direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C8○D12, C40.84D6, D12.2C20, Dic6.2C20, C60.285C23, C120.111C22, (S3×C8)⋊6C10, (C2×C40)⋊15S3, (S3×C40)⋊15C2, C8⋊S3⋊7C10, (C2×C24)⋊12C10, (C2×C120)⋊28C2, C15⋊23(C8○D4), C4.10(S3×C20), C8.22(S3×C10), C20.99(C4×S3), D6.1(C2×C20), (C5×D12).6C4, C3⋊D4.2C20, C12.20(C2×C20), C60.207(C2×C4), C24.27(C2×C10), C4○D12.6C10, (C2×C20).431D6, C22.2(S3×C20), (C5×Dic6).6C4, C4.Dic3⋊11C10, C6.14(C22×C20), Dic3.3(C2×C20), (S3×C20).64C22, C30.205(C22×C4), C20.243(C22×S3), (C2×C60).536C22, C12.37(C22×C10), C3⋊1(C5×C8○D4), (C2×C8)⋊7(C5×S3), C2.15(S3×C2×C20), C4.37(S3×C2×C10), C3⋊C8.11(C2×C10), C10.141(S3×C2×C4), (C5×C3⋊D4).6C4, (C5×C8⋊S3)⋊15C2, (C2×C6).16(C2×C20), (C2×C10).50(C4×S3), (C2×C4).79(S3×C10), (C5×C3⋊C8).47C22, (S3×C10).31(C2×C4), (C4×S3).15(C2×C10), (C2×C30).161(C2×C4), (C5×C4○D12).12C2, (C5×C4.Dic3)⋊23C2, (C2×C12).103(C2×C10), (C5×Dic3).39(C2×C4), SmallGroup(480,780)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8○D12
G = < a,b,c,d | a5=b8=d2=1, c6=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c5 >
Subgroups: 228 in 124 conjugacy classes, 74 normal (46 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, C2×C8, M4(2), C4○D4, C20, C20, C2×C10, C2×C10, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C5×S3, C30, C30, C8○D4, C40, C40, C2×C20, C2×C20, C5×D4, C5×Q8, S3×C8, C8⋊S3, C4.Dic3, C2×C24, C4○D12, C5×Dic3, C60, S3×C10, C2×C30, C2×C40, C2×C40, C5×M4(2), C5×C4○D4, C8○D12, C5×C3⋊C8, C120, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, C2×C60, C5×C8○D4, S3×C40, C5×C8⋊S3, C5×C4.Dic3, C2×C120, C5×C4○D12, C5×C8○D12
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, C23, C10, D6, C22×C4, C20, C2×C10, C4×S3, C22×S3, C5×S3, C8○D4, C2×C20, C22×C10, S3×C2×C4, S3×C10, C22×C20, C8○D12, S3×C20, S3×C2×C10, C5×C8○D4, S3×C2×C20, C5×C8○D12
►On 240 points
Generators in S240
(1 230 159 92 71)(2 231 160 93 72)(3 232 161 94 61)(4 233 162 95 62)(5 234 163 96 63)(6 235 164 85 64)(7 236 165 86 65)(8 237 166 87 66)(9 238 167 88 67)(10 239 168 89 68)(11 240 157 90 69)(12 229 158 91 70)(13 55 35 184 46)(14 56 36 185 47)(15 57 25 186 48)(16 58 26 187 37)(17 59 27 188 38)(18 60 28 189 39)(19 49 29 190 40)(20 50 30 191 41)(21 51 31 192 42)(22 52 32 181 43)(23 53 33 182 44)(24 54 34 183 45)(73 143 104 129 173)(74 144 105 130 174)(75 133 106 131 175)(76 134 107 132 176)(77 135 108 121 177)(78 136 97 122 178)(79 137 98 123 179)(80 138 99 124 180)(81 139 100 125 169)(82 140 101 126 170)(83 141 102 127 171)(84 142 103 128 172)(109 223 212 145 203)(110 224 213 146 204)(111 225 214 147 193)(112 226 215 148 194)(113 227 216 149 195)(114 228 205 150 196)(115 217 206 151 197)(116 218 207 152 198)(117 219 208 153 199)(118 220 209 154 200)(119 221 210 155 201)(120 222 211 156 202)
(1 54 103 111 7 60 97 117)(2 55 104 112 8 49 98 118)(3 56 105 113 9 50 99 119)(4 57 106 114 10 51 100 120)(5 58 107 115 11 52 101 109)(6 59 108 116 12 53 102 110)(13 143 194 66 19 137 200 72)(14 144 195 67 20 138 201 61)(15 133 196 68 21 139 202 62)(16 134 197 69 22 140 203 63)(17 135 198 70 23 141 204 64)(18 136 199 71 24 142 193 65)(25 131 228 239 31 125 222 233)(26 132 217 240 32 126 223 234)(27 121 218 229 33 127 224 235)(28 122 219 230 34 128 225 236)(29 123 220 231 35 129 226 237)(30 124 221 232 36 130 227 238)(37 76 151 90 43 82 145 96)(38 77 152 91 44 83 146 85)(39 78 153 92 45 84 147 86)(40 79 154 93 46 73 148 87)(41 80 155 94 47 74 149 88)(42 81 156 95 48 75 150 89)(157 181 170 212 163 187 176 206)(158 182 171 213 164 188 177 207)(159 183 172 214 165 189 178 208)(160 184 173 215 166 190 179 209)(161 185 174 216 167 191 180 210)(162 186 175 205 168 192 169 211)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 22)(14 21)(15 20)(16 19)(17 18)(23 24)(25 30)(26 29)(27 28)(31 36)(32 35)(33 34)(37 40)(38 39)(41 48)(42 47)(43 46)(44 45)(49 58)(50 57)(51 56)(52 55)(53 54)(59 60)(61 68)(62 67)(63 66)(64 65)(69 72)(70 71)(73 82)(74 81)(75 80)(76 79)(77 78)(83 84)(85 86)(87 96)(88 95)(89 94)(90 93)(91 92)(97 108)(98 107)(99 106)(100 105)(101 104)(102 103)(109 112)(110 111)(113 120)(114 119)(115 118)(116 117)(121 122)(123 132)(124 131)(125 130)(126 129)(127 128)(133 138)(134 137)(135 136)(139 144)(140 143)(141 142)(145 148)(146 147)(149 156)(150 155)(151 154)(152 153)(157 160)(158 159)(161 168)(162 167)(163 166)(164 165)(169 174)(170 173)(171 172)(175 180)(176 179)(177 178)(181 184)(182 183)(185 192)(186 191)(187 190)(188 189)(193 204)(194 203)(195 202)(196 201)(197 200)(198 199)(205 210)(206 209)(207 208)(211 216)(212 215)(213 214)(217 220)(218 219)(221 228)(222 227)(223 226)(224 225)(229 230)(231 240)(232 239)(233 238)(234 237)(235 236)
G:=sub<Sym(240)| (1,230,159,92,71)(2,231,160,93,72)(3,232,161,94,61)(4,233,162,95,62)(5,234,163,96,63)(6,235,164,85,64)(7,236,165,86,65)(8,237,166,87,66)(9,238,167,88,67)(10,239,168,89,68)(11,240,157,90,69)(12,229,158,91,70)(13,55,35,184,46)(14,56,36,185,47)(15,57,25,186,48)(16,58,26,187,37)(17,59,27,188,38)(18,60,28,189,39)(19,49,29,190,40)(20,50,30,191,41)(21,51,31,192,42)(22,52,32,181,43)(23,53,33,182,44)(24,54,34,183,45)(73,143,104,129,173)(74,144,105,130,174)(75,133,106,131,175)(76,134,107,132,176)(77,135,108,121,177)(78,136,97,122,178)(79,137,98,123,179)(80,138,99,124,180)(81,139,100,125,169)(82,140,101,126,170)(83,141,102,127,171)(84,142,103,128,172)(109,223,212,145,203)(110,224,213,146,204)(111,225,214,147,193)(112,226,215,148,194)(113,227,216,149,195)(114,228,205,150,196)(115,217,206,151,197)(116,218,207,152,198)(117,219,208,153,199)(118,220,209,154,200)(119,221,210,155,201)(120,222,211,156,202), (1,54,103,111,7,60,97,117)(2,55,104,112,8,49,98,118)(3,56,105,113,9,50,99,119)(4,57,106,114,10,51,100,120)(5,58,107,115,11,52,101,109)(6,59,108,116,12,53,102,110)(13,143,194,66,19,137,200,72)(14,144,195,67,20,138,201,61)(15,133,196,68,21,139,202,62)(16,134,197,69,22,140,203,63)(17,135,198,70,23,141,204,64)(18,136,199,71,24,142,193,65)(25,131,228,239,31,125,222,233)(26,132,217,240,32,126,223,234)(27,121,218,229,33,127,224,235)(28,122,219,230,34,128,225,236)(29,123,220,231,35,129,226,237)(30,124,221,232,36,130,227,238)(37,76,151,90,43,82,145,96)(38,77,152,91,44,83,146,85)(39,78,153,92,45,84,147,86)(40,79,154,93,46,73,148,87)(41,80,155,94,47,74,149,88)(42,81,156,95,48,75,150,89)(157,181,170,212,163,187,176,206)(158,182,171,213,164,188,177,207)(159,183,172,214,165,189,178,208)(160,184,173,215,166,190,179,209)(161,185,174,216,167,191,180,210)(162,186,175,205,168,192,169,211), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,22)(14,21)(15,20)(16,19)(17,18)(23,24)(25,30)(26,29)(27,28)(31,36)(32,35)(33,34)(37,40)(38,39)(41,48)(42,47)(43,46)(44,45)(49,58)(50,57)(51,56)(52,55)(53,54)(59,60)(61,68)(62,67)(63,66)(64,65)(69,72)(70,71)(73,82)(74,81)(75,80)(76,79)(77,78)(83,84)(85,86)(87,96)(88,95)(89,94)(90,93)(91,92)(97,108)(98,107)(99,106)(100,105)(101,104)(102,103)(109,112)(110,111)(113,120)(114,119)(115,118)(116,117)(121,122)(123,132)(124,131)(125,130)(126,129)(127,128)(133,138)(134,137)(135,136)(139,144)(140,143)(141,142)(145,148)(146,147)(149,156)(150,155)(151,154)(152,153)(157,160)(158,159)(161,168)(162,167)(163,166)(164,165)(169,174)(170,173)(171,172)(175,180)(176,179)(177,178)(181,184)(182,183)(185,192)(186,191)(187,190)(188,189)(193,204)(194,203)(195,202)(196,201)(197,200)(198,199)(205,210)(206,209)(207,208)(211,216)(212,215)(213,214)(217,220)(218,219)(221,228)(222,227)(223,226)(224,225)(229,230)(231,240)(232,239)(233,238)(234,237)(235,236)>;
G:=Group( (1,230,159,92,71)(2,231,160,93,72)(3,232,161,94,61)(4,233,162,95,62)(5,234,163,96,63)(6,235,164,85,64)(7,236,165,86,65)(8,237,166,87,66)(9,238,167,88,67)(10,239,168,89,68)(11,240,157,90,69)(12,229,158,91,70)(13,55,35,184,46)(14,56,36,185,47)(15,57,25,186,48)(16,58,26,187,37)(17,59,27,188,38)(18,60,28,189,39)(19,49,29,190,40)(20,50,30,191,41)(21,51,31,192,42)(22,52,32,181,43)(23,53,33,182,44)(24,54,34,183,45)(73,143,104,129,173)(74,144,105,130,174)(75,133,106,131,175)(76,134,107,132,176)(77,135,108,121,177)(78,136,97,122,178)(79,137,98,123,179)(80,138,99,124,180)(81,139,100,125,169)(82,140,101,126,170)(83,141,102,127,171)(84,142,103,128,172)(109,223,212,145,203)(110,224,213,146,204)(111,225,214,147,193)(112,226,215,148,194)(113,227,216,149,195)(114,228,205,150,196)(115,217,206,151,197)(116,218,207,152,198)(117,219,208,153,199)(118,220,209,154,200)(119,221,210,155,201)(120,222,211,156,202), (1,54,103,111,7,60,97,117)(2,55,104,112,8,49,98,118)(3,56,105,113,9,50,99,119)(4,57,106,114,10,51,100,120)(5,58,107,115,11,52,101,109)(6,59,108,116,12,53,102,110)(13,143,194,66,19,137,200,72)(14,144,195,67,20,138,201,61)(15,133,196,68,21,139,202,62)(16,134,197,69,22,140,203,63)(17,135,198,70,23,141,204,64)(18,136,199,71,24,142,193,65)(25,131,228,239,31,125,222,233)(26,132,217,240,32,126,223,234)(27,121,218,229,33,127,224,235)(28,122,219,230,34,128,225,236)(29,123,220,231,35,129,226,237)(30,124,221,232,36,130,227,238)(37,76,151,90,43,82,145,96)(38,77,152,91,44,83,146,85)(39,78,153,92,45,84,147,86)(40,79,154,93,46,73,148,87)(41,80,155,94,47,74,149,88)(42,81,156,95,48,75,150,89)(157,181,170,212,163,187,176,206)(158,182,171,213,164,188,177,207)(159,183,172,214,165,189,178,208)(160,184,173,215,166,190,179,209)(161,185,174,216,167,191,180,210)(162,186,175,205,168,192,169,211), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,22)(14,21)(15,20)(16,19)(17,18)(23,24)(25,30)(26,29)(27,28)(31,36)(32,35)(33,34)(37,40)(38,39)(41,48)(42,47)(43,46)(44,45)(49,58)(50,57)(51,56)(52,55)(53,54)(59,60)(61,68)(62,67)(63,66)(64,65)(69,72)(70,71)(73,82)(74,81)(75,80)(76,79)(77,78)(83,84)(85,86)(87,96)(88,95)(89,94)(90,93)(91,92)(97,108)(98,107)(99,106)(100,105)(101,104)(102,103)(109,112)(110,111)(113,120)(114,119)(115,118)(116,117)(121,122)(123,132)(124,131)(125,130)(126,129)(127,128)(133,138)(134,137)(135,136)(139,144)(140,143)(141,142)(145,148)(146,147)(149,156)(150,155)(151,154)(152,153)(157,160)(158,159)(161,168)(162,167)(163,166)(164,165)(169,174)(170,173)(171,172)(175,180)(176,179)(177,178)(181,184)(182,183)(185,192)(186,191)(187,190)(188,189)(193,204)(194,203)(195,202)(196,201)(197,200)(198,199)(205,210)(206,209)(207,208)(211,216)(212,215)(213,214)(217,220)(218,219)(221,228)(222,227)(223,226)(224,225)(229,230)(231,240)(232,239)(233,238)(234,237)(235,236) );
G=PermutationGroup([[(1,230,159,92,71),(2,231,160,93,72),(3,232,161,94,61),(4,233,162,95,62),(5,234,163,96,63),(6,235,164,85,64),(7,236,165,86,65),(8,237,166,87,66),(9,238,167,88,67),(10,239,168,89,68),(11,240,157,90,69),(12,229,158,91,70),(13,55,35,184,46),(14,56,36,185,47),(15,57,25,186,48),(16,58,26,187,37),(17,59,27,188,38),(18,60,28,189,39),(19,49,29,190,40),(20,50,30,191,41),(21,51,31,192,42),(22,52,32,181,43),(23,53,33,182,44),(24,54,34,183,45),(73,143,104,129,173),(74,144,105,130,174),(75,133,106,131,175),(76,134,107,132,176),(77,135,108,121,177),(78,136,97,122,178),(79,137,98,123,179),(80,138,99,124,180),(81,139,100,125,169),(82,140,101,126,170),(83,141,102,127,171),(84,142,103,128,172),(109,223,212,145,203),(110,224,213,146,204),(111,225,214,147,193),(112,226,215,148,194),(113,227,216,149,195),(114,228,205,150,196),(115,217,206,151,197),(116,218,207,152,198),(117,219,208,153,199),(118,220,209,154,200),(119,221,210,155,201),(120,222,211,156,202)], [(1,54,103,111,7,60,97,117),(2,55,104,112,8,49,98,118),(3,56,105,113,9,50,99,119),(4,57,106,114,10,51,100,120),(5,58,107,115,11,52,101,109),(6,59,108,116,12,53,102,110),(13,143,194,66,19,137,200,72),(14,144,195,67,20,138,201,61),(15,133,196,68,21,139,202,62),(16,134,197,69,22,140,203,63),(17,135,198,70,23,141,204,64),(18,136,199,71,24,142,193,65),(25,131,228,239,31,125,222,233),(26,132,217,240,32,126,223,234),(27,121,218,229,33,127,224,235),(28,122,219,230,34,128,225,236),(29,123,220,231,35,129,226,237),(30,124,221,232,36,130,227,238),(37,76,151,90,43,82,145,96),(38,77,152,91,44,83,146,85),(39,78,153,92,45,84,147,86),(40,79,154,93,46,73,148,87),(41,80,155,94,47,74,149,88),(42,81,156,95,48,75,150,89),(157,181,170,212,163,187,176,206),(158,182,171,213,164,188,177,207),(159,183,172,214,165,189,178,208),(160,184,173,215,166,190,179,209),(161,185,174,216,167,191,180,210),(162,186,175,205,168,192,169,211)], [(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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,22),(14,21),(15,20),(16,19),(17,18),(23,24),(25,30),(26,29),(27,28),(31,36),(32,35),(33,34),(37,40),(38,39),(41,48),(42,47),(43,46),(44,45),(49,58),(50,57),(51,56),(52,55),(53,54),(59,60),(61,68),(62,67),(63,66),(64,65),(69,72),(70,71),(73,82),(74,81),(75,80),(76,79),(77,78),(83,84),(85,86),(87,96),(88,95),(89,94),(90,93),(91,92),(97,108),(98,107),(99,106),(100,105),(101,104),(102,103),(109,112),(110,111),(113,120),(114,119),(115,118),(116,117),(121,122),(123,132),(124,131),(125,130),(126,129),(127,128),(133,138),(134,137),(135,136),(139,144),(140,143),(141,142),(145,148),(146,147),(149,156),(150,155),(151,154),(152,153),(157,160),(158,159),(161,168),(162,167),(163,166),(164,165),(169,174),(170,173),(171,172),(175,180),(176,179),(177,178),(181,184),(182,183),(185,192),(186,191),(187,190),(188,189),(193,204),(194,203),(195,202),(196,201),(197,200),(198,199),(205,210),(206,209),(207,208),(211,216),(212,215),(213,214),(217,220),(218,219),(221,228),(222,227),(223,226),(224,225),(229,230),(231,240),(232,239),(233,238),(234,237),(235,236)]])
180 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10P | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 20M | ··· | 20T | 24A | ··· | 24H | 30A | ··· | 30L | 40A | ··· | 40P | 40Q | ··· | 40X | 40Y | ··· | 40AN | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 40 | ··· | 40 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 6 | 6 | 2 | 1 | 1 | 2 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
180 irreducible representations
| dim | 1 | 1 | 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 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C5 | C10 | C10 | C10 | C10 | C10 | C20 | C20 | C20 | S3 | D6 | D6 | C4×S3 | C4×S3 | C5×S3 | C8○D4 | S3×C10 | S3×C10 | C8○D12 | S3×C20 | S3×C20 | C5×C8○D4 | C5×C8○D12 |
| kernel | C5×C8○D12 | S3×C40 | C5×C8⋊S3 | C5×C4.Dic3 | C2×C120 | C5×C4○D12 | C5×Dic6 | C5×D12 | C5×C3⋊D4 | C8○D12 | S3×C8 | C8⋊S3 | C4.Dic3 | C2×C24 | C4○D12 | Dic6 | D12 | C3⋊D4 | C2×C40 | C40 | C2×C20 | C20 | C2×C10 | C2×C8 | C15 | C8 | C2×C4 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 8 | 8 | 16 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 8 | 4 | 8 | 8 | 8 | 16 | 32 |
Matrix representation of C5×C8○D12 ►in GL2(𝔽241) generated by
| 87 | 0 |
| 0 | 87 |
| 8 | 0 |
| 0 | 8 |
| 142 | 142 |
| 99 | 43 |
| 99 | 43 |
| 142 | 142 |
G:=sub<GL(2,GF(241))| [87,0,0,87],[8,0,0,8],[142,99,142,43],[99,142,43,142] >;
C5×C8○D12 in GAP, Magma, Sage, TeX
C_5\times C_8\circ D_{12} % in TeX
G:=Group("C5xC8oD12"); // GroupNames label
G:=SmallGroup(480,780);
// by ID
G=gap.SmallGroup(480,780);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,226,102,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=d^2=1,c^6=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c^5>;
// generators/relations