊D4, (C6×D5).D4
►On 240 points
Generators in S240
(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 59 49 19 30)(2 60 50 20 25)(3 55 51 21 26)(4 56 52 22 27)(5 57 53 23 28)(6 58 54 24 29)(7 34 41 221 195)(8 35 42 222 196)(9 36 37 217 197)(10 31 38 218 198)(11 32 39 219 193)(12 33 40 220 194)(13 48 94 62 71)(14 43 95 63 72)(15 44 96 64 67)(16 45 91 65 68)(17 46 92 66 69)(18 47 93 61 70)(73 102 154 127 126)(74 97 155 128 121)(75 98 156 129 122)(76 99 151 130 123)(77 100 152 131 124)(78 101 153 132 125)(79 104 86 119 109)(80 105 87 120 110)(81 106 88 115 111)(82 107 89 116 112)(83 108 90 117 113)(84 103 85 118 114)(133 212 189 186 159)(134 213 190 181 160)(135 214 191 182 161)(136 215 192 183 162)(137 216 187 184 157)(138 211 188 185 158)(139 175 171 166 146)(140 176 172 167 147)(141 177 173 168 148)(142 178 174 163 149)(143 179 169 164 150)(144 180 170 165 145)(199 235 231 226 206)(200 236 232 227 207)(201 237 233 228 208)(202 238 234 223 209)(203 239 229 224 210)(204 240 230 225 205)
(1 69)(2 70)(3 71)(4 72)(5 67)(6 68)(7 230)(8 231)(9 232)(10 233)(11 234)(12 229)(13 26)(14 27)(15 28)(16 29)(17 30)(18 25)(19 46)(20 47)(21 48)(22 43)(23 44)(24 45)(31 237)(32 238)(33 239)(34 240)(35 235)(36 236)(37 200)(38 201)(39 202)(40 203)(41 204)(42 199)(49 92)(50 93)(51 94)(52 95)(53 96)(54 91)(55 62)(56 63)(57 64)(58 65)(59 66)(60 61)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 97)(80 98)(81 99)(82 100)(83 101)(84 102)(85 126)(86 121)(87 122)(88 123)(89 124)(90 125)(109 155)(110 156)(111 151)(112 152)(113 153)(114 154)(115 130)(116 131)(117 132)(118 127)(119 128)(120 129)(133 166)(134 167)(135 168)(136 163)(137 164)(138 165)(139 186)(140 181)(141 182)(142 183)(143 184)(144 185)(145 158)(146 159)(147 160)(148 161)(149 162)(150 157)(169 216)(170 211)(171 212)(172 213)(173 214)(174 215)(175 189)(176 190)(177 191)(178 192)(179 187)(180 188)(193 223)(194 224)(195 225)(196 226)(197 227)(198 228)(205 221)(206 222)(207 217)(208 218)(209 219)(210 220)
(1 222 14 199)(2 221 15 204)(3 220 16 203)(4 219 17 202)(5 218 18 201)(6 217 13 200)(7 96 230 50)(8 95 231 49)(9 94 232 54)(10 93 233 53)(11 92 234 52)(12 91 229 51)(19 35 63 226)(20 34 64 225)(21 33 65 224)(22 32 66 223)(23 31 61 228)(24 36 62 227)(25 41 67 205)(26 40 68 210)(27 39 69 209)(28 38 70 208)(29 37 71 207)(30 42 72 206)(43 235 59 196)(44 240 60 195)(45 239 55 194)(46 238 56 193)(47 237 57 198)(48 236 58 197)(73 143 85 160)(74 142 86 159)(75 141 87 158)(76 140 88 157)(77 139 89 162)(78 144 90 161)(79 189 128 163)(80 188 129 168)(81 187 130 167)(82 192 131 166)(83 191 132 165)(84 190 127 164)(97 178 119 133)(98 177 120 138)(99 176 115 137)(100 175 116 136)(101 180 117 135)(102 179 118 134)(103 181 126 150)(104 186 121 149)(105 185 122 148)(106 184 123 147)(107 183 124 146)(108 182 125 145)(109 212 155 174)(110 211 156 173)(111 216 151 172)(112 215 152 171)(113 214 153 170)(114 213 154 169)
(1 142 4 139)(2 141 5 144)(3 140 6 143)(7 156 10 153)(8 155 11 152)(9 154 12 151)(13 160 16 157)(14 159 17 162)(15 158 18 161)(19 163 22 166)(20 168 23 165)(21 167 24 164)(25 148 28 145)(26 147 29 150)(27 146 30 149)(31 132 34 129)(32 131 35 128)(33 130 36 127)(37 126 40 123)(38 125 41 122)(39 124 42 121)(43 133 46 136)(44 138 47 135)(45 137 48 134)(49 174 52 171)(50 173 53 170)(51 172 54 169)(55 176 58 179)(56 175 59 178)(57 180 60 177)(61 191 64 188)(62 190 65 187)(63 189 66 192)(67 185 70 182)(68 184 71 181)(69 183 72 186)(73 220 76 217)(74 219 77 222)(75 218 78 221)(79 223 82 226)(80 228 83 225)(81 227 84 224)(85 203 88 200)(86 202 89 199)(87 201 90 204)(91 216 94 213)(92 215 95 212)(93 214 96 211)(97 193 100 196)(98 198 101 195)(99 197 102 194)(103 210 106 207)(104 209 107 206)(105 208 108 205)(109 234 112 231)(110 233 113 230)(111 232 114 229)(115 236 118 239)(116 235 119 238)(117 240 120 237)
G:=sub<Sym(240)| (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), (1,59,49,19,30)(2,60,50,20,25)(3,55,51,21,26)(4,56,52,22,27)(5,57,53,23,28)(6,58,54,24,29)(7,34,41,221,195)(8,35,42,222,196)(9,36,37,217,197)(10,31,38,218,198)(11,32,39,219,193)(12,33,40,220,194)(13,48,94,62,71)(14,43,95,63,72)(15,44,96,64,67)(16,45,91,65,68)(17,46,92,66,69)(18,47,93,61,70)(73,102,154,127,126)(74,97,155,128,121)(75,98,156,129,122)(76,99,151,130,123)(77,100,152,131,124)(78,101,153,132,125)(79,104,86,119,109)(80,105,87,120,110)(81,106,88,115,111)(82,107,89,116,112)(83,108,90,117,113)(84,103,85,118,114)(133,212,189,186,159)(134,213,190,181,160)(135,214,191,182,161)(136,215,192,183,162)(137,216,187,184,157)(138,211,188,185,158)(139,175,171,166,146)(140,176,172,167,147)(141,177,173,168,148)(142,178,174,163,149)(143,179,169,164,150)(144,180,170,165,145)(199,235,231,226,206)(200,236,232,227,207)(201,237,233,228,208)(202,238,234,223,209)(203,239,229,224,210)(204,240,230,225,205), (1,69)(2,70)(3,71)(4,72)(5,67)(6,68)(7,230)(8,231)(9,232)(10,233)(11,234)(12,229)(13,26)(14,27)(15,28)(16,29)(17,30)(18,25)(19,46)(20,47)(21,48)(22,43)(23,44)(24,45)(31,237)(32,238)(33,239)(34,240)(35,235)(36,236)(37,200)(38,201)(39,202)(40,203)(41,204)(42,199)(49,92)(50,93)(51,94)(52,95)(53,96)(54,91)(55,62)(56,63)(57,64)(58,65)(59,66)(60,61)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,126)(86,121)(87,122)(88,123)(89,124)(90,125)(109,155)(110,156)(111,151)(112,152)(113,153)(114,154)(115,130)(116,131)(117,132)(118,127)(119,128)(120,129)(133,166)(134,167)(135,168)(136,163)(137,164)(138,165)(139,186)(140,181)(141,182)(142,183)(143,184)(144,185)(145,158)(146,159)(147,160)(148,161)(149,162)(150,157)(169,216)(170,211)(171,212)(172,213)(173,214)(174,215)(175,189)(176,190)(177,191)(178,192)(179,187)(180,188)(193,223)(194,224)(195,225)(196,226)(197,227)(198,228)(205,221)(206,222)(207,217)(208,218)(209,219)(210,220), (1,222,14,199)(2,221,15,204)(3,220,16,203)(4,219,17,202)(5,218,18,201)(6,217,13,200)(7,96,230,50)(8,95,231,49)(9,94,232,54)(10,93,233,53)(11,92,234,52)(12,91,229,51)(19,35,63,226)(20,34,64,225)(21,33,65,224)(22,32,66,223)(23,31,61,228)(24,36,62,227)(25,41,67,205)(26,40,68,210)(27,39,69,209)(28,38,70,208)(29,37,71,207)(30,42,72,206)(43,235,59,196)(44,240,60,195)(45,239,55,194)(46,238,56,193)(47,237,57,198)(48,236,58,197)(73,143,85,160)(74,142,86,159)(75,141,87,158)(76,140,88,157)(77,139,89,162)(78,144,90,161)(79,189,128,163)(80,188,129,168)(81,187,130,167)(82,192,131,166)(83,191,132,165)(84,190,127,164)(97,178,119,133)(98,177,120,138)(99,176,115,137)(100,175,116,136)(101,180,117,135)(102,179,118,134)(103,181,126,150)(104,186,121,149)(105,185,122,148)(106,184,123,147)(107,183,124,146)(108,182,125,145)(109,212,155,174)(110,211,156,173)(111,216,151,172)(112,215,152,171)(113,214,153,170)(114,213,154,169), (1,142,4,139)(2,141,5,144)(3,140,6,143)(7,156,10,153)(8,155,11,152)(9,154,12,151)(13,160,16,157)(14,159,17,162)(15,158,18,161)(19,163,22,166)(20,168,23,165)(21,167,24,164)(25,148,28,145)(26,147,29,150)(27,146,30,149)(31,132,34,129)(32,131,35,128)(33,130,36,127)(37,126,40,123)(38,125,41,122)(39,124,42,121)(43,133,46,136)(44,138,47,135)(45,137,48,134)(49,174,52,171)(50,173,53,170)(51,172,54,169)(55,176,58,179)(56,175,59,178)(57,180,60,177)(61,191,64,188)(62,190,65,187)(63,189,66,192)(67,185,70,182)(68,184,71,181)(69,183,72,186)(73,220,76,217)(74,219,77,222)(75,218,78,221)(79,223,82,226)(80,228,83,225)(81,227,84,224)(85,203,88,200)(86,202,89,199)(87,201,90,204)(91,216,94,213)(92,215,95,212)(93,214,96,211)(97,193,100,196)(98,198,101,195)(99,197,102,194)(103,210,106,207)(104,209,107,206)(105,208,108,205)(109,234,112,231)(110,233,113,230)(111,232,114,229)(115,236,118,239)(116,235,119,238)(117,240,120,237)>;
G:=Group( (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), (1,59,49,19,30)(2,60,50,20,25)(3,55,51,21,26)(4,56,52,22,27)(5,57,53,23,28)(6,58,54,24,29)(7,34,41,221,195)(8,35,42,222,196)(9,36,37,217,197)(10,31,38,218,198)(11,32,39,219,193)(12,33,40,220,194)(13,48,94,62,71)(14,43,95,63,72)(15,44,96,64,67)(16,45,91,65,68)(17,46,92,66,69)(18,47,93,61,70)(73,102,154,127,126)(74,97,155,128,121)(75,98,156,129,122)(76,99,151,130,123)(77,100,152,131,124)(78,101,153,132,125)(79,104,86,119,109)(80,105,87,120,110)(81,106,88,115,111)(82,107,89,116,112)(83,108,90,117,113)(84,103,85,118,114)(133,212,189,186,159)(134,213,190,181,160)(135,214,191,182,161)(136,215,192,183,162)(137,216,187,184,157)(138,211,188,185,158)(139,175,171,166,146)(140,176,172,167,147)(141,177,173,168,148)(142,178,174,163,149)(143,179,169,164,150)(144,180,170,165,145)(199,235,231,226,206)(200,236,232,227,207)(201,237,233,228,208)(202,238,234,223,209)(203,239,229,224,210)(204,240,230,225,205), (1,69)(2,70)(3,71)(4,72)(5,67)(6,68)(7,230)(8,231)(9,232)(10,233)(11,234)(12,229)(13,26)(14,27)(15,28)(16,29)(17,30)(18,25)(19,46)(20,47)(21,48)(22,43)(23,44)(24,45)(31,237)(32,238)(33,239)(34,240)(35,235)(36,236)(37,200)(38,201)(39,202)(40,203)(41,204)(42,199)(49,92)(50,93)(51,94)(52,95)(53,96)(54,91)(55,62)(56,63)(57,64)(58,65)(59,66)(60,61)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,126)(86,121)(87,122)(88,123)(89,124)(90,125)(109,155)(110,156)(111,151)(112,152)(113,153)(114,154)(115,130)(116,131)(117,132)(118,127)(119,128)(120,129)(133,166)(134,167)(135,168)(136,163)(137,164)(138,165)(139,186)(140,181)(141,182)(142,183)(143,184)(144,185)(145,158)(146,159)(147,160)(148,161)(149,162)(150,157)(169,216)(170,211)(171,212)(172,213)(173,214)(174,215)(175,189)(176,190)(177,191)(178,192)(179,187)(180,188)(193,223)(194,224)(195,225)(196,226)(197,227)(198,228)(205,221)(206,222)(207,217)(208,218)(209,219)(210,220), (1,222,14,199)(2,221,15,204)(3,220,16,203)(4,219,17,202)(5,218,18,201)(6,217,13,200)(7,96,230,50)(8,95,231,49)(9,94,232,54)(10,93,233,53)(11,92,234,52)(12,91,229,51)(19,35,63,226)(20,34,64,225)(21,33,65,224)(22,32,66,223)(23,31,61,228)(24,36,62,227)(25,41,67,205)(26,40,68,210)(27,39,69,209)(28,38,70,208)(29,37,71,207)(30,42,72,206)(43,235,59,196)(44,240,60,195)(45,239,55,194)(46,238,56,193)(47,237,57,198)(48,236,58,197)(73,143,85,160)(74,142,86,159)(75,141,87,158)(76,140,88,157)(77,139,89,162)(78,144,90,161)(79,189,128,163)(80,188,129,168)(81,187,130,167)(82,192,131,166)(83,191,132,165)(84,190,127,164)(97,178,119,133)(98,177,120,138)(99,176,115,137)(100,175,116,136)(101,180,117,135)(102,179,118,134)(103,181,126,150)(104,186,121,149)(105,185,122,148)(106,184,123,147)(107,183,124,146)(108,182,125,145)(109,212,155,174)(110,211,156,173)(111,216,151,172)(112,215,152,171)(113,214,153,170)(114,213,154,169), (1,142,4,139)(2,141,5,144)(3,140,6,143)(7,156,10,153)(8,155,11,152)(9,154,12,151)(13,160,16,157)(14,159,17,162)(15,158,18,161)(19,163,22,166)(20,168,23,165)(21,167,24,164)(25,148,28,145)(26,147,29,150)(27,146,30,149)(31,132,34,129)(32,131,35,128)(33,130,36,127)(37,126,40,123)(38,125,41,122)(39,124,42,121)(43,133,46,136)(44,138,47,135)(45,137,48,134)(49,174,52,171)(50,173,53,170)(51,172,54,169)(55,176,58,179)(56,175,59,178)(57,180,60,177)(61,191,64,188)(62,190,65,187)(63,189,66,192)(67,185,70,182)(68,184,71,181)(69,183,72,186)(73,220,76,217)(74,219,77,222)(75,218,78,221)(79,223,82,226)(80,228,83,225)(81,227,84,224)(85,203,88,200)(86,202,89,199)(87,201,90,204)(91,216,94,213)(92,215,95,212)(93,214,96,211)(97,193,100,196)(98,198,101,195)(99,197,102,194)(103,210,106,207)(104,209,107,206)(105,208,108,205)(109,234,112,231)(110,233,113,230)(111,232,114,229)(115,236,118,239)(116,235,119,238)(117,240,120,237) );
G=PermutationGroup([[(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)], [(1,59,49,19,30),(2,60,50,20,25),(3,55,51,21,26),(4,56,52,22,27),(5,57,53,23,28),(6,58,54,24,29),(7,34,41,221,195),(8,35,42,222,196),(9,36,37,217,197),(10,31,38,218,198),(11,32,39,219,193),(12,33,40,220,194),(13,48,94,62,71),(14,43,95,63,72),(15,44,96,64,67),(16,45,91,65,68),(17,46,92,66,69),(18,47,93,61,70),(73,102,154,127,126),(74,97,155,128,121),(75,98,156,129,122),(76,99,151,130,123),(77,100,152,131,124),(78,101,153,132,125),(79,104,86,119,109),(80,105,87,120,110),(81,106,88,115,111),(82,107,89,116,112),(83,108,90,117,113),(84,103,85,118,114),(133,212,189,186,159),(134,213,190,181,160),(135,214,191,182,161),(136,215,192,183,162),(137,216,187,184,157),(138,211,188,185,158),(139,175,171,166,146),(140,176,172,167,147),(141,177,173,168,148),(142,178,174,163,149),(143,179,169,164,150),(144,180,170,165,145),(199,235,231,226,206),(200,236,232,227,207),(201,237,233,228,208),(202,238,234,223,209),(203,239,229,224,210),(204,240,230,225,205)], [(1,69),(2,70),(3,71),(4,72),(5,67),(6,68),(7,230),(8,231),(9,232),(10,233),(11,234),(12,229),(13,26),(14,27),(15,28),(16,29),(17,30),(18,25),(19,46),(20,47),(21,48),(22,43),(23,44),(24,45),(31,237),(32,238),(33,239),(34,240),(35,235),(36,236),(37,200),(38,201),(39,202),(40,203),(41,204),(42,199),(49,92),(50,93),(51,94),(52,95),(53,96),(54,91),(55,62),(56,63),(57,64),(58,65),(59,66),(60,61),(73,103),(74,104),(75,105),(76,106),(77,107),(78,108),(79,97),(80,98),(81,99),(82,100),(83,101),(84,102),(85,126),(86,121),(87,122),(88,123),(89,124),(90,125),(109,155),(110,156),(111,151),(112,152),(113,153),(114,154),(115,130),(116,131),(117,132),(118,127),(119,128),(120,129),(133,166),(134,167),(135,168),(136,163),(137,164),(138,165),(139,186),(140,181),(141,182),(142,183),(143,184),(144,185),(145,158),(146,159),(147,160),(148,161),(149,162),(150,157),(169,216),(170,211),(171,212),(172,213),(173,214),(174,215),(175,189),(176,190),(177,191),(178,192),(179,187),(180,188),(193,223),(194,224),(195,225),(196,226),(197,227),(198,228),(205,221),(206,222),(207,217),(208,218),(209,219),(210,220)], [(1,222,14,199),(2,221,15,204),(3,220,16,203),(4,219,17,202),(5,218,18,201),(6,217,13,200),(7,96,230,50),(8,95,231,49),(9,94,232,54),(10,93,233,53),(11,92,234,52),(12,91,229,51),(19,35,63,226),(20,34,64,225),(21,33,65,224),(22,32,66,223),(23,31,61,228),(24,36,62,227),(25,41,67,205),(26,40,68,210),(27,39,69,209),(28,38,70,208),(29,37,71,207),(30,42,72,206),(43,235,59,196),(44,240,60,195),(45,239,55,194),(46,238,56,193),(47,237,57,198),(48,236,58,197),(73,143,85,160),(74,142,86,159),(75,141,87,158),(76,140,88,157),(77,139,89,162),(78,144,90,161),(79,189,128,163),(80,188,129,168),(81,187,130,167),(82,192,131,166),(83,191,132,165),(84,190,127,164),(97,178,119,133),(98,177,120,138),(99,176,115,137),(100,175,116,136),(101,180,117,135),(102,179,118,134),(103,181,126,150),(104,186,121,149),(105,185,122,148),(106,184,123,147),(107,183,124,146),(108,182,125,145),(109,212,155,174),(110,211,156,173),(111,216,151,172),(112,215,152,171),(113,214,153,170),(114,213,154,169)], [(1,142,4,139),(2,141,5,144),(3,140,6,143),(7,156,10,153),(8,155,11,152),(9,154,12,151),(13,160,16,157),(14,159,17,162),(15,158,18,161),(19,163,22,166),(20,168,23,165),(21,167,24,164),(25,148,28,145),(26,147,29,150),(27,146,30,149),(31,132,34,129),(32,131,35,128),(33,130,36,127),(37,126,40,123),(38,125,41,122),(39,124,42,121),(43,133,46,136),(44,138,47,135),(45,137,48,134),(49,174,52,171),(50,173,53,170),(51,172,54,169),(55,176,58,179),(56,175,59,178),(57,180,60,177),(61,191,64,188),(62,190,65,187),(63,189,66,192),(67,185,70,182),(68,184,71,181),(69,183,72,186),(73,220,76,217),(74,219,77,222),(75,218,78,221),(79,223,82,226),(80,228,83,225),(81,227,84,224),(85,203,88,200),(86,202,89,199),(87,201,90,204),(91,216,94,213),(92,215,95,212),(93,214,96,211),(97,193,100,196),(98,198,101,195),(99,197,102,194),(103,210,106,207),(104,209,107,206),(105,208,108,205),(109,234,112,231),(110,233,113,230),(111,232,114,229),(115,236,118,239),(116,235,119,238),(117,240,120,237)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 20 | 2 | 4 | 6 | 6 | 12 | 30 | 30 | 60 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | C3⋊D4 | C4○D20 | D4⋊2S3 | S3×D5 | D4×D5 | Q8⋊2D5 | C2×S3×D5 | D20⋊5S3 | D20⋊S3 | D5×C3⋊D4 |
| kernel | (C6×D5).D4 | D10⋊Dic3 | C5×Dic3⋊C4 | C30.4Q8 | C2×D5×Dic3 | C6×D20 | C2×D20 | C6×D5 | Dic3⋊C4 | C2×C20 | C22×D5 | C30 | C2×Dic3 | C2×C12 | D10 | C6 | C10 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 3 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 4 | 4 | 2 | 4 | 8 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of (C6×D5).D4 ►in GL6(𝔽61)
| 60 | 1 | 0 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 43 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 1 | 0 | 0 |
| 0 | 0 | 43 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 1 | 60 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 44 | 0 | 0 |
| 0 | 0 | 17 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 7 |
| 0 | 0 | 0 | 0 | 54 | 29 |
| 60 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 54 | 29 |
| 0 | 0 | 0 | 0 | 32 | 7 |
G:=sub<GL(6,GF(61))| [60,60,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,60,43,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,43,43,0,0,0,0,1,18,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,60,60,0,0,0,0,0,0,31,17,0,0,0,0,44,30,0,0,0,0,0,0,32,54,0,0,0,0,7,29],[60,0,0,0,0,0,1,1,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,54,32,0,0,0,0,29,7] >;
(C6×D5).D4 in GAP, Magma, Sage, TeX
(C_6\times D_5).D_4
% in TeX
G:=Group("(C6xD5).D4"); // GroupNames label
G:=SmallGroup(480,483);
// by ID
G=gap.SmallGroup(480,483);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,254,219,58,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^5=c^2=d^4=1,e^2=a^3,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e^-1=a^-1,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=a^3*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations