direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×D4.D6, C40.38D6, Dic12⋊6C10, C120.70C22, C60.222C23, C8.2(S3×C10), (S3×Q8)⋊2C10, D6.8(C5×D4), C8⋊S3⋊2C10, C24.9(C2×C10), D4⋊2S3.C10, D4.S3⋊4C10, SD16⋊2(C5×S3), (C5×SD16)⋊6S3, D4.4(S3×C10), C3⋊Q16⋊1C10, (C5×D4).28D6, C6.32(D4×C10), Q8.6(S3×C10), (C5×Q8).44D6, (C3×SD16)⋊2C10, (C15×SD16)⋊8C2, (S3×C10).44D4, C10.186(S3×D4), C30.368(C2×D4), (C5×Dic12)⋊14C2, C15⋊30(C8.C22), C12.6(C22×C10), Dic6.2(C2×C10), (C5×Dic3).47D4, Dic3.10(C5×D4), (S3×C20).38C22, C20.195(C22×S3), (D4×C15).33C22, (Q8×C15).32C22, (C5×Dic6).33C22, (C5×S3×Q8)⋊9C2, C4.6(S3×C2×C10), C2.20(C5×S3×D4), C3⋊C8.1(C2×C10), C3⋊2(C5×C8.C22), (C5×C3⋊Q16)⋊9C2, (C5×C8⋊S3)⋊10C2, (C4×S3).3(C2×C10), (C5×D4.S3)⋊12C2, (C3×D4).4(C2×C10), (C5×C3⋊C8).27C22, (C3×Q8).1(C2×C10), (C5×D4⋊2S3).2C2, SmallGroup(480,794)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D4.D6
G = < a,b,c,d,e | a5=b4=c2=1, d6=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=bc, ece-1=b-1c, ede-1=d5 >
Subgroups: 276 in 120 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, Q8, C10, C10, Dic3, Dic3, C12, C12, D6, C2×C6, C15, M4(2), SD16, SD16, Q16, C2×Q8, C4○D4, C20, C20, C2×C10, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C5×S3, C30, C30, C8.C22, C40, C40, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C8⋊S3, Dic12, D4.S3, C3⋊Q16, C3×SD16, D4⋊2S3, S3×Q8, C5×Dic3, C5×Dic3, C60, C60, S3×C10, C2×C30, C5×M4(2), C5×SD16, C5×SD16, C5×Q16, Q8×C10, C5×C4○D4, D4.D6, C5×C3⋊C8, C120, C5×Dic6, C5×Dic6, S3×C20, S3×C20, C10×Dic3, C5×C3⋊D4, D4×C15, Q8×C15, C5×C8.C22, C5×C8⋊S3, C5×Dic12, C5×D4.S3, C5×C3⋊Q16, C15×SD16, C5×D4⋊2S3, C5×S3×Q8, C5×D4.D6
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C2×C10, C22×S3, C5×S3, C8.C22, C5×D4, C22×C10, S3×D4, S3×C10, D4×C10, D4.D6, S3×C2×C10, C5×C8.C22, C5×S3×D4, C5×D4.D6
►On 240 points
Generators in S240
(1 218 26 50 202)(2 219 27 51 203)(3 220 28 52 204)(4 221 29 53 193)(5 222 30 54 194)(6 223 31 55 195)(7 224 32 56 196)(8 225 33 57 197)(9 226 34 58 198)(10 227 35 59 199)(11 228 36 60 200)(12 217 25 49 201)(13 109 78 98 42)(14 110 79 99 43)(15 111 80 100 44)(16 112 81 101 45)(17 113 82 102 46)(18 114 83 103 47)(19 115 84 104 48)(20 116 73 105 37)(21 117 74 106 38)(22 118 75 107 39)(23 119 76 108 40)(24 120 77 97 41)(61 230 186 92 166)(62 231 187 93 167)(63 232 188 94 168)(64 233 189 95 157)(65 234 190 96 158)(66 235 191 85 159)(67 236 192 86 160)(68 237 181 87 161)(69 238 182 88 162)(70 239 183 89 163)(71 240 184 90 164)(72 229 185 91 165)(121 170 135 208 150)(122 171 136 209 151)(123 172 137 210 152)(124 173 138 211 153)(125 174 139 212 154)(126 175 140 213 155)(127 176 141 214 156)(128 177 142 215 145)(129 178 143 216 146)(130 179 144 205 147)(131 180 133 206 148)(132 169 134 207 149)
(1 122 7 128)(2 129 8 123)(3 124 9 130)(4 131 10 125)(5 126 11 132)(6 121 12 127)(13 89 19 95)(14 96 20 90)(15 91 21 85)(16 86 22 92)(17 93 23 87)(18 88 24 94)(25 141 31 135)(26 136 32 142)(27 143 33 137)(28 138 34 144)(29 133 35 139)(30 140 36 134)(37 184 43 190)(38 191 44 185)(39 186 45 192)(40 181 46 187)(41 188 47 182)(42 183 48 189)(49 214 55 208)(50 209 56 215)(51 216 57 210)(52 211 58 205)(53 206 59 212)(54 213 60 207)(61 81 67 75)(62 76 68 82)(63 83 69 77)(64 78 70 84)(65 73 71 79)(66 80 72 74)(97 232 103 238)(98 239 104 233)(99 234 105 240)(100 229 106 235)(101 236 107 230)(102 231 108 237)(109 163 115 157)(110 158 116 164)(111 165 117 159)(112 160 118 166)(113 167 119 161)(114 162 120 168)(145 202 151 196)(146 197 152 203)(147 204 153 198)(148 199 154 193)(149 194 155 200)(150 201 156 195)(169 222 175 228)(170 217 176 223)(171 224 177 218)(172 219 178 225)(173 226 179 220)(174 221 180 227)
(1 118)(2 161)(3 120)(4 163)(5 110)(6 165)(7 112)(8 167)(9 114)(10 157)(11 116)(12 159)(13 148)(14 194)(15 150)(16 196)(17 152)(18 198)(19 154)(20 200)(21 156)(22 202)(23 146)(24 204)(25 235)(26 107)(27 237)(28 97)(29 239)(30 99)(31 229)(32 101)(33 231)(34 103)(35 233)(36 105)(37 60)(38 214)(39 50)(40 216)(41 52)(42 206)(43 54)(44 208)(45 56)(46 210)(47 58)(48 212)(49 191)(51 181)(53 183)(55 185)(57 187)(59 189)(61 177)(62 225)(63 179)(64 227)(65 169)(66 217)(67 171)(68 219)(69 173)(70 221)(71 175)(72 223)(73 228)(74 176)(75 218)(76 178)(77 220)(78 180)(79 222)(80 170)(81 224)(82 172)(83 226)(84 174)(85 201)(86 151)(87 203)(88 153)(89 193)(90 155)(91 195)(92 145)(93 197)(94 147)(95 199)(96 149)(98 133)(100 135)(102 137)(104 139)(106 141)(108 143)(109 131)(111 121)(113 123)(115 125)(117 127)(119 129)(122 160)(124 162)(126 164)(128 166)(130 168)(132 158)(134 234)(136 236)(138 238)(140 240)(142 230)(144 232)(182 211)(184 213)(186 215)(188 205)(190 207)(192 209)
(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 6 7 12)(2 11 8 5)(3 4 9 10)(13 24 19 18)(14 17 20 23)(15 22 21 16)(25 26 31 32)(27 36 33 30)(28 29 34 35)(37 40 43 46)(38 45 44 39)(41 48 47 42)(49 50 55 56)(51 60 57 54)(52 53 58 59)(61 72 67 66)(62 65 68 71)(63 70 69 64)(73 76 79 82)(74 81 80 75)(77 84 83 78)(85 92 91 86)(87 90 93 96)(88 95 94 89)(97 104 103 98)(99 102 105 108)(100 107 106 101)(109 120 115 114)(110 113 116 119)(111 118 117 112)(121 122 127 128)(123 132 129 126)(124 125 130 131)(133 138 139 144)(134 143 140 137)(135 136 141 142)(145 150 151 156)(146 155 152 149)(147 148 153 154)(157 168 163 162)(158 161 164 167)(159 166 165 160)(169 178 175 172)(170 171 176 177)(173 174 179 180)(181 184 187 190)(182 189 188 183)(185 192 191 186)(193 198 199 204)(194 203 200 197)(195 196 201 202)(205 206 211 212)(207 216 213 210)(208 209 214 215)(217 218 223 224)(219 228 225 222)(220 221 226 227)(229 236 235 230)(231 234 237 240)(232 239 238 233)
G:=sub<Sym(240)| (1,218,26,50,202)(2,219,27,51,203)(3,220,28,52,204)(4,221,29,53,193)(5,222,30,54,194)(6,223,31,55,195)(7,224,32,56,196)(8,225,33,57,197)(9,226,34,58,198)(10,227,35,59,199)(11,228,36,60,200)(12,217,25,49,201)(13,109,78,98,42)(14,110,79,99,43)(15,111,80,100,44)(16,112,81,101,45)(17,113,82,102,46)(18,114,83,103,47)(19,115,84,104,48)(20,116,73,105,37)(21,117,74,106,38)(22,118,75,107,39)(23,119,76,108,40)(24,120,77,97,41)(61,230,186,92,166)(62,231,187,93,167)(63,232,188,94,168)(64,233,189,95,157)(65,234,190,96,158)(66,235,191,85,159)(67,236,192,86,160)(68,237,181,87,161)(69,238,182,88,162)(70,239,183,89,163)(71,240,184,90,164)(72,229,185,91,165)(121,170,135,208,150)(122,171,136,209,151)(123,172,137,210,152)(124,173,138,211,153)(125,174,139,212,154)(126,175,140,213,155)(127,176,141,214,156)(128,177,142,215,145)(129,178,143,216,146)(130,179,144,205,147)(131,180,133,206,148)(132,169,134,207,149), (1,122,7,128)(2,129,8,123)(3,124,9,130)(4,131,10,125)(5,126,11,132)(6,121,12,127)(13,89,19,95)(14,96,20,90)(15,91,21,85)(16,86,22,92)(17,93,23,87)(18,88,24,94)(25,141,31,135)(26,136,32,142)(27,143,33,137)(28,138,34,144)(29,133,35,139)(30,140,36,134)(37,184,43,190)(38,191,44,185)(39,186,45,192)(40,181,46,187)(41,188,47,182)(42,183,48,189)(49,214,55,208)(50,209,56,215)(51,216,57,210)(52,211,58,205)(53,206,59,212)(54,213,60,207)(61,81,67,75)(62,76,68,82)(63,83,69,77)(64,78,70,84)(65,73,71,79)(66,80,72,74)(97,232,103,238)(98,239,104,233)(99,234,105,240)(100,229,106,235)(101,236,107,230)(102,231,108,237)(109,163,115,157)(110,158,116,164)(111,165,117,159)(112,160,118,166)(113,167,119,161)(114,162,120,168)(145,202,151,196)(146,197,152,203)(147,204,153,198)(148,199,154,193)(149,194,155,200)(150,201,156,195)(169,222,175,228)(170,217,176,223)(171,224,177,218)(172,219,178,225)(173,226,179,220)(174,221,180,227), (1,118)(2,161)(3,120)(4,163)(5,110)(6,165)(7,112)(8,167)(9,114)(10,157)(11,116)(12,159)(13,148)(14,194)(15,150)(16,196)(17,152)(18,198)(19,154)(20,200)(21,156)(22,202)(23,146)(24,204)(25,235)(26,107)(27,237)(28,97)(29,239)(30,99)(31,229)(32,101)(33,231)(34,103)(35,233)(36,105)(37,60)(38,214)(39,50)(40,216)(41,52)(42,206)(43,54)(44,208)(45,56)(46,210)(47,58)(48,212)(49,191)(51,181)(53,183)(55,185)(57,187)(59,189)(61,177)(62,225)(63,179)(64,227)(65,169)(66,217)(67,171)(68,219)(69,173)(70,221)(71,175)(72,223)(73,228)(74,176)(75,218)(76,178)(77,220)(78,180)(79,222)(80,170)(81,224)(82,172)(83,226)(84,174)(85,201)(86,151)(87,203)(88,153)(89,193)(90,155)(91,195)(92,145)(93,197)(94,147)(95,199)(96,149)(98,133)(100,135)(102,137)(104,139)(106,141)(108,143)(109,131)(111,121)(113,123)(115,125)(117,127)(119,129)(122,160)(124,162)(126,164)(128,166)(130,168)(132,158)(134,234)(136,236)(138,238)(140,240)(142,230)(144,232)(182,211)(184,213)(186,215)(188,205)(190,207)(192,209), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,24,19,18)(14,17,20,23)(15,22,21,16)(25,26,31,32)(27,36,33,30)(28,29,34,35)(37,40,43,46)(38,45,44,39)(41,48,47,42)(49,50,55,56)(51,60,57,54)(52,53,58,59)(61,72,67,66)(62,65,68,71)(63,70,69,64)(73,76,79,82)(74,81,80,75)(77,84,83,78)(85,92,91,86)(87,90,93,96)(88,95,94,89)(97,104,103,98)(99,102,105,108)(100,107,106,101)(109,120,115,114)(110,113,116,119)(111,118,117,112)(121,122,127,128)(123,132,129,126)(124,125,130,131)(133,138,139,144)(134,143,140,137)(135,136,141,142)(145,150,151,156)(146,155,152,149)(147,148,153,154)(157,168,163,162)(158,161,164,167)(159,166,165,160)(169,178,175,172)(170,171,176,177)(173,174,179,180)(181,184,187,190)(182,189,188,183)(185,192,191,186)(193,198,199,204)(194,203,200,197)(195,196,201,202)(205,206,211,212)(207,216,213,210)(208,209,214,215)(217,218,223,224)(219,228,225,222)(220,221,226,227)(229,236,235,230)(231,234,237,240)(232,239,238,233)>;
G:=Group( (1,218,26,50,202)(2,219,27,51,203)(3,220,28,52,204)(4,221,29,53,193)(5,222,30,54,194)(6,223,31,55,195)(7,224,32,56,196)(8,225,33,57,197)(9,226,34,58,198)(10,227,35,59,199)(11,228,36,60,200)(12,217,25,49,201)(13,109,78,98,42)(14,110,79,99,43)(15,111,80,100,44)(16,112,81,101,45)(17,113,82,102,46)(18,114,83,103,47)(19,115,84,104,48)(20,116,73,105,37)(21,117,74,106,38)(22,118,75,107,39)(23,119,76,108,40)(24,120,77,97,41)(61,230,186,92,166)(62,231,187,93,167)(63,232,188,94,168)(64,233,189,95,157)(65,234,190,96,158)(66,235,191,85,159)(67,236,192,86,160)(68,237,181,87,161)(69,238,182,88,162)(70,239,183,89,163)(71,240,184,90,164)(72,229,185,91,165)(121,170,135,208,150)(122,171,136,209,151)(123,172,137,210,152)(124,173,138,211,153)(125,174,139,212,154)(126,175,140,213,155)(127,176,141,214,156)(128,177,142,215,145)(129,178,143,216,146)(130,179,144,205,147)(131,180,133,206,148)(132,169,134,207,149), (1,122,7,128)(2,129,8,123)(3,124,9,130)(4,131,10,125)(5,126,11,132)(6,121,12,127)(13,89,19,95)(14,96,20,90)(15,91,21,85)(16,86,22,92)(17,93,23,87)(18,88,24,94)(25,141,31,135)(26,136,32,142)(27,143,33,137)(28,138,34,144)(29,133,35,139)(30,140,36,134)(37,184,43,190)(38,191,44,185)(39,186,45,192)(40,181,46,187)(41,188,47,182)(42,183,48,189)(49,214,55,208)(50,209,56,215)(51,216,57,210)(52,211,58,205)(53,206,59,212)(54,213,60,207)(61,81,67,75)(62,76,68,82)(63,83,69,77)(64,78,70,84)(65,73,71,79)(66,80,72,74)(97,232,103,238)(98,239,104,233)(99,234,105,240)(100,229,106,235)(101,236,107,230)(102,231,108,237)(109,163,115,157)(110,158,116,164)(111,165,117,159)(112,160,118,166)(113,167,119,161)(114,162,120,168)(145,202,151,196)(146,197,152,203)(147,204,153,198)(148,199,154,193)(149,194,155,200)(150,201,156,195)(169,222,175,228)(170,217,176,223)(171,224,177,218)(172,219,178,225)(173,226,179,220)(174,221,180,227), (1,118)(2,161)(3,120)(4,163)(5,110)(6,165)(7,112)(8,167)(9,114)(10,157)(11,116)(12,159)(13,148)(14,194)(15,150)(16,196)(17,152)(18,198)(19,154)(20,200)(21,156)(22,202)(23,146)(24,204)(25,235)(26,107)(27,237)(28,97)(29,239)(30,99)(31,229)(32,101)(33,231)(34,103)(35,233)(36,105)(37,60)(38,214)(39,50)(40,216)(41,52)(42,206)(43,54)(44,208)(45,56)(46,210)(47,58)(48,212)(49,191)(51,181)(53,183)(55,185)(57,187)(59,189)(61,177)(62,225)(63,179)(64,227)(65,169)(66,217)(67,171)(68,219)(69,173)(70,221)(71,175)(72,223)(73,228)(74,176)(75,218)(76,178)(77,220)(78,180)(79,222)(80,170)(81,224)(82,172)(83,226)(84,174)(85,201)(86,151)(87,203)(88,153)(89,193)(90,155)(91,195)(92,145)(93,197)(94,147)(95,199)(96,149)(98,133)(100,135)(102,137)(104,139)(106,141)(108,143)(109,131)(111,121)(113,123)(115,125)(117,127)(119,129)(122,160)(124,162)(126,164)(128,166)(130,168)(132,158)(134,234)(136,236)(138,238)(140,240)(142,230)(144,232)(182,211)(184,213)(186,215)(188,205)(190,207)(192,209), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,24,19,18)(14,17,20,23)(15,22,21,16)(25,26,31,32)(27,36,33,30)(28,29,34,35)(37,40,43,46)(38,45,44,39)(41,48,47,42)(49,50,55,56)(51,60,57,54)(52,53,58,59)(61,72,67,66)(62,65,68,71)(63,70,69,64)(73,76,79,82)(74,81,80,75)(77,84,83,78)(85,92,91,86)(87,90,93,96)(88,95,94,89)(97,104,103,98)(99,102,105,108)(100,107,106,101)(109,120,115,114)(110,113,116,119)(111,118,117,112)(121,122,127,128)(123,132,129,126)(124,125,130,131)(133,138,139,144)(134,143,140,137)(135,136,141,142)(145,150,151,156)(146,155,152,149)(147,148,153,154)(157,168,163,162)(158,161,164,167)(159,166,165,160)(169,178,175,172)(170,171,176,177)(173,174,179,180)(181,184,187,190)(182,189,188,183)(185,192,191,186)(193,198,199,204)(194,203,200,197)(195,196,201,202)(205,206,211,212)(207,216,213,210)(208,209,214,215)(217,218,223,224)(219,228,225,222)(220,221,226,227)(229,236,235,230)(231,234,237,240)(232,239,238,233) );
G=PermutationGroup([[(1,218,26,50,202),(2,219,27,51,203),(3,220,28,52,204),(4,221,29,53,193),(5,222,30,54,194),(6,223,31,55,195),(7,224,32,56,196),(8,225,33,57,197),(9,226,34,58,198),(10,227,35,59,199),(11,228,36,60,200),(12,217,25,49,201),(13,109,78,98,42),(14,110,79,99,43),(15,111,80,100,44),(16,112,81,101,45),(17,113,82,102,46),(18,114,83,103,47),(19,115,84,104,48),(20,116,73,105,37),(21,117,74,106,38),(22,118,75,107,39),(23,119,76,108,40),(24,120,77,97,41),(61,230,186,92,166),(62,231,187,93,167),(63,232,188,94,168),(64,233,189,95,157),(65,234,190,96,158),(66,235,191,85,159),(67,236,192,86,160),(68,237,181,87,161),(69,238,182,88,162),(70,239,183,89,163),(71,240,184,90,164),(72,229,185,91,165),(121,170,135,208,150),(122,171,136,209,151),(123,172,137,210,152),(124,173,138,211,153),(125,174,139,212,154),(126,175,140,213,155),(127,176,141,214,156),(128,177,142,215,145),(129,178,143,216,146),(130,179,144,205,147),(131,180,133,206,148),(132,169,134,207,149)], [(1,122,7,128),(2,129,8,123),(3,124,9,130),(4,131,10,125),(5,126,11,132),(6,121,12,127),(13,89,19,95),(14,96,20,90),(15,91,21,85),(16,86,22,92),(17,93,23,87),(18,88,24,94),(25,141,31,135),(26,136,32,142),(27,143,33,137),(28,138,34,144),(29,133,35,139),(30,140,36,134),(37,184,43,190),(38,191,44,185),(39,186,45,192),(40,181,46,187),(41,188,47,182),(42,183,48,189),(49,214,55,208),(50,209,56,215),(51,216,57,210),(52,211,58,205),(53,206,59,212),(54,213,60,207),(61,81,67,75),(62,76,68,82),(63,83,69,77),(64,78,70,84),(65,73,71,79),(66,80,72,74),(97,232,103,238),(98,239,104,233),(99,234,105,240),(100,229,106,235),(101,236,107,230),(102,231,108,237),(109,163,115,157),(110,158,116,164),(111,165,117,159),(112,160,118,166),(113,167,119,161),(114,162,120,168),(145,202,151,196),(146,197,152,203),(147,204,153,198),(148,199,154,193),(149,194,155,200),(150,201,156,195),(169,222,175,228),(170,217,176,223),(171,224,177,218),(172,219,178,225),(173,226,179,220),(174,221,180,227)], [(1,118),(2,161),(3,120),(4,163),(5,110),(6,165),(7,112),(8,167),(9,114),(10,157),(11,116),(12,159),(13,148),(14,194),(15,150),(16,196),(17,152),(18,198),(19,154),(20,200),(21,156),(22,202),(23,146),(24,204),(25,235),(26,107),(27,237),(28,97),(29,239),(30,99),(31,229),(32,101),(33,231),(34,103),(35,233),(36,105),(37,60),(38,214),(39,50),(40,216),(41,52),(42,206),(43,54),(44,208),(45,56),(46,210),(47,58),(48,212),(49,191),(51,181),(53,183),(55,185),(57,187),(59,189),(61,177),(62,225),(63,179),(64,227),(65,169),(66,217),(67,171),(68,219),(69,173),(70,221),(71,175),(72,223),(73,228),(74,176),(75,218),(76,178),(77,220),(78,180),(79,222),(80,170),(81,224),(82,172),(83,226),(84,174),(85,201),(86,151),(87,203),(88,153),(89,193),(90,155),(91,195),(92,145),(93,197),(94,147),(95,199),(96,149),(98,133),(100,135),(102,137),(104,139),(106,141),(108,143),(109,131),(111,121),(113,123),(115,125),(117,127),(119,129),(122,160),(124,162),(126,164),(128,166),(130,168),(132,158),(134,234),(136,236),(138,238),(140,240),(142,230),(144,232),(182,211),(184,213),(186,215),(188,205),(190,207),(192,209)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,24,19,18),(14,17,20,23),(15,22,21,16),(25,26,31,32),(27,36,33,30),(28,29,34,35),(37,40,43,46),(38,45,44,39),(41,48,47,42),(49,50,55,56),(51,60,57,54),(52,53,58,59),(61,72,67,66),(62,65,68,71),(63,70,69,64),(73,76,79,82),(74,81,80,75),(77,84,83,78),(85,92,91,86),(87,90,93,96),(88,95,94,89),(97,104,103,98),(99,102,105,108),(100,107,106,101),(109,120,115,114),(110,113,116,119),(111,118,117,112),(121,122,127,128),(123,132,129,126),(124,125,130,131),(133,138,139,144),(134,143,140,137),(135,136,141,142),(145,150,151,156),(146,155,152,149),(147,148,153,154),(157,168,163,162),(158,161,164,167),(159,166,165,160),(169,178,175,172),(170,171,176,177),(173,174,179,180),(181,184,187,190),(182,189,188,183),(185,192,191,186),(193,198,199,204),(194,203,200,197),(195,196,201,202),(205,206,211,212),(207,216,213,210),(208,209,214,215),(217,218,223,224),(219,228,225,222),(220,221,226,227),(229,236,235,230),(231,234,237,240),(232,239,238,233)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 6A | 6B | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 20K | 20L | 20M | ··· | 20T | 24A | 24B | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | 40A | 40B | 40C | 40D | 40E | 40F | 40G | 40H | 60A | 60B | 60C | 60D | 60E | 60F | 60G | 60H | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | 40 | 40 | 40 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 4 | 6 | 2 | 2 | 4 | 6 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 8 | 4 | 12 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 4 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 4 | 4 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | S3 | D4 | D4 | D6 | D6 | D6 | C5×S3 | C5×D4 | C5×D4 | S3×C10 | S3×C10 | S3×C10 | C8.C22 | S3×D4 | D4.D6 | C5×C8.C22 | C5×S3×D4 | C5×D4.D6 |
| kernel | C5×D4.D6 | C5×C8⋊S3 | C5×Dic12 | C5×D4.S3 | C5×C3⋊Q16 | C15×SD16 | C5×D4⋊2S3 | C5×S3×Q8 | D4.D6 | C8⋊S3 | Dic12 | D4.S3 | C3⋊Q16 | C3×SD16 | D4⋊2S3 | S3×Q8 | C5×SD16 | C5×Dic3 | S3×C10 | C40 | C5×D4 | C5×Q8 | SD16 | Dic3 | D6 | C8 | D4 | Q8 | C15 | C10 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | 4 | 4 | 8 |
Matrix representation of C5×D4.D6 ►in GL4(𝔽241) generated by
| 98 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
| 0 | 0 | 240 | 1 |
| 240 | 240 | 239 | 240 |
| 81 | 81 | 1 | 0 |
| 80 | 81 | 1 | 0 |
| 87 | 213 | 73 | 73 |
| 101 | 227 | 0 | 73 |
| 112 | 157 | 14 | 28 |
| 140 | 112 | 140 | 154 |
| 170 | 66 | 118 | 236 |
| 52 | 0 | 123 | 118 |
| 186 | 115 | 0 | 175 |
| 11 | 186 | 189 | 71 |
| 52 | 0 | 123 | 118 |
| 170 | 66 | 118 | 236 |
| 186 | 115 | 0 | 175 |
| 63 | 115 | 52 | 123 |
G:=sub<GL(4,GF(241))| [98,0,0,0,0,98,0,0,0,0,98,0,0,0,0,98],[0,240,81,80,0,240,81,81,240,239,1,1,1,240,0,0],[87,101,112,140,213,227,157,112,73,0,14,140,73,73,28,154],[170,52,186,11,66,0,115,186,118,123,0,189,236,118,175,71],[52,170,186,63,0,66,115,115,123,118,0,52,118,236,175,123] >;
C5×D4.D6 in GAP, Magma, Sage, TeX
C_5\times D_4.D_6
% in TeX
G:=Group("C5xD4.D6"); // GroupNames label
G:=SmallGroup(480,794);
// by ID
G=gap.SmallGroup(480,794);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,568,1766,471,2111,1068,102,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=c^2=1,d^6=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=b*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^5>;
// generators/relations