direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6×D4⋊D5, C30⋊8D8, C60.124D4, C60.200C23, C5⋊3(C6×D8), D4⋊3(C6×D5), (C6×D4)⋊8D5, C10⋊2(C3×D8), C15⋊17(C2×D8), D20⋊5(C2×C6), (D4×C30)⋊8C2, (D4×C10)⋊1C6, (C2×D20)⋊8C6, (C6×D20)⋊24C2, (C3×D4)⋊25D10, C20.14(C3×D4), C10.48(C6×D4), C30.402(C2×D4), (C2×C30).163D4, (C2×C12).359D10, (D4×C15)⋊27C22, (C3×D20)⋊35C22, C12.73(C5⋊D4), C20.11(C22×C6), (C2×C60).289C22, C12.200(C22×D5), (C2×C5⋊2C8)⋊4C6, C4.11(D5×C2×C6), C5⋊2C8⋊7(C2×C6), (C5×D4)⋊3(C2×C6), (C2×D4)⋊1(C3×D5), (C6×C5⋊2C8)⋊18C2, C4.5(C3×C5⋊D4), C2.8(C6×C5⋊D4), (C2×C4).46(C6×D5), (C2×C20).26(C2×C6), (C2×C10).38(C3×D4), C6.129(C2×C5⋊D4), (C3×C5⋊2C8)⋊40C22, (C2×C6).93(C5⋊D4), C22.21(C3×C5⋊D4), SmallGroup(480,724)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×D4⋊D5
G = < a,b,c,d,e | a6=b4=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=d-1 >
Subgroups: 560 in 152 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, C6, C6, C6, C8, C2×C4, D4, D4, C23, D5, C10, C10, C10, C12, C2×C6, C2×C6, C15, C2×C8, D8, C2×D4, C2×D4, C20, D10, C2×C10, C2×C10, C24, C2×C12, C3×D4, C3×D4, C22×C6, C3×D5, C30, C30, C30, C2×D8, C5⋊2C8, D20, D20, C2×C20, C5×D4, C5×D4, C22×D5, C22×C10, C2×C24, C3×D8, C6×D4, C6×D4, C60, C6×D5, C2×C30, C2×C30, C2×C5⋊2C8, D4⋊D5, C2×D20, D4×C10, C6×D8, C3×C5⋊2C8, C3×D20, C3×D20, C2×C60, D4×C15, D4×C15, D5×C2×C6, C22×C30, C2×D4⋊D5, C6×C5⋊2C8, C3×D4⋊D5, C6×D20, D4×C30, C6×D4⋊D5
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, D8, C2×D4, D10, C3×D4, C22×C6, C3×D5, C2×D8, C5⋊D4, C22×D5, C3×D8, C6×D4, C6×D5, D4⋊D5, C2×C5⋊D4, C6×D8, C3×C5⋊D4, D5×C2×C6, C2×D4⋊D5, C3×D4⋊D5, C6×C5⋊D4, C6×D4⋊D5
►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)
(1 213 85 167)(2 214 86 168)(3 215 87 163)(4 216 88 164)(5 211 89 165)(6 212 90 166)(7 115 193 35)(8 116 194 36)(9 117 195 31)(10 118 196 32)(11 119 197 33)(12 120 198 34)(13 63 199 122)(14 64 200 123)(15 65 201 124)(16 66 202 125)(17 61 203 126)(18 62 204 121)(19 220 174 104)(20 221 169 105)(21 222 170 106)(22 217 171 107)(23 218 172 108)(24 219 173 103)(25 155 229 113)(26 156 230 114)(27 151 231 109)(28 152 232 110)(29 153 233 111)(30 154 234 112)(37 98 190 58)(38 99 191 59)(39 100 192 60)(40 101 187 55)(41 102 188 56)(42 97 189 57)(43 239 142 181)(44 240 143 182)(45 235 144 183)(46 236 139 184)(47 237 140 185)(48 238 141 186)(49 127 67 91)(50 128 68 92)(51 129 69 93)(52 130 70 94)(53 131 71 95)(54 132 72 96)(73 149 133 223)(74 150 134 224)(75 145 135 225)(76 146 136 226)(77 147 137 227)(78 148 138 228)(79 162 209 177)(80 157 210 178)(81 158 205 179)(82 159 206 180)(83 160 207 175)(84 161 208 176)
(1 123)(2 124)(3 125)(4 126)(5 121)(6 122)(7 139)(8 140)(9 141)(10 142)(11 143)(12 144)(13 166)(14 167)(15 168)(16 163)(17 164)(18 165)(19 71)(20 72)(21 67)(22 68)(23 69)(24 70)(25 206)(26 207)(27 208)(28 209)(29 210)(30 205)(31 186)(32 181)(33 182)(34 183)(35 184)(36 185)(37 223)(38 224)(39 225)(40 226)(41 227)(42 228)(43 196)(44 197)(45 198)(46 193)(47 194)(48 195)(49 170)(50 171)(51 172)(52 173)(53 174)(54 169)(55 76)(56 77)(57 78)(58 73)(59 74)(60 75)(61 88)(62 89)(63 90)(64 85)(65 86)(66 87)(79 232)(80 233)(81 234)(82 229)(83 230)(84 231)(91 106)(92 107)(93 108)(94 103)(95 104)(96 105)(97 138)(98 133)(99 134)(100 135)(101 136)(102 137)(109 176)(110 177)(111 178)(112 179)(113 180)(114 175)(115 236)(116 237)(117 238)(118 239)(119 240)(120 235)(127 222)(128 217)(129 218)(130 219)(131 220)(132 221)(145 192)(146 187)(147 188)(148 189)(149 190)(150 191)(151 161)(152 162)(153 157)(154 158)(155 159)(156 160)(199 212)(200 213)(201 214)(202 215)(203 216)(204 211)
(1 95 135 113 7)(2 96 136 114 8)(3 91 137 109 9)(4 92 138 110 10)(5 93 133 111 11)(6 94 134 112 12)(13 24 191 81 183)(14 19 192 82 184)(15 20 187 83 185)(16 21 188 84 186)(17 22 189 79 181)(18 23 190 80 182)(25 115 213 53 225)(26 116 214 54 226)(27 117 215 49 227)(28 118 216 50 228)(29 119 211 51 223)(30 120 212 52 224)(31 163 67 147 231)(32 164 68 148 232)(33 165 69 149 233)(34 166 70 150 234)(35 167 71 145 229)(36 168 72 146 230)(37 210 240 204 172)(38 205 235 199 173)(39 206 236 200 174)(40 207 237 201 169)(41 208 238 202 170)(42 209 239 203 171)(43 61 217 57 162)(44 62 218 58 157)(45 63 219 59 158)(46 64 220 60 159)(47 65 221 55 160)(48 66 222 56 161)(73 153 197 89 129)(74 154 198 90 130)(75 155 193 85 131)(76 156 194 86 132)(77 151 195 87 127)(78 152 196 88 128)(97 177 142 126 107)(98 178 143 121 108)(99 179 144 122 103)(100 180 139 123 104)(101 175 140 124 105)(102 176 141 125 106)
(1 10)(2 11)(3 12)(4 7)(5 8)(6 9)(13 48)(14 43)(15 44)(16 45)(17 46)(18 47)(19 162)(20 157)(21 158)(22 159)(23 160)(24 161)(25 68)(26 69)(27 70)(28 71)(29 72)(30 67)(31 212)(32 213)(33 214)(34 215)(35 216)(36 211)(37 101)(38 102)(39 97)(40 98)(41 99)(42 100)(49 234)(50 229)(51 230)(52 231)(53 232)(54 233)(55 190)(56 191)(57 192)(58 187)(59 188)(60 189)(61 184)(62 185)(63 186)(64 181)(65 182)(66 183)(73 76)(74 77)(75 78)(79 220)(80 221)(81 222)(82 217)(83 218)(84 219)(85 196)(86 197)(87 198)(88 193)(89 194)(90 195)(91 112)(92 113)(93 114)(94 109)(95 110)(96 111)(103 208)(104 209)(105 210)(106 205)(107 206)(108 207)(115 164)(116 165)(117 166)(118 167)(119 168)(120 163)(121 237)(122 238)(123 239)(124 240)(125 235)(126 236)(127 154)(128 155)(129 156)(130 151)(131 152)(132 153)(133 136)(134 137)(135 138)(139 203)(140 204)(141 199)(142 200)(143 201)(144 202)(145 228)(146 223)(147 224)(148 225)(149 226)(150 227)(169 178)(170 179)(171 180)(172 175)(173 176)(174 177)
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), (1,213,85,167)(2,214,86,168)(3,215,87,163)(4,216,88,164)(5,211,89,165)(6,212,90,166)(7,115,193,35)(8,116,194,36)(9,117,195,31)(10,118,196,32)(11,119,197,33)(12,120,198,34)(13,63,199,122)(14,64,200,123)(15,65,201,124)(16,66,202,125)(17,61,203,126)(18,62,204,121)(19,220,174,104)(20,221,169,105)(21,222,170,106)(22,217,171,107)(23,218,172,108)(24,219,173,103)(25,155,229,113)(26,156,230,114)(27,151,231,109)(28,152,232,110)(29,153,233,111)(30,154,234,112)(37,98,190,58)(38,99,191,59)(39,100,192,60)(40,101,187,55)(41,102,188,56)(42,97,189,57)(43,239,142,181)(44,240,143,182)(45,235,144,183)(46,236,139,184)(47,237,140,185)(48,238,141,186)(49,127,67,91)(50,128,68,92)(51,129,69,93)(52,130,70,94)(53,131,71,95)(54,132,72,96)(73,149,133,223)(74,150,134,224)(75,145,135,225)(76,146,136,226)(77,147,137,227)(78,148,138,228)(79,162,209,177)(80,157,210,178)(81,158,205,179)(82,159,206,180)(83,160,207,175)(84,161,208,176), (1,123)(2,124)(3,125)(4,126)(5,121)(6,122)(7,139)(8,140)(9,141)(10,142)(11,143)(12,144)(13,166)(14,167)(15,168)(16,163)(17,164)(18,165)(19,71)(20,72)(21,67)(22,68)(23,69)(24,70)(25,206)(26,207)(27,208)(28,209)(29,210)(30,205)(31,186)(32,181)(33,182)(34,183)(35,184)(36,185)(37,223)(38,224)(39,225)(40,226)(41,227)(42,228)(43,196)(44,197)(45,198)(46,193)(47,194)(48,195)(49,170)(50,171)(51,172)(52,173)(53,174)(54,169)(55,76)(56,77)(57,78)(58,73)(59,74)(60,75)(61,88)(62,89)(63,90)(64,85)(65,86)(66,87)(79,232)(80,233)(81,234)(82,229)(83,230)(84,231)(91,106)(92,107)(93,108)(94,103)(95,104)(96,105)(97,138)(98,133)(99,134)(100,135)(101,136)(102,137)(109,176)(110,177)(111,178)(112,179)(113,180)(114,175)(115,236)(116,237)(117,238)(118,239)(119,240)(120,235)(127,222)(128,217)(129,218)(130,219)(131,220)(132,221)(145,192)(146,187)(147,188)(148,189)(149,190)(150,191)(151,161)(152,162)(153,157)(154,158)(155,159)(156,160)(199,212)(200,213)(201,214)(202,215)(203,216)(204,211), (1,95,135,113,7)(2,96,136,114,8)(3,91,137,109,9)(4,92,138,110,10)(5,93,133,111,11)(6,94,134,112,12)(13,24,191,81,183)(14,19,192,82,184)(15,20,187,83,185)(16,21,188,84,186)(17,22,189,79,181)(18,23,190,80,182)(25,115,213,53,225)(26,116,214,54,226)(27,117,215,49,227)(28,118,216,50,228)(29,119,211,51,223)(30,120,212,52,224)(31,163,67,147,231)(32,164,68,148,232)(33,165,69,149,233)(34,166,70,150,234)(35,167,71,145,229)(36,168,72,146,230)(37,210,240,204,172)(38,205,235,199,173)(39,206,236,200,174)(40,207,237,201,169)(41,208,238,202,170)(42,209,239,203,171)(43,61,217,57,162)(44,62,218,58,157)(45,63,219,59,158)(46,64,220,60,159)(47,65,221,55,160)(48,66,222,56,161)(73,153,197,89,129)(74,154,198,90,130)(75,155,193,85,131)(76,156,194,86,132)(77,151,195,87,127)(78,152,196,88,128)(97,177,142,126,107)(98,178,143,121,108)(99,179,144,122,103)(100,180,139,123,104)(101,175,140,124,105)(102,176,141,125,106), (1,10)(2,11)(3,12)(4,7)(5,8)(6,9)(13,48)(14,43)(15,44)(16,45)(17,46)(18,47)(19,162)(20,157)(21,158)(22,159)(23,160)(24,161)(25,68)(26,69)(27,70)(28,71)(29,72)(30,67)(31,212)(32,213)(33,214)(34,215)(35,216)(36,211)(37,101)(38,102)(39,97)(40,98)(41,99)(42,100)(49,234)(50,229)(51,230)(52,231)(53,232)(54,233)(55,190)(56,191)(57,192)(58,187)(59,188)(60,189)(61,184)(62,185)(63,186)(64,181)(65,182)(66,183)(73,76)(74,77)(75,78)(79,220)(80,221)(81,222)(82,217)(83,218)(84,219)(85,196)(86,197)(87,198)(88,193)(89,194)(90,195)(91,112)(92,113)(93,114)(94,109)(95,110)(96,111)(103,208)(104,209)(105,210)(106,205)(107,206)(108,207)(115,164)(116,165)(117,166)(118,167)(119,168)(120,163)(121,237)(122,238)(123,239)(124,240)(125,235)(126,236)(127,154)(128,155)(129,156)(130,151)(131,152)(132,153)(133,136)(134,137)(135,138)(139,203)(140,204)(141,199)(142,200)(143,201)(144,202)(145,228)(146,223)(147,224)(148,225)(149,226)(150,227)(169,178)(170,179)(171,180)(172,175)(173,176)(174,177)>;
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), (1,213,85,167)(2,214,86,168)(3,215,87,163)(4,216,88,164)(5,211,89,165)(6,212,90,166)(7,115,193,35)(8,116,194,36)(9,117,195,31)(10,118,196,32)(11,119,197,33)(12,120,198,34)(13,63,199,122)(14,64,200,123)(15,65,201,124)(16,66,202,125)(17,61,203,126)(18,62,204,121)(19,220,174,104)(20,221,169,105)(21,222,170,106)(22,217,171,107)(23,218,172,108)(24,219,173,103)(25,155,229,113)(26,156,230,114)(27,151,231,109)(28,152,232,110)(29,153,233,111)(30,154,234,112)(37,98,190,58)(38,99,191,59)(39,100,192,60)(40,101,187,55)(41,102,188,56)(42,97,189,57)(43,239,142,181)(44,240,143,182)(45,235,144,183)(46,236,139,184)(47,237,140,185)(48,238,141,186)(49,127,67,91)(50,128,68,92)(51,129,69,93)(52,130,70,94)(53,131,71,95)(54,132,72,96)(73,149,133,223)(74,150,134,224)(75,145,135,225)(76,146,136,226)(77,147,137,227)(78,148,138,228)(79,162,209,177)(80,157,210,178)(81,158,205,179)(82,159,206,180)(83,160,207,175)(84,161,208,176), (1,123)(2,124)(3,125)(4,126)(5,121)(6,122)(7,139)(8,140)(9,141)(10,142)(11,143)(12,144)(13,166)(14,167)(15,168)(16,163)(17,164)(18,165)(19,71)(20,72)(21,67)(22,68)(23,69)(24,70)(25,206)(26,207)(27,208)(28,209)(29,210)(30,205)(31,186)(32,181)(33,182)(34,183)(35,184)(36,185)(37,223)(38,224)(39,225)(40,226)(41,227)(42,228)(43,196)(44,197)(45,198)(46,193)(47,194)(48,195)(49,170)(50,171)(51,172)(52,173)(53,174)(54,169)(55,76)(56,77)(57,78)(58,73)(59,74)(60,75)(61,88)(62,89)(63,90)(64,85)(65,86)(66,87)(79,232)(80,233)(81,234)(82,229)(83,230)(84,231)(91,106)(92,107)(93,108)(94,103)(95,104)(96,105)(97,138)(98,133)(99,134)(100,135)(101,136)(102,137)(109,176)(110,177)(111,178)(112,179)(113,180)(114,175)(115,236)(116,237)(117,238)(118,239)(119,240)(120,235)(127,222)(128,217)(129,218)(130,219)(131,220)(132,221)(145,192)(146,187)(147,188)(148,189)(149,190)(150,191)(151,161)(152,162)(153,157)(154,158)(155,159)(156,160)(199,212)(200,213)(201,214)(202,215)(203,216)(204,211), (1,95,135,113,7)(2,96,136,114,8)(3,91,137,109,9)(4,92,138,110,10)(5,93,133,111,11)(6,94,134,112,12)(13,24,191,81,183)(14,19,192,82,184)(15,20,187,83,185)(16,21,188,84,186)(17,22,189,79,181)(18,23,190,80,182)(25,115,213,53,225)(26,116,214,54,226)(27,117,215,49,227)(28,118,216,50,228)(29,119,211,51,223)(30,120,212,52,224)(31,163,67,147,231)(32,164,68,148,232)(33,165,69,149,233)(34,166,70,150,234)(35,167,71,145,229)(36,168,72,146,230)(37,210,240,204,172)(38,205,235,199,173)(39,206,236,200,174)(40,207,237,201,169)(41,208,238,202,170)(42,209,239,203,171)(43,61,217,57,162)(44,62,218,58,157)(45,63,219,59,158)(46,64,220,60,159)(47,65,221,55,160)(48,66,222,56,161)(73,153,197,89,129)(74,154,198,90,130)(75,155,193,85,131)(76,156,194,86,132)(77,151,195,87,127)(78,152,196,88,128)(97,177,142,126,107)(98,178,143,121,108)(99,179,144,122,103)(100,180,139,123,104)(101,175,140,124,105)(102,176,141,125,106), (1,10)(2,11)(3,12)(4,7)(5,8)(6,9)(13,48)(14,43)(15,44)(16,45)(17,46)(18,47)(19,162)(20,157)(21,158)(22,159)(23,160)(24,161)(25,68)(26,69)(27,70)(28,71)(29,72)(30,67)(31,212)(32,213)(33,214)(34,215)(35,216)(36,211)(37,101)(38,102)(39,97)(40,98)(41,99)(42,100)(49,234)(50,229)(51,230)(52,231)(53,232)(54,233)(55,190)(56,191)(57,192)(58,187)(59,188)(60,189)(61,184)(62,185)(63,186)(64,181)(65,182)(66,183)(73,76)(74,77)(75,78)(79,220)(80,221)(81,222)(82,217)(83,218)(84,219)(85,196)(86,197)(87,198)(88,193)(89,194)(90,195)(91,112)(92,113)(93,114)(94,109)(95,110)(96,111)(103,208)(104,209)(105,210)(106,205)(107,206)(108,207)(115,164)(116,165)(117,166)(118,167)(119,168)(120,163)(121,237)(122,238)(123,239)(124,240)(125,235)(126,236)(127,154)(128,155)(129,156)(130,151)(131,152)(132,153)(133,136)(134,137)(135,138)(139,203)(140,204)(141,199)(142,200)(143,201)(144,202)(145,228)(146,223)(147,224)(148,225)(149,226)(150,227)(169,178)(170,179)(171,180)(172,175)(173,176)(174,177) );
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)], [(1,213,85,167),(2,214,86,168),(3,215,87,163),(4,216,88,164),(5,211,89,165),(6,212,90,166),(7,115,193,35),(8,116,194,36),(9,117,195,31),(10,118,196,32),(11,119,197,33),(12,120,198,34),(13,63,199,122),(14,64,200,123),(15,65,201,124),(16,66,202,125),(17,61,203,126),(18,62,204,121),(19,220,174,104),(20,221,169,105),(21,222,170,106),(22,217,171,107),(23,218,172,108),(24,219,173,103),(25,155,229,113),(26,156,230,114),(27,151,231,109),(28,152,232,110),(29,153,233,111),(30,154,234,112),(37,98,190,58),(38,99,191,59),(39,100,192,60),(40,101,187,55),(41,102,188,56),(42,97,189,57),(43,239,142,181),(44,240,143,182),(45,235,144,183),(46,236,139,184),(47,237,140,185),(48,238,141,186),(49,127,67,91),(50,128,68,92),(51,129,69,93),(52,130,70,94),(53,131,71,95),(54,132,72,96),(73,149,133,223),(74,150,134,224),(75,145,135,225),(76,146,136,226),(77,147,137,227),(78,148,138,228),(79,162,209,177),(80,157,210,178),(81,158,205,179),(82,159,206,180),(83,160,207,175),(84,161,208,176)], [(1,123),(2,124),(3,125),(4,126),(5,121),(6,122),(7,139),(8,140),(9,141),(10,142),(11,143),(12,144),(13,166),(14,167),(15,168),(16,163),(17,164),(18,165),(19,71),(20,72),(21,67),(22,68),(23,69),(24,70),(25,206),(26,207),(27,208),(28,209),(29,210),(30,205),(31,186),(32,181),(33,182),(34,183),(35,184),(36,185),(37,223),(38,224),(39,225),(40,226),(41,227),(42,228),(43,196),(44,197),(45,198),(46,193),(47,194),(48,195),(49,170),(50,171),(51,172),(52,173),(53,174),(54,169),(55,76),(56,77),(57,78),(58,73),(59,74),(60,75),(61,88),(62,89),(63,90),(64,85),(65,86),(66,87),(79,232),(80,233),(81,234),(82,229),(83,230),(84,231),(91,106),(92,107),(93,108),(94,103),(95,104),(96,105),(97,138),(98,133),(99,134),(100,135),(101,136),(102,137),(109,176),(110,177),(111,178),(112,179),(113,180),(114,175),(115,236),(116,237),(117,238),(118,239),(119,240),(120,235),(127,222),(128,217),(129,218),(130,219),(131,220),(132,221),(145,192),(146,187),(147,188),(148,189),(149,190),(150,191),(151,161),(152,162),(153,157),(154,158),(155,159),(156,160),(199,212),(200,213),(201,214),(202,215),(203,216),(204,211)], [(1,95,135,113,7),(2,96,136,114,8),(3,91,137,109,9),(4,92,138,110,10),(5,93,133,111,11),(6,94,134,112,12),(13,24,191,81,183),(14,19,192,82,184),(15,20,187,83,185),(16,21,188,84,186),(17,22,189,79,181),(18,23,190,80,182),(25,115,213,53,225),(26,116,214,54,226),(27,117,215,49,227),(28,118,216,50,228),(29,119,211,51,223),(30,120,212,52,224),(31,163,67,147,231),(32,164,68,148,232),(33,165,69,149,233),(34,166,70,150,234),(35,167,71,145,229),(36,168,72,146,230),(37,210,240,204,172),(38,205,235,199,173),(39,206,236,200,174),(40,207,237,201,169),(41,208,238,202,170),(42,209,239,203,171),(43,61,217,57,162),(44,62,218,58,157),(45,63,219,59,158),(46,64,220,60,159),(47,65,221,55,160),(48,66,222,56,161),(73,153,197,89,129),(74,154,198,90,130),(75,155,193,85,131),(76,156,194,86,132),(77,151,195,87,127),(78,152,196,88,128),(97,177,142,126,107),(98,178,143,121,108),(99,179,144,122,103),(100,180,139,123,104),(101,175,140,124,105),(102,176,141,125,106)], [(1,10),(2,11),(3,12),(4,7),(5,8),(6,9),(13,48),(14,43),(15,44),(16,45),(17,46),(18,47),(19,162),(20,157),(21,158),(22,159),(23,160),(24,161),(25,68),(26,69),(27,70),(28,71),(29,72),(30,67),(31,212),(32,213),(33,214),(34,215),(35,216),(36,211),(37,101),(38,102),(39,97),(40,98),(41,99),(42,100),(49,234),(50,229),(51,230),(52,231),(53,232),(54,233),(55,190),(56,191),(57,192),(58,187),(59,188),(60,189),(61,184),(62,185),(63,186),(64,181),(65,182),(66,183),(73,76),(74,77),(75,78),(79,220),(80,221),(81,222),(82,217),(83,218),(84,219),(85,196),(86,197),(87,198),(88,193),(89,194),(90,195),(91,112),(92,113),(93,114),(94,109),(95,110),(96,111),(103,208),(104,209),(105,210),(106,205),(107,206),(108,207),(115,164),(116,165),(117,166),(118,167),(119,168),(120,163),(121,237),(122,238),(123,239),(124,240),(125,235),(126,236),(127,154),(128,155),(129,156),(130,151),(131,152),(132,153),(133,136),(134,137),(135,138),(139,203),(140,204),(141,199),(142,200),(143,201),(144,202),(145,228),(146,223),(147,224),(148,225),(149,226),(150,227),(169,178),(170,179),(171,180),(172,175),(173,176),(174,177)]])
102 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 5A | 5B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 24A | ··· | 24H | 30A | ··· | 30L | 30M | ··· | 30AB | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 20 | 20 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
102 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | D4 | D4 | D5 | D8 | D10 | D10 | C3×D4 | C3×D4 | C3×D5 | C5⋊D4 | C5⋊D4 | C3×D8 | C6×D5 | C6×D5 | C3×C5⋊D4 | C3×C5⋊D4 | D4⋊D5 | C3×D4⋊D5 |
| kernel | C6×D4⋊D5 | C6×C5⋊2C8 | C3×D4⋊D5 | C6×D20 | D4×C30 | C2×D4⋊D5 | C2×C5⋊2C8 | D4⋊D5 | C2×D20 | D4×C10 | C60 | C2×C30 | C6×D4 | C30 | C2×C12 | C3×D4 | C20 | C2×C10 | C2×D4 | C12 | C2×C6 | C10 | C2×C4 | D4 | C4 | C22 | C6 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 1 | 1 | 2 | 4 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 8 | 4 | 8 | 8 | 8 | 4 | 8 |
Matrix representation of C6×D4⋊D5 ►in GL4(𝔽241) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 192 | 0 | 0 |
| 123 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 57 | 0 | 0 |
| 148 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 51 | 240 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 123 | 240 | 0 | 0 |
| 0 | 0 | 190 | 1 |
| 0 | 0 | 51 | 51 |
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[1,123,0,0,192,240,0,0,0,0,1,0,0,0,0,1],[0,148,0,0,57,0,0,0,0,0,240,0,0,0,0,240],[1,0,0,0,0,1,0,0,0,0,51,1,0,0,240,0],[1,123,0,0,0,240,0,0,0,0,190,51,0,0,1,51] >;
C6×D4⋊D5 in GAP, Magma, Sage, TeX
C_6\times D_4\rtimes D_5
% in TeX
G:=Group("C6xD4:D5"); // GroupNames label
G:=SmallGroup(480,724);
// by ID
G=gap.SmallGroup(480,724);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,590,2524,648,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=c^2=d^5=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations