direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6×D4⋊2D5, C30.74C24, C60.209C23, D4⋊5(C6×D5), (D4×C10)⋊6C6, (C6×D4)⋊17D5, (C3×D4)⋊27D10, (D4×C30)⋊13C2, C30⋊15(C4○D4), Dic10⋊7(C2×C6), C10.6(C23×C6), C23.24(C6×D5), C6.74(C23×D5), (C6×Dic10)⋊28C2, (C2×Dic10)⋊12C6, (C2×C12).370D10, (D4×C15)⋊30C22, (D5×C12)⋊23C22, C20.20(C22×C6), (C6×D5).53C23, D10.2(C22×C6), (C22×C6).78D10, (C2×C60).305C22, (C2×C30).383C23, (C6×Dic5)⋊36C22, (C22×Dic5)⋊11C6, C12.209(C22×D5), Dic5.3(C22×C6), (C3×Dic10)⋊34C22, (C3×Dic5).55C23, (C22×C30).109C22, (C2×C4×D5)⋊4C6, C5⋊2(C6×C4○D4), C4.20(D5×C2×C6), (D5×C2×C12)⋊14C2, (C2×D4)⋊8(C3×D5), (C4×D5)⋊4(C2×C6), (C5×D4)⋊6(C2×C6), C10⋊2(C3×C4○D4), C15⋊24(C2×C4○D4), C5⋊D4⋊2(C2×C6), (C2×C5⋊D4)⋊10C6, (C6×C5⋊D4)⋊25C2, C22.1(D5×C2×C6), C2.7(D5×C22×C6), (C2×C6×Dic5)⋊19C2, (C2×C4).60(C6×D5), (C2×C20).42(C2×C6), (C2×Dic5)⋊9(C2×C6), (C3×C5⋊D4)⋊18C22, (C2×C10).1(C22×C6), (D5×C2×C6).139C22, (C2×C6).20(C22×D5), (C22×C10).28(C2×C6), (C22×D5).34(C2×C6), SmallGroup(480,1140)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×D4⋊2D5
G = < a,b,c,d,e | a6=b4=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >
Subgroups: 848 in 328 conjugacy classes, 178 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, D4, D4, Q8, C23, C23, D5, C10, C10, C10, C12, C12, C2×C6, C2×C6, C2×C6, C15, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×C6, C22×C6, C3×D5, C30, C30, C30, C2×C4○D4, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, C2×C30, C2×C30, C2×Dic10, C2×C4×D5, D4⋊2D5, C22×Dic5, C2×C5⋊D4, D4×C10, C6×C4○D4, C3×Dic10, D5×C12, C6×Dic5, C6×Dic5, C3×C5⋊D4, C2×C60, D4×C15, D5×C2×C6, C22×C30, C2×D4⋊2D5, C6×Dic10, D5×C2×C12, C3×D4⋊2D5, C2×C6×Dic5, C6×C5⋊D4, D4×C30, C6×D4⋊2D5
Quotients: C1, C2, C3, C22, C6, C23, D5, C2×C6, C4○D4, C24, D10, C22×C6, C3×D5, C2×C4○D4, C22×D5, C3×C4○D4, C23×C6, C6×D5, D4⋊2D5, C23×D5, C6×C4○D4, D5×C2×C6, C2×D4⋊2D5, C3×D4⋊2D5, D5×C22×C6, C6×D4⋊2D5
►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)
(1 137 77 115)(2 138 78 116)(3 133 73 117)(4 134 74 118)(5 135 75 119)(6 136 76 120)(7 216 221 161)(8 211 222 162)(9 212 217 157)(10 213 218 158)(11 214 219 159)(12 215 220 160)(13 165 105 89)(14 166 106 90)(15 167 107 85)(16 168 108 86)(17 163 103 87)(18 164 104 88)(19 182 230 170)(20 183 231 171)(21 184 232 172)(22 185 233 173)(23 186 234 174)(24 181 229 169)(25 143 83 71)(26 144 84 72)(27 139 79 67)(28 140 80 68)(29 141 81 69)(30 142 82 70)(31 224 208 148)(32 225 209 149)(33 226 210 150)(34 227 205 145)(35 228 206 146)(36 223 207 147)(37 202 190 130)(38 203 191 131)(39 204 192 132)(40 199 187 127)(41 200 188 128)(42 201 189 129)(43 156 96 101)(44 151 91 102)(45 152 92 97)(46 153 93 98)(47 154 94 99)(48 155 95 100)(49 198 236 176)(50 193 237 177)(51 194 238 178)(52 195 239 179)(53 196 240 180)(54 197 235 175)(55 125 65 113)(56 126 66 114)(57 121 61 109)(58 122 62 110)(59 123 63 111)(60 124 64 112)
(1 118)(2 119)(3 120)(4 115)(5 116)(6 117)(7 158)(8 159)(9 160)(10 161)(11 162)(12 157)(13 86)(14 87)(15 88)(16 89)(17 90)(18 85)(19 173)(20 174)(21 169)(22 170)(23 171)(24 172)(25 68)(26 69)(27 70)(28 71)(29 72)(30 67)(31 145)(32 146)(33 147)(34 148)(35 149)(36 150)(37 127)(38 128)(39 129)(40 130)(41 131)(42 132)(43 98)(44 99)(45 100)(46 101)(47 102)(48 97)(49 179)(50 180)(51 175)(52 176)(53 177)(54 178)(55 110)(56 111)(57 112)(58 113)(59 114)(60 109)(61 124)(62 125)(63 126)(64 121)(65 122)(66 123)(73 136)(74 137)(75 138)(76 133)(77 134)(78 135)(79 142)(80 143)(81 144)(82 139)(83 140)(84 141)(91 154)(92 155)(93 156)(94 151)(95 152)(96 153)(103 166)(104 167)(105 168)(106 163)(107 164)(108 165)(181 232)(182 233)(183 234)(184 229)(185 230)(186 231)(187 202)(188 203)(189 204)(190 199)(191 200)(192 201)(193 240)(194 235)(195 236)(196 237)(197 238)(198 239)(205 224)(206 225)(207 226)(208 227)(209 228)(210 223)(211 219)(212 220)(213 221)(214 222)(215 217)(216 218)
(1 93 17 65 29)(2 94 18 66 30)(3 95 13 61 25)(4 96 14 62 26)(5 91 15 63 27)(6 92 16 64 28)(7 238 192 19 210)(8 239 187 20 205)(9 240 188 21 206)(10 235 189 22 207)(11 236 190 23 208)(12 237 191 24 209)(31 219 49 37 234)(32 220 50 38 229)(33 221 51 39 230)(34 222 52 40 231)(35 217 53 41 232)(36 218 54 42 233)(43 106 58 84 74)(44 107 59 79 75)(45 108 60 80 76)(46 103 55 81 77)(47 104 56 82 78)(48 105 57 83 73)(67 119 151 85 123)(68 120 152 86 124)(69 115 153 87 125)(70 116 154 88 126)(71 117 155 89 121)(72 118 156 90 122)(97 168 112 140 136)(98 163 113 141 137)(99 164 114 142 138)(100 165 109 143 133)(101 166 110 144 134)(102 167 111 139 135)(127 183 145 211 179)(128 184 146 212 180)(129 185 147 213 175)(130 186 148 214 176)(131 181 149 215 177)(132 182 150 216 178)(157 196 200 172 228)(158 197 201 173 223)(159 198 202 174 224)(160 193 203 169 225)(161 194 204 170 226)(162 195 199 171 227)
(1 213)(2 214)(3 215)(4 216)(5 211)(6 212)(7 118)(8 119)(9 120)(10 115)(11 116)(12 117)(13 181)(14 182)(15 183)(16 184)(17 185)(18 186)(19 90)(20 85)(21 86)(22 87)(23 88)(24 89)(25 177)(26 178)(27 179)(28 180)(29 175)(30 176)(31 99)(32 100)(33 101)(34 102)(35 97)(36 98)(37 114)(38 109)(39 110)(40 111)(41 112)(42 113)(43 226)(44 227)(45 228)(46 223)(47 224)(48 225)(49 142)(50 143)(51 144)(52 139)(53 140)(54 141)(55 201)(56 202)(57 203)(58 204)(59 199)(60 200)(61 131)(62 132)(63 127)(64 128)(65 129)(66 130)(67 239)(68 240)(69 235)(70 236)(71 237)(72 238)(73 160)(74 161)(75 162)(76 157)(77 158)(78 159)(79 195)(80 196)(81 197)(82 198)(83 193)(84 194)(91 145)(92 146)(93 147)(94 148)(95 149)(96 150)(103 173)(104 174)(105 169)(106 170)(107 171)(108 172)(121 191)(122 192)(123 187)(124 188)(125 189)(126 190)(133 220)(134 221)(135 222)(136 217)(137 218)(138 219)(151 205)(152 206)(153 207)(154 208)(155 209)(156 210)(163 233)(164 234)(165 229)(166 230)(167 231)(168 232)
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), (1,137,77,115)(2,138,78,116)(3,133,73,117)(4,134,74,118)(5,135,75,119)(6,136,76,120)(7,216,221,161)(8,211,222,162)(9,212,217,157)(10,213,218,158)(11,214,219,159)(12,215,220,160)(13,165,105,89)(14,166,106,90)(15,167,107,85)(16,168,108,86)(17,163,103,87)(18,164,104,88)(19,182,230,170)(20,183,231,171)(21,184,232,172)(22,185,233,173)(23,186,234,174)(24,181,229,169)(25,143,83,71)(26,144,84,72)(27,139,79,67)(28,140,80,68)(29,141,81,69)(30,142,82,70)(31,224,208,148)(32,225,209,149)(33,226,210,150)(34,227,205,145)(35,228,206,146)(36,223,207,147)(37,202,190,130)(38,203,191,131)(39,204,192,132)(40,199,187,127)(41,200,188,128)(42,201,189,129)(43,156,96,101)(44,151,91,102)(45,152,92,97)(46,153,93,98)(47,154,94,99)(48,155,95,100)(49,198,236,176)(50,193,237,177)(51,194,238,178)(52,195,239,179)(53,196,240,180)(54,197,235,175)(55,125,65,113)(56,126,66,114)(57,121,61,109)(58,122,62,110)(59,123,63,111)(60,124,64,112), (1,118)(2,119)(3,120)(4,115)(5,116)(6,117)(7,158)(8,159)(9,160)(10,161)(11,162)(12,157)(13,86)(14,87)(15,88)(16,89)(17,90)(18,85)(19,173)(20,174)(21,169)(22,170)(23,171)(24,172)(25,68)(26,69)(27,70)(28,71)(29,72)(30,67)(31,145)(32,146)(33,147)(34,148)(35,149)(36,150)(37,127)(38,128)(39,129)(40,130)(41,131)(42,132)(43,98)(44,99)(45,100)(46,101)(47,102)(48,97)(49,179)(50,180)(51,175)(52,176)(53,177)(54,178)(55,110)(56,111)(57,112)(58,113)(59,114)(60,109)(61,124)(62,125)(63,126)(64,121)(65,122)(66,123)(73,136)(74,137)(75,138)(76,133)(77,134)(78,135)(79,142)(80,143)(81,144)(82,139)(83,140)(84,141)(91,154)(92,155)(93,156)(94,151)(95,152)(96,153)(103,166)(104,167)(105,168)(106,163)(107,164)(108,165)(181,232)(182,233)(183,234)(184,229)(185,230)(186,231)(187,202)(188,203)(189,204)(190,199)(191,200)(192,201)(193,240)(194,235)(195,236)(196,237)(197,238)(198,239)(205,224)(206,225)(207,226)(208,227)(209,228)(210,223)(211,219)(212,220)(213,221)(214,222)(215,217)(216,218), (1,93,17,65,29)(2,94,18,66,30)(3,95,13,61,25)(4,96,14,62,26)(5,91,15,63,27)(6,92,16,64,28)(7,238,192,19,210)(8,239,187,20,205)(9,240,188,21,206)(10,235,189,22,207)(11,236,190,23,208)(12,237,191,24,209)(31,219,49,37,234)(32,220,50,38,229)(33,221,51,39,230)(34,222,52,40,231)(35,217,53,41,232)(36,218,54,42,233)(43,106,58,84,74)(44,107,59,79,75)(45,108,60,80,76)(46,103,55,81,77)(47,104,56,82,78)(48,105,57,83,73)(67,119,151,85,123)(68,120,152,86,124)(69,115,153,87,125)(70,116,154,88,126)(71,117,155,89,121)(72,118,156,90,122)(97,168,112,140,136)(98,163,113,141,137)(99,164,114,142,138)(100,165,109,143,133)(101,166,110,144,134)(102,167,111,139,135)(127,183,145,211,179)(128,184,146,212,180)(129,185,147,213,175)(130,186,148,214,176)(131,181,149,215,177)(132,182,150,216,178)(157,196,200,172,228)(158,197,201,173,223)(159,198,202,174,224)(160,193,203,169,225)(161,194,204,170,226)(162,195,199,171,227), (1,213)(2,214)(3,215)(4,216)(5,211)(6,212)(7,118)(8,119)(9,120)(10,115)(11,116)(12,117)(13,181)(14,182)(15,183)(16,184)(17,185)(18,186)(19,90)(20,85)(21,86)(22,87)(23,88)(24,89)(25,177)(26,178)(27,179)(28,180)(29,175)(30,176)(31,99)(32,100)(33,101)(34,102)(35,97)(36,98)(37,114)(38,109)(39,110)(40,111)(41,112)(42,113)(43,226)(44,227)(45,228)(46,223)(47,224)(48,225)(49,142)(50,143)(51,144)(52,139)(53,140)(54,141)(55,201)(56,202)(57,203)(58,204)(59,199)(60,200)(61,131)(62,132)(63,127)(64,128)(65,129)(66,130)(67,239)(68,240)(69,235)(70,236)(71,237)(72,238)(73,160)(74,161)(75,162)(76,157)(77,158)(78,159)(79,195)(80,196)(81,197)(82,198)(83,193)(84,194)(91,145)(92,146)(93,147)(94,148)(95,149)(96,150)(103,173)(104,174)(105,169)(106,170)(107,171)(108,172)(121,191)(122,192)(123,187)(124,188)(125,189)(126,190)(133,220)(134,221)(135,222)(136,217)(137,218)(138,219)(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(163,233)(164,234)(165,229)(166,230)(167,231)(168,232)>;
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), (1,137,77,115)(2,138,78,116)(3,133,73,117)(4,134,74,118)(5,135,75,119)(6,136,76,120)(7,216,221,161)(8,211,222,162)(9,212,217,157)(10,213,218,158)(11,214,219,159)(12,215,220,160)(13,165,105,89)(14,166,106,90)(15,167,107,85)(16,168,108,86)(17,163,103,87)(18,164,104,88)(19,182,230,170)(20,183,231,171)(21,184,232,172)(22,185,233,173)(23,186,234,174)(24,181,229,169)(25,143,83,71)(26,144,84,72)(27,139,79,67)(28,140,80,68)(29,141,81,69)(30,142,82,70)(31,224,208,148)(32,225,209,149)(33,226,210,150)(34,227,205,145)(35,228,206,146)(36,223,207,147)(37,202,190,130)(38,203,191,131)(39,204,192,132)(40,199,187,127)(41,200,188,128)(42,201,189,129)(43,156,96,101)(44,151,91,102)(45,152,92,97)(46,153,93,98)(47,154,94,99)(48,155,95,100)(49,198,236,176)(50,193,237,177)(51,194,238,178)(52,195,239,179)(53,196,240,180)(54,197,235,175)(55,125,65,113)(56,126,66,114)(57,121,61,109)(58,122,62,110)(59,123,63,111)(60,124,64,112), (1,118)(2,119)(3,120)(4,115)(5,116)(6,117)(7,158)(8,159)(9,160)(10,161)(11,162)(12,157)(13,86)(14,87)(15,88)(16,89)(17,90)(18,85)(19,173)(20,174)(21,169)(22,170)(23,171)(24,172)(25,68)(26,69)(27,70)(28,71)(29,72)(30,67)(31,145)(32,146)(33,147)(34,148)(35,149)(36,150)(37,127)(38,128)(39,129)(40,130)(41,131)(42,132)(43,98)(44,99)(45,100)(46,101)(47,102)(48,97)(49,179)(50,180)(51,175)(52,176)(53,177)(54,178)(55,110)(56,111)(57,112)(58,113)(59,114)(60,109)(61,124)(62,125)(63,126)(64,121)(65,122)(66,123)(73,136)(74,137)(75,138)(76,133)(77,134)(78,135)(79,142)(80,143)(81,144)(82,139)(83,140)(84,141)(91,154)(92,155)(93,156)(94,151)(95,152)(96,153)(103,166)(104,167)(105,168)(106,163)(107,164)(108,165)(181,232)(182,233)(183,234)(184,229)(185,230)(186,231)(187,202)(188,203)(189,204)(190,199)(191,200)(192,201)(193,240)(194,235)(195,236)(196,237)(197,238)(198,239)(205,224)(206,225)(207,226)(208,227)(209,228)(210,223)(211,219)(212,220)(213,221)(214,222)(215,217)(216,218), (1,93,17,65,29)(2,94,18,66,30)(3,95,13,61,25)(4,96,14,62,26)(5,91,15,63,27)(6,92,16,64,28)(7,238,192,19,210)(8,239,187,20,205)(9,240,188,21,206)(10,235,189,22,207)(11,236,190,23,208)(12,237,191,24,209)(31,219,49,37,234)(32,220,50,38,229)(33,221,51,39,230)(34,222,52,40,231)(35,217,53,41,232)(36,218,54,42,233)(43,106,58,84,74)(44,107,59,79,75)(45,108,60,80,76)(46,103,55,81,77)(47,104,56,82,78)(48,105,57,83,73)(67,119,151,85,123)(68,120,152,86,124)(69,115,153,87,125)(70,116,154,88,126)(71,117,155,89,121)(72,118,156,90,122)(97,168,112,140,136)(98,163,113,141,137)(99,164,114,142,138)(100,165,109,143,133)(101,166,110,144,134)(102,167,111,139,135)(127,183,145,211,179)(128,184,146,212,180)(129,185,147,213,175)(130,186,148,214,176)(131,181,149,215,177)(132,182,150,216,178)(157,196,200,172,228)(158,197,201,173,223)(159,198,202,174,224)(160,193,203,169,225)(161,194,204,170,226)(162,195,199,171,227), (1,213)(2,214)(3,215)(4,216)(5,211)(6,212)(7,118)(8,119)(9,120)(10,115)(11,116)(12,117)(13,181)(14,182)(15,183)(16,184)(17,185)(18,186)(19,90)(20,85)(21,86)(22,87)(23,88)(24,89)(25,177)(26,178)(27,179)(28,180)(29,175)(30,176)(31,99)(32,100)(33,101)(34,102)(35,97)(36,98)(37,114)(38,109)(39,110)(40,111)(41,112)(42,113)(43,226)(44,227)(45,228)(46,223)(47,224)(48,225)(49,142)(50,143)(51,144)(52,139)(53,140)(54,141)(55,201)(56,202)(57,203)(58,204)(59,199)(60,200)(61,131)(62,132)(63,127)(64,128)(65,129)(66,130)(67,239)(68,240)(69,235)(70,236)(71,237)(72,238)(73,160)(74,161)(75,162)(76,157)(77,158)(78,159)(79,195)(80,196)(81,197)(82,198)(83,193)(84,194)(91,145)(92,146)(93,147)(94,148)(95,149)(96,150)(103,173)(104,174)(105,169)(106,170)(107,171)(108,172)(121,191)(122,192)(123,187)(124,188)(125,189)(126,190)(133,220)(134,221)(135,222)(136,217)(137,218)(138,219)(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(163,233)(164,234)(165,229)(166,230)(167,231)(168,232) );
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)], [(1,137,77,115),(2,138,78,116),(3,133,73,117),(4,134,74,118),(5,135,75,119),(6,136,76,120),(7,216,221,161),(8,211,222,162),(9,212,217,157),(10,213,218,158),(11,214,219,159),(12,215,220,160),(13,165,105,89),(14,166,106,90),(15,167,107,85),(16,168,108,86),(17,163,103,87),(18,164,104,88),(19,182,230,170),(20,183,231,171),(21,184,232,172),(22,185,233,173),(23,186,234,174),(24,181,229,169),(25,143,83,71),(26,144,84,72),(27,139,79,67),(28,140,80,68),(29,141,81,69),(30,142,82,70),(31,224,208,148),(32,225,209,149),(33,226,210,150),(34,227,205,145),(35,228,206,146),(36,223,207,147),(37,202,190,130),(38,203,191,131),(39,204,192,132),(40,199,187,127),(41,200,188,128),(42,201,189,129),(43,156,96,101),(44,151,91,102),(45,152,92,97),(46,153,93,98),(47,154,94,99),(48,155,95,100),(49,198,236,176),(50,193,237,177),(51,194,238,178),(52,195,239,179),(53,196,240,180),(54,197,235,175),(55,125,65,113),(56,126,66,114),(57,121,61,109),(58,122,62,110),(59,123,63,111),(60,124,64,112)], [(1,118),(2,119),(3,120),(4,115),(5,116),(6,117),(7,158),(8,159),(9,160),(10,161),(11,162),(12,157),(13,86),(14,87),(15,88),(16,89),(17,90),(18,85),(19,173),(20,174),(21,169),(22,170),(23,171),(24,172),(25,68),(26,69),(27,70),(28,71),(29,72),(30,67),(31,145),(32,146),(33,147),(34,148),(35,149),(36,150),(37,127),(38,128),(39,129),(40,130),(41,131),(42,132),(43,98),(44,99),(45,100),(46,101),(47,102),(48,97),(49,179),(50,180),(51,175),(52,176),(53,177),(54,178),(55,110),(56,111),(57,112),(58,113),(59,114),(60,109),(61,124),(62,125),(63,126),(64,121),(65,122),(66,123),(73,136),(74,137),(75,138),(76,133),(77,134),(78,135),(79,142),(80,143),(81,144),(82,139),(83,140),(84,141),(91,154),(92,155),(93,156),(94,151),(95,152),(96,153),(103,166),(104,167),(105,168),(106,163),(107,164),(108,165),(181,232),(182,233),(183,234),(184,229),(185,230),(186,231),(187,202),(188,203),(189,204),(190,199),(191,200),(192,201),(193,240),(194,235),(195,236),(196,237),(197,238),(198,239),(205,224),(206,225),(207,226),(208,227),(209,228),(210,223),(211,219),(212,220),(213,221),(214,222),(215,217),(216,218)], [(1,93,17,65,29),(2,94,18,66,30),(3,95,13,61,25),(4,96,14,62,26),(5,91,15,63,27),(6,92,16,64,28),(7,238,192,19,210),(8,239,187,20,205),(9,240,188,21,206),(10,235,189,22,207),(11,236,190,23,208),(12,237,191,24,209),(31,219,49,37,234),(32,220,50,38,229),(33,221,51,39,230),(34,222,52,40,231),(35,217,53,41,232),(36,218,54,42,233),(43,106,58,84,74),(44,107,59,79,75),(45,108,60,80,76),(46,103,55,81,77),(47,104,56,82,78),(48,105,57,83,73),(67,119,151,85,123),(68,120,152,86,124),(69,115,153,87,125),(70,116,154,88,126),(71,117,155,89,121),(72,118,156,90,122),(97,168,112,140,136),(98,163,113,141,137),(99,164,114,142,138),(100,165,109,143,133),(101,166,110,144,134),(102,167,111,139,135),(127,183,145,211,179),(128,184,146,212,180),(129,185,147,213,175),(130,186,148,214,176),(131,181,149,215,177),(132,182,150,216,178),(157,196,200,172,228),(158,197,201,173,223),(159,198,202,174,224),(160,193,203,169,225),(161,194,204,170,226),(162,195,199,171,227)], [(1,213),(2,214),(3,215),(4,216),(5,211),(6,212),(7,118),(8,119),(9,120),(10,115),(11,116),(12,117),(13,181),(14,182),(15,183),(16,184),(17,185),(18,186),(19,90),(20,85),(21,86),(22,87),(23,88),(24,89),(25,177),(26,178),(27,179),(28,180),(29,175),(30,176),(31,99),(32,100),(33,101),(34,102),(35,97),(36,98),(37,114),(38,109),(39,110),(40,111),(41,112),(42,113),(43,226),(44,227),(45,228),(46,223),(47,224),(48,225),(49,142),(50,143),(51,144),(52,139),(53,140),(54,141),(55,201),(56,202),(57,203),(58,204),(59,199),(60,200),(61,131),(62,132),(63,127),(64,128),(65,129),(66,130),(67,239),(68,240),(69,235),(70,236),(71,237),(72,238),(73,160),(74,161),(75,162),(76,157),(77,158),(78,159),(79,195),(80,196),(81,197),(82,198),(83,193),(84,194),(91,145),(92,146),(93,147),(94,148),(95,149),(96,150),(103,173),(104,174),(105,169),(106,170),(107,171),(108,172),(121,191),(122,192),(123,187),(124,188),(125,189),(126,190),(133,220),(134,221),(135,222),(136,217),(137,218),(138,219),(151,205),(152,206),(153,207),(154,208),(155,209),(156,210),(163,233),(164,234),(165,229),(166,230),(167,231),(168,232)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | ··· | 6F | 6G | ··· | 6N | 6O | 6P | 6Q | 6R | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 12M | ··· | 12T | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 30A | ··· | 30L | 30M | ··· | 30AB | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 10 | 10 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 5 | ··· | 5 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | D5 | C4○D4 | D10 | D10 | D10 | C3×D5 | C3×C4○D4 | C6×D5 | C6×D5 | C6×D5 | D4⋊2D5 | C3×D4⋊2D5 |
| kernel | C6×D4⋊2D5 | C6×Dic10 | D5×C2×C12 | C3×D4⋊2D5 | C2×C6×Dic5 | C6×C5⋊D4 | D4×C30 | C2×D4⋊2D5 | C2×Dic10 | C2×C4×D5 | D4⋊2D5 | C22×Dic5 | C2×C5⋊D4 | D4×C10 | C6×D4 | C30 | C2×C12 | C3×D4 | C22×C6 | C2×D4 | C10 | C2×C4 | D4 | C23 | C6 | C2 |
| # reps | 1 | 1 | 1 | 8 | 2 | 2 | 1 | 2 | 2 | 2 | 16 | 4 | 4 | 2 | 2 | 4 | 2 | 8 | 4 | 4 | 8 | 4 | 16 | 8 | 4 | 8 |
Matrix representation of C6×D4⋊2D5 ►in GL5(𝔽61)
| 60 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 14 | 0 |
| 0 | 0 | 0 | 0 | 14 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 59 |
| 0 | 0 | 0 | 1 | 60 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 60 | 2 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 17 | 60 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 55 | 53 | 0 | 0 |
| 0 | 12 | 6 | 0 | 0 |
| 0 | 0 | 0 | 50 | 22 |
| 0 | 0 | 0 | 50 | 11 |
G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,13,0,0,0,0,0,13,0,0,0,0,0,14,0,0,0,0,0,14],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,59,60],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,2,1],[1,0,0,0,0,0,17,1,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,0,55,12,0,0,0,53,6,0,0,0,0,0,50,50,0,0,0,22,11] >;
C6×D4⋊2D5 in GAP, Magma, Sage, TeX
C_6\times D_4\rtimes_2D_5
% in TeX
G:=Group("C6xD4:2D5"); // GroupNames label
G:=SmallGroup(480,1140);
// by ID
G=gap.SmallGroup(480,1140);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-5,268,1571,409,18822]);
// 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=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations