direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×D12.C4, D12.C20, C40.64D6, Dic6.C20, C60.287C23, C120.77C22, C3⋊D4.C20, (S3×C8)⋊8C10, C4.5(S3×C20), (S3×C40)⋊17C2, C8⋊S3⋊6C10, C15⋊26(C8○D4), C8.12(S3×C10), C20.80(C4×S3), (C5×D12).5C4, D6.2(C2×C20), C24.13(C2×C10), C60.183(C2×C4), C12.13(C2×C20), C4○D12.3C10, (C2×C20).357D6, M4(2)⋊5(C5×S3), C22.1(S3×C20), (C5×Dic6).5C4, (C3×M4(2))⋊6C10, (C5×M4(2))⋊11S3, C6.16(C22×C20), Dic3.4(C2×C20), (C15×M4(2))⋊14C2, (S3×C20).66C22, (C2×C60).354C22, C12.39(C22×C10), C30.207(C22×C4), C20.245(C22×S3), C3⋊2(C5×C8○D4), (C2×C3⋊C8)⋊3C10, (C10×C3⋊C8)⋊17C2, C2.17(S3×C2×C20), C4.39(S3×C2×C10), C3⋊C8.12(C2×C10), C10.143(S3×C2×C4), (C2×C6).6(C2×C20), (C5×C3⋊D4).5C4, (C5×C8⋊S3)⋊14C2, (C2×C10).38(C4×S3), (C2×C4).46(S3×C10), (C5×C4○D12).9C2, (C5×C3⋊C8).48C22, (S3×C10).32(C2×C4), (C4×S3).17(C2×C10), (C2×C12).27(C2×C10), (C2×C30).130(C2×C4), (C5×Dic3).40(C2×C4), SmallGroup(480,786)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D12.C4
G = < a,b,c,d | a5=b12=c2=1, d4=b6, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b7, dcd-1=b6c >
Subgroups: 228 in 124 conjugacy classes, 74 normal (42 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, M4(2), 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, C2×C3⋊C8, C3×M4(2), C4○D12, C5×Dic3, C60, S3×C10, C2×C30, C2×C40, C5×M4(2), C5×M4(2), C5×C4○D4, D12.C4, C5×C3⋊C8, C120, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, C2×C60, C5×C8○D4, S3×C40, C5×C8⋊S3, C10×C3⋊C8, C15×M4(2), C5×C4○D12, C5×D12.C4
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, D12.C4, S3×C20, S3×C2×C10, C5×C8○D4, S3×C2×C20, C5×D12.C4
►On 240 points
Generators in S240
(1 84 191 204 14)(2 73 192 193 15)(3 74 181 194 16)(4 75 182 195 17)(5 76 183 196 18)(6 77 184 197 19)(7 78 185 198 20)(8 79 186 199 21)(9 80 187 200 22)(10 81 188 201 23)(11 82 189 202 24)(12 83 190 203 13)(25 115 54 167 128)(26 116 55 168 129)(27 117 56 157 130)(28 118 57 158 131)(29 119 58 159 132)(30 120 59 160 121)(31 109 60 161 122)(32 110 49 162 123)(33 111 50 163 124)(34 112 51 164 125)(35 113 52 165 126)(36 114 53 166 127)(37 153 231 96 67)(38 154 232 85 68)(39 155 233 86 69)(40 156 234 87 70)(41 145 235 88 71)(42 146 236 89 72)(43 147 237 90 61)(44 148 238 91 62)(45 149 239 92 63)(46 150 240 93 64)(47 151 229 94 65)(48 152 230 95 66)(97 210 222 144 180)(98 211 223 133 169)(99 212 224 134 170)(100 213 225 135 171)(101 214 226 136 172)(102 215 227 137 173)(103 216 228 138 174)(104 205 217 139 175)(105 206 218 140 176)(106 207 219 141 177)(107 208 220 142 178)(108 209 221 143 179)
(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 14)(15 24)(16 23)(17 22)(18 21)(19 20)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(37 38)(39 48)(40 47)(41 46)(42 45)(43 44)(49 58)(50 57)(51 56)(52 55)(53 54)(59 60)(61 62)(63 72)(64 71)(65 70)(66 69)(67 68)(73 82)(74 81)(75 80)(76 79)(77 78)(83 84)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(97 104)(98 103)(99 102)(100 101)(105 108)(106 107)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)(121 122)(123 132)(124 131)(125 130)(126 129)(127 128)(133 138)(134 137)(135 136)(139 144)(140 143)(141 142)(145 150)(146 149)(147 148)(151 156)(152 155)(153 154)(157 164)(158 163)(159 162)(160 161)(165 168)(166 167)(169 174)(170 173)(171 172)(175 180)(176 179)(177 178)(181 188)(182 187)(183 186)(184 185)(189 192)(190 191)(193 202)(194 201)(195 200)(196 199)(197 198)(203 204)(205 210)(206 209)(207 208)(211 216)(212 215)(213 214)(217 222)(218 221)(219 220)(223 228)(224 227)(225 226)(229 234)(230 233)(231 232)(235 240)(236 239)(237 238)
(1 136 161 85 7 142 167 91)(2 143 162 92 8 137 168 86)(3 138 163 87 9 144 157 93)(4 133 164 94 10 139 158 88)(5 140 165 89 11 134 159 95)(6 135 166 96 12 141 160 90)(13 219 59 237 19 225 53 231)(14 226 60 232 20 220 54 238)(15 221 49 239 21 227 55 233)(16 228 50 234 22 222 56 240)(17 223 51 229 23 217 57 235)(18 218 52 236 24 224 58 230)(25 44 191 101 31 38 185 107)(26 39 192 108 32 45 186 102)(27 46 181 103 33 40 187 97)(28 41 182 98 34 47 188 104)(29 48 183 105 35 42 189 99)(30 43 184 100 36 37 190 106)(61 77 171 127 67 83 177 121)(62 84 172 122 68 78 178 128)(63 79 173 129 69 73 179 123)(64 74 174 124 70 80 180 130)(65 81 175 131 71 75 169 125)(66 76 176 126 72 82 170 132)(109 154 198 208 115 148 204 214)(110 149 199 215 116 155 193 209)(111 156 200 210 117 150 194 216)(112 151 201 205 118 145 195 211)(113 146 202 212 119 152 196 206)(114 153 203 207 120 147 197 213)
G:=sub<Sym(240)| (1,84,191,204,14)(2,73,192,193,15)(3,74,181,194,16)(4,75,182,195,17)(5,76,183,196,18)(6,77,184,197,19)(7,78,185,198,20)(8,79,186,199,21)(9,80,187,200,22)(10,81,188,201,23)(11,82,189,202,24)(12,83,190,203,13)(25,115,54,167,128)(26,116,55,168,129)(27,117,56,157,130)(28,118,57,158,131)(29,119,58,159,132)(30,120,59,160,121)(31,109,60,161,122)(32,110,49,162,123)(33,111,50,163,124)(34,112,51,164,125)(35,113,52,165,126)(36,114,53,166,127)(37,153,231,96,67)(38,154,232,85,68)(39,155,233,86,69)(40,156,234,87,70)(41,145,235,88,71)(42,146,236,89,72)(43,147,237,90,61)(44,148,238,91,62)(45,149,239,92,63)(46,150,240,93,64)(47,151,229,94,65)(48,152,230,95,66)(97,210,222,144,180)(98,211,223,133,169)(99,212,224,134,170)(100,213,225,135,171)(101,214,226,136,172)(102,215,227,137,173)(103,216,228,138,174)(104,205,217,139,175)(105,206,218,140,176)(106,207,219,141,177)(107,208,220,142,178)(108,209,221,143,179), (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)(13,14)(15,24)(16,23)(17,22)(18,21)(19,20)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(37,38)(39,48)(40,47)(41,46)(42,45)(43,44)(49,58)(50,57)(51,56)(52,55)(53,54)(59,60)(61,62)(63,72)(64,71)(65,70)(66,69)(67,68)(73,82)(74,81)(75,80)(76,79)(77,78)(83,84)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(97,104)(98,103)(99,102)(100,101)(105,108)(106,107)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115)(121,122)(123,132)(124,131)(125,130)(126,129)(127,128)(133,138)(134,137)(135,136)(139,144)(140,143)(141,142)(145,150)(146,149)(147,148)(151,156)(152,155)(153,154)(157,164)(158,163)(159,162)(160,161)(165,168)(166,167)(169,174)(170,173)(171,172)(175,180)(176,179)(177,178)(181,188)(182,187)(183,186)(184,185)(189,192)(190,191)(193,202)(194,201)(195,200)(196,199)(197,198)(203,204)(205,210)(206,209)(207,208)(211,216)(212,215)(213,214)(217,222)(218,221)(219,220)(223,228)(224,227)(225,226)(229,234)(230,233)(231,232)(235,240)(236,239)(237,238), (1,136,161,85,7,142,167,91)(2,143,162,92,8,137,168,86)(3,138,163,87,9,144,157,93)(4,133,164,94,10,139,158,88)(5,140,165,89,11,134,159,95)(6,135,166,96,12,141,160,90)(13,219,59,237,19,225,53,231)(14,226,60,232,20,220,54,238)(15,221,49,239,21,227,55,233)(16,228,50,234,22,222,56,240)(17,223,51,229,23,217,57,235)(18,218,52,236,24,224,58,230)(25,44,191,101,31,38,185,107)(26,39,192,108,32,45,186,102)(27,46,181,103,33,40,187,97)(28,41,182,98,34,47,188,104)(29,48,183,105,35,42,189,99)(30,43,184,100,36,37,190,106)(61,77,171,127,67,83,177,121)(62,84,172,122,68,78,178,128)(63,79,173,129,69,73,179,123)(64,74,174,124,70,80,180,130)(65,81,175,131,71,75,169,125)(66,76,176,126,72,82,170,132)(109,154,198,208,115,148,204,214)(110,149,199,215,116,155,193,209)(111,156,200,210,117,150,194,216)(112,151,201,205,118,145,195,211)(113,146,202,212,119,152,196,206)(114,153,203,207,120,147,197,213)>;
G:=Group( (1,84,191,204,14)(2,73,192,193,15)(3,74,181,194,16)(4,75,182,195,17)(5,76,183,196,18)(6,77,184,197,19)(7,78,185,198,20)(8,79,186,199,21)(9,80,187,200,22)(10,81,188,201,23)(11,82,189,202,24)(12,83,190,203,13)(25,115,54,167,128)(26,116,55,168,129)(27,117,56,157,130)(28,118,57,158,131)(29,119,58,159,132)(30,120,59,160,121)(31,109,60,161,122)(32,110,49,162,123)(33,111,50,163,124)(34,112,51,164,125)(35,113,52,165,126)(36,114,53,166,127)(37,153,231,96,67)(38,154,232,85,68)(39,155,233,86,69)(40,156,234,87,70)(41,145,235,88,71)(42,146,236,89,72)(43,147,237,90,61)(44,148,238,91,62)(45,149,239,92,63)(46,150,240,93,64)(47,151,229,94,65)(48,152,230,95,66)(97,210,222,144,180)(98,211,223,133,169)(99,212,224,134,170)(100,213,225,135,171)(101,214,226,136,172)(102,215,227,137,173)(103,216,228,138,174)(104,205,217,139,175)(105,206,218,140,176)(106,207,219,141,177)(107,208,220,142,178)(108,209,221,143,179), (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)(13,14)(15,24)(16,23)(17,22)(18,21)(19,20)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(37,38)(39,48)(40,47)(41,46)(42,45)(43,44)(49,58)(50,57)(51,56)(52,55)(53,54)(59,60)(61,62)(63,72)(64,71)(65,70)(66,69)(67,68)(73,82)(74,81)(75,80)(76,79)(77,78)(83,84)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(97,104)(98,103)(99,102)(100,101)(105,108)(106,107)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115)(121,122)(123,132)(124,131)(125,130)(126,129)(127,128)(133,138)(134,137)(135,136)(139,144)(140,143)(141,142)(145,150)(146,149)(147,148)(151,156)(152,155)(153,154)(157,164)(158,163)(159,162)(160,161)(165,168)(166,167)(169,174)(170,173)(171,172)(175,180)(176,179)(177,178)(181,188)(182,187)(183,186)(184,185)(189,192)(190,191)(193,202)(194,201)(195,200)(196,199)(197,198)(203,204)(205,210)(206,209)(207,208)(211,216)(212,215)(213,214)(217,222)(218,221)(219,220)(223,228)(224,227)(225,226)(229,234)(230,233)(231,232)(235,240)(236,239)(237,238), (1,136,161,85,7,142,167,91)(2,143,162,92,8,137,168,86)(3,138,163,87,9,144,157,93)(4,133,164,94,10,139,158,88)(5,140,165,89,11,134,159,95)(6,135,166,96,12,141,160,90)(13,219,59,237,19,225,53,231)(14,226,60,232,20,220,54,238)(15,221,49,239,21,227,55,233)(16,228,50,234,22,222,56,240)(17,223,51,229,23,217,57,235)(18,218,52,236,24,224,58,230)(25,44,191,101,31,38,185,107)(26,39,192,108,32,45,186,102)(27,46,181,103,33,40,187,97)(28,41,182,98,34,47,188,104)(29,48,183,105,35,42,189,99)(30,43,184,100,36,37,190,106)(61,77,171,127,67,83,177,121)(62,84,172,122,68,78,178,128)(63,79,173,129,69,73,179,123)(64,74,174,124,70,80,180,130)(65,81,175,131,71,75,169,125)(66,76,176,126,72,82,170,132)(109,154,198,208,115,148,204,214)(110,149,199,215,116,155,193,209)(111,156,200,210,117,150,194,216)(112,151,201,205,118,145,195,211)(113,146,202,212,119,152,196,206)(114,153,203,207,120,147,197,213) );
G=PermutationGroup([[(1,84,191,204,14),(2,73,192,193,15),(3,74,181,194,16),(4,75,182,195,17),(5,76,183,196,18),(6,77,184,197,19),(7,78,185,198,20),(8,79,186,199,21),(9,80,187,200,22),(10,81,188,201,23),(11,82,189,202,24),(12,83,190,203,13),(25,115,54,167,128),(26,116,55,168,129),(27,117,56,157,130),(28,118,57,158,131),(29,119,58,159,132),(30,120,59,160,121),(31,109,60,161,122),(32,110,49,162,123),(33,111,50,163,124),(34,112,51,164,125),(35,113,52,165,126),(36,114,53,166,127),(37,153,231,96,67),(38,154,232,85,68),(39,155,233,86,69),(40,156,234,87,70),(41,145,235,88,71),(42,146,236,89,72),(43,147,237,90,61),(44,148,238,91,62),(45,149,239,92,63),(46,150,240,93,64),(47,151,229,94,65),(48,152,230,95,66),(97,210,222,144,180),(98,211,223,133,169),(99,212,224,134,170),(100,213,225,135,171),(101,214,226,136,172),(102,215,227,137,173),(103,216,228,138,174),(104,205,217,139,175),(105,206,218,140,176),(106,207,219,141,177),(107,208,220,142,178),(108,209,221,143,179)], [(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),(13,14),(15,24),(16,23),(17,22),(18,21),(19,20),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(37,38),(39,48),(40,47),(41,46),(42,45),(43,44),(49,58),(50,57),(51,56),(52,55),(53,54),(59,60),(61,62),(63,72),(64,71),(65,70),(66,69),(67,68),(73,82),(74,81),(75,80),(76,79),(77,78),(83,84),(85,96),(86,95),(87,94),(88,93),(89,92),(90,91),(97,104),(98,103),(99,102),(100,101),(105,108),(106,107),(109,120),(110,119),(111,118),(112,117),(113,116),(114,115),(121,122),(123,132),(124,131),(125,130),(126,129),(127,128),(133,138),(134,137),(135,136),(139,144),(140,143),(141,142),(145,150),(146,149),(147,148),(151,156),(152,155),(153,154),(157,164),(158,163),(159,162),(160,161),(165,168),(166,167),(169,174),(170,173),(171,172),(175,180),(176,179),(177,178),(181,188),(182,187),(183,186),(184,185),(189,192),(190,191),(193,202),(194,201),(195,200),(196,199),(197,198),(203,204),(205,210),(206,209),(207,208),(211,216),(212,215),(213,214),(217,222),(218,221),(219,220),(223,228),(224,227),(225,226),(229,234),(230,233),(231,232),(235,240),(236,239),(237,238)], [(1,136,161,85,7,142,167,91),(2,143,162,92,8,137,168,86),(3,138,163,87,9,144,157,93),(4,133,164,94,10,139,158,88),(5,140,165,89,11,134,159,95),(6,135,166,96,12,141,160,90),(13,219,59,237,19,225,53,231),(14,226,60,232,20,220,54,238),(15,221,49,239,21,227,55,233),(16,228,50,234,22,222,56,240),(17,223,51,229,23,217,57,235),(18,218,52,236,24,224,58,230),(25,44,191,101,31,38,185,107),(26,39,192,108,32,45,186,102),(27,46,181,103,33,40,187,97),(28,41,182,98,34,47,188,104),(29,48,183,105,35,42,189,99),(30,43,184,100,36,37,190,106),(61,77,171,127,67,83,177,121),(62,84,172,122,68,78,178,128),(63,79,173,129,69,73,179,123),(64,74,174,124,70,80,180,130),(65,81,175,131,71,75,169,125),(66,76,176,126,72,82,170,132),(109,154,198,208,115,148,204,214),(110,149,199,215,116,155,193,209),(111,156,200,210,117,150,194,216),(112,151,201,205,118,145,195,211),(113,146,202,212,119,152,196,206),(114,153,203,207,120,147,197,213)]])
150 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10P | 12A | 12B | 12C | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 20M | ··· | 20T | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | 40A | ··· | 40P | 40Q | ··· | 40AF | 40AG | ··· | 40AN | 60A | ··· | 60H | 60I | 60J | 60K | 60L | 120A | ··· | 120P |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 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 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 40 | ··· | 40 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 6 | 6 | 2 | 1 | 1 | 2 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
150 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 | 4 | 4 |
| 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 | S3×C20 | S3×C20 | C5×C8○D4 | D12.C4 | C5×D12.C4 |
| kernel | C5×D12.C4 | S3×C40 | C5×C8⋊S3 | C10×C3⋊C8 | C15×M4(2) | C5×C4○D12 | C5×Dic6 | C5×D12 | C5×C3⋊D4 | D12.C4 | S3×C8 | C8⋊S3 | C2×C3⋊C8 | C3×M4(2) | C4○D12 | Dic6 | D12 | C3⋊D4 | C5×M4(2) | C40 | C2×C20 | C20 | C2×C10 | M4(2) | C15 | C8 | C2×C4 | C4 | C22 | C3 | C5 | 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 | 16 | 2 | 8 |
Matrix representation of C5×D12.C4 ►in GL4(𝔽241) generated by
| 91 | 0 | 0 | 0 |
| 0 | 91 | 0 | 0 |
| 0 | 0 | 87 | 0 |
| 0 | 0 | 0 | 87 |
| 0 | 64 | 0 | 0 |
| 64 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 |
| 0 | 0 | 240 | 0 |
| 0 | 64 | 0 | 0 |
| 177 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 240 | 1 |
| 233 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,87,0,0,0,0,87],[0,64,0,0,64,0,0,0,0,0,240,240,0,0,1,0],[0,177,0,0,64,0,0,0,0,0,240,240,0,0,0,1],[233,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1] >;
C5×D12.C4 in GAP, Magma, Sage, TeX
C_5\times D_{12}.C_4 % in TeX
G:=Group("C5xD12.C4"); // GroupNames label
G:=SmallGroup(480,786);
// by ID
G=gap.SmallGroup(480,786);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,891,226,102,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^12=c^2=1,d^4=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^7,d*c*d^-1=b^6*c>;
// generators/relations