direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C4○D24, D24⋊7C10, C40.78D6, C20.72D12, C60.192D4, Dic12⋊7C10, C60.269C23, C120.96C22, (C2×C24)⋊6C10, (C2×C40)⋊12S3, C24⋊C2⋊7C10, (C2×C120)⋊18C2, C4○D12⋊1C10, (C5×D24)⋊15C2, C15⋊27(C4○D8), C8.17(S3×C10), C6.11(D4×C10), C12.35(C5×D4), C4.20(C5×D12), C24.17(C2×C10), D12.7(C2×C10), C30.298(C2×D4), (C2×C20).434D6, (C2×C10).12D12, (C2×C30).125D4, C2.13(C10×D12), C10.82(C2×D12), C22.1(C5×D12), (C5×Dic12)⋊15C2, Dic6.6(C2×C10), C12.30(C22×C10), (C2×C60).537C22, C20.233(C22×S3), (C5×D12).46C22, (C5×Dic6).48C22, C3⋊1(C5×C4○D8), (C2×C8)⋊4(C5×S3), C4.30(S3×C2×C10), (C2×C6).18(C5×D4), (C5×C24⋊C2)⋊15C2, (C5×C4○D12)⋊11C2, (C2×C4).82(S3×C10), (C2×C12).104(C2×C10), SmallGroup(480,783)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C4○D24
G = < a,b,c,d | a5=b4=d2=1, c12=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c11 >
Subgroups: 324 in 124 conjugacy classes, 58 normal (42 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, D8, SD16, Q16, C4○D4, C20, C20, C2×C10, C2×C10, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C5×S3, C30, C30, C4○D8, C40, C2×C20, C2×C20, C5×D4, C5×Q8, C24⋊C2, D24, Dic12, C2×C24, C4○D12, C5×Dic3, C60, S3×C10, C2×C30, C2×C40, C5×D8, C5×SD16, C5×Q16, C5×C4○D4, C4○D24, C120, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, C2×C60, C5×C4○D8, C5×C24⋊C2, C5×D24, C5×Dic12, C2×C120, C5×C4○D12, C5×C4○D24
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C2×C10, D12, C22×S3, C5×S3, C4○D8, C5×D4, C22×C10, C2×D12, S3×C10, D4×C10, C4○D24, C5×D12, S3×C2×C10, C5×C4○D8, C10×D12, C5×C4○D24
►On 240 points
Generators in S240
(1 50 93 214 160)(2 51 94 215 161)(3 52 95 216 162)(4 53 96 193 163)(5 54 73 194 164)(6 55 74 195 165)(7 56 75 196 166)(8 57 76 197 167)(9 58 77 198 168)(10 59 78 199 145)(11 60 79 200 146)(12 61 80 201 147)(13 62 81 202 148)(14 63 82 203 149)(15 64 83 204 150)(16 65 84 205 151)(17 66 85 206 152)(18 67 86 207 153)(19 68 87 208 154)(20 69 88 209 155)(21 70 89 210 156)(22 71 90 211 157)(23 72 91 212 158)(24 49 92 213 159)(25 238 116 125 189)(26 239 117 126 190)(27 240 118 127 191)(28 217 119 128 192)(29 218 120 129 169)(30 219 97 130 170)(31 220 98 131 171)(32 221 99 132 172)(33 222 100 133 173)(34 223 101 134 174)(35 224 102 135 175)(36 225 103 136 176)(37 226 104 137 177)(38 227 105 138 178)(39 228 106 139 179)(40 229 107 140 180)(41 230 108 141 181)(42 231 109 142 182)(43 232 110 143 183)(44 233 111 144 184)(45 234 112 121 185)(46 235 113 122 186)(47 236 114 123 187)(48 237 115 124 188)
(1 19 13 7)(2 20 14 8)(3 21 15 9)(4 22 16 10)(5 23 17 11)(6 24 18 12)(25 31 37 43)(26 32 38 44)(27 33 39 45)(28 34 40 46)(29 35 41 47)(30 36 42 48)(49 67 61 55)(50 68 62 56)(51 69 63 57)(52 70 64 58)(53 71 65 59)(54 72 66 60)(73 91 85 79)(74 92 86 80)(75 93 87 81)(76 94 88 82)(77 95 89 83)(78 96 90 84)(97 103 109 115)(98 104 110 116)(99 105 111 117)(100 106 112 118)(101 107 113 119)(102 108 114 120)(121 127 133 139)(122 128 134 140)(123 129 135 141)(124 130 136 142)(125 131 137 143)(126 132 138 144)(145 163 157 151)(146 164 158 152)(147 165 159 153)(148 166 160 154)(149 167 161 155)(150 168 162 156)(169 175 181 187)(170 176 182 188)(171 177 183 189)(172 178 184 190)(173 179 185 191)(174 180 186 192)(193 211 205 199)(194 212 206 200)(195 213 207 201)(196 214 208 202)(197 215 209 203)(198 216 210 204)(217 223 229 235)(218 224 230 236)(219 225 231 237)(220 226 232 238)(221 227 233 239)(222 228 234 240)
(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)
(1 125)(2 124)(3 123)(4 122)(5 121)(6 144)(7 143)(8 142)(9 141)(10 140)(11 139)(12 138)(13 137)(14 136)(15 135)(16 134)(17 133)(18 132)(19 131)(20 130)(21 129)(22 128)(23 127)(24 126)(25 93)(26 92)(27 91)(28 90)(29 89)(30 88)(31 87)(32 86)(33 85)(34 84)(35 83)(36 82)(37 81)(38 80)(39 79)(40 78)(41 77)(42 76)(43 75)(44 74)(45 73)(46 96)(47 95)(48 94)(49 190)(50 189)(51 188)(52 187)(53 186)(54 185)(55 184)(56 183)(57 182)(58 181)(59 180)(60 179)(61 178)(62 177)(63 176)(64 175)(65 174)(66 173)(67 172)(68 171)(69 170)(70 169)(71 192)(72 191)(97 155)(98 154)(99 153)(100 152)(101 151)(102 150)(103 149)(104 148)(105 147)(106 146)(107 145)(108 168)(109 167)(110 166)(111 165)(112 164)(113 163)(114 162)(115 161)(116 160)(117 159)(118 158)(119 157)(120 156)(193 235)(194 234)(195 233)(196 232)(197 231)(198 230)(199 229)(200 228)(201 227)(202 226)(203 225)(204 224)(205 223)(206 222)(207 221)(208 220)(209 219)(210 218)(211 217)(212 240)(213 239)(214 238)(215 237)(216 236)
G:=sub<Sym(240)| (1,50,93,214,160)(2,51,94,215,161)(3,52,95,216,162)(4,53,96,193,163)(5,54,73,194,164)(6,55,74,195,165)(7,56,75,196,166)(8,57,76,197,167)(9,58,77,198,168)(10,59,78,199,145)(11,60,79,200,146)(12,61,80,201,147)(13,62,81,202,148)(14,63,82,203,149)(15,64,83,204,150)(16,65,84,205,151)(17,66,85,206,152)(18,67,86,207,153)(19,68,87,208,154)(20,69,88,209,155)(21,70,89,210,156)(22,71,90,211,157)(23,72,91,212,158)(24,49,92,213,159)(25,238,116,125,189)(26,239,117,126,190)(27,240,118,127,191)(28,217,119,128,192)(29,218,120,129,169)(30,219,97,130,170)(31,220,98,131,171)(32,221,99,132,172)(33,222,100,133,173)(34,223,101,134,174)(35,224,102,135,175)(36,225,103,136,176)(37,226,104,137,177)(38,227,105,138,178)(39,228,106,139,179)(40,229,107,140,180)(41,230,108,141,181)(42,231,109,142,182)(43,232,110,143,183)(44,233,111,144,184)(45,234,112,121,185)(46,235,113,122,186)(47,236,114,123,187)(48,237,115,124,188), (1,19,13,7)(2,20,14,8)(3,21,15,9)(4,22,16,10)(5,23,17,11)(6,24,18,12)(25,31,37,43)(26,32,38,44)(27,33,39,45)(28,34,40,46)(29,35,41,47)(30,36,42,48)(49,67,61,55)(50,68,62,56)(51,69,63,57)(52,70,64,58)(53,71,65,59)(54,72,66,60)(73,91,85,79)(74,92,86,80)(75,93,87,81)(76,94,88,82)(77,95,89,83)(78,96,90,84)(97,103,109,115)(98,104,110,116)(99,105,111,117)(100,106,112,118)(101,107,113,119)(102,108,114,120)(121,127,133,139)(122,128,134,140)(123,129,135,141)(124,130,136,142)(125,131,137,143)(126,132,138,144)(145,163,157,151)(146,164,158,152)(147,165,159,153)(148,166,160,154)(149,167,161,155)(150,168,162,156)(169,175,181,187)(170,176,182,188)(171,177,183,189)(172,178,184,190)(173,179,185,191)(174,180,186,192)(193,211,205,199)(194,212,206,200)(195,213,207,201)(196,214,208,202)(197,215,209,203)(198,216,210,204)(217,223,229,235)(218,224,230,236)(219,225,231,237)(220,226,232,238)(221,227,233,239)(222,228,234,240), (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), (1,125)(2,124)(3,123)(4,122)(5,121)(6,144)(7,143)(8,142)(9,141)(10,140)(11,139)(12,138)(13,137)(14,136)(15,135)(16,134)(17,133)(18,132)(19,131)(20,130)(21,129)(22,128)(23,127)(24,126)(25,93)(26,92)(27,91)(28,90)(29,89)(30,88)(31,87)(32,86)(33,85)(34,84)(35,83)(36,82)(37,81)(38,80)(39,79)(40,78)(41,77)(42,76)(43,75)(44,74)(45,73)(46,96)(47,95)(48,94)(49,190)(50,189)(51,188)(52,187)(53,186)(54,185)(55,184)(56,183)(57,182)(58,181)(59,180)(60,179)(61,178)(62,177)(63,176)(64,175)(65,174)(66,173)(67,172)(68,171)(69,170)(70,169)(71,192)(72,191)(97,155)(98,154)(99,153)(100,152)(101,151)(102,150)(103,149)(104,148)(105,147)(106,146)(107,145)(108,168)(109,167)(110,166)(111,165)(112,164)(113,163)(114,162)(115,161)(116,160)(117,159)(118,158)(119,157)(120,156)(193,235)(194,234)(195,233)(196,232)(197,231)(198,230)(199,229)(200,228)(201,227)(202,226)(203,225)(204,224)(205,223)(206,222)(207,221)(208,220)(209,219)(210,218)(211,217)(212,240)(213,239)(214,238)(215,237)(216,236)>;
G:=Group( (1,50,93,214,160)(2,51,94,215,161)(3,52,95,216,162)(4,53,96,193,163)(5,54,73,194,164)(6,55,74,195,165)(7,56,75,196,166)(8,57,76,197,167)(9,58,77,198,168)(10,59,78,199,145)(11,60,79,200,146)(12,61,80,201,147)(13,62,81,202,148)(14,63,82,203,149)(15,64,83,204,150)(16,65,84,205,151)(17,66,85,206,152)(18,67,86,207,153)(19,68,87,208,154)(20,69,88,209,155)(21,70,89,210,156)(22,71,90,211,157)(23,72,91,212,158)(24,49,92,213,159)(25,238,116,125,189)(26,239,117,126,190)(27,240,118,127,191)(28,217,119,128,192)(29,218,120,129,169)(30,219,97,130,170)(31,220,98,131,171)(32,221,99,132,172)(33,222,100,133,173)(34,223,101,134,174)(35,224,102,135,175)(36,225,103,136,176)(37,226,104,137,177)(38,227,105,138,178)(39,228,106,139,179)(40,229,107,140,180)(41,230,108,141,181)(42,231,109,142,182)(43,232,110,143,183)(44,233,111,144,184)(45,234,112,121,185)(46,235,113,122,186)(47,236,114,123,187)(48,237,115,124,188), (1,19,13,7)(2,20,14,8)(3,21,15,9)(4,22,16,10)(5,23,17,11)(6,24,18,12)(25,31,37,43)(26,32,38,44)(27,33,39,45)(28,34,40,46)(29,35,41,47)(30,36,42,48)(49,67,61,55)(50,68,62,56)(51,69,63,57)(52,70,64,58)(53,71,65,59)(54,72,66,60)(73,91,85,79)(74,92,86,80)(75,93,87,81)(76,94,88,82)(77,95,89,83)(78,96,90,84)(97,103,109,115)(98,104,110,116)(99,105,111,117)(100,106,112,118)(101,107,113,119)(102,108,114,120)(121,127,133,139)(122,128,134,140)(123,129,135,141)(124,130,136,142)(125,131,137,143)(126,132,138,144)(145,163,157,151)(146,164,158,152)(147,165,159,153)(148,166,160,154)(149,167,161,155)(150,168,162,156)(169,175,181,187)(170,176,182,188)(171,177,183,189)(172,178,184,190)(173,179,185,191)(174,180,186,192)(193,211,205,199)(194,212,206,200)(195,213,207,201)(196,214,208,202)(197,215,209,203)(198,216,210,204)(217,223,229,235)(218,224,230,236)(219,225,231,237)(220,226,232,238)(221,227,233,239)(222,228,234,240), (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), (1,125)(2,124)(3,123)(4,122)(5,121)(6,144)(7,143)(8,142)(9,141)(10,140)(11,139)(12,138)(13,137)(14,136)(15,135)(16,134)(17,133)(18,132)(19,131)(20,130)(21,129)(22,128)(23,127)(24,126)(25,93)(26,92)(27,91)(28,90)(29,89)(30,88)(31,87)(32,86)(33,85)(34,84)(35,83)(36,82)(37,81)(38,80)(39,79)(40,78)(41,77)(42,76)(43,75)(44,74)(45,73)(46,96)(47,95)(48,94)(49,190)(50,189)(51,188)(52,187)(53,186)(54,185)(55,184)(56,183)(57,182)(58,181)(59,180)(60,179)(61,178)(62,177)(63,176)(64,175)(65,174)(66,173)(67,172)(68,171)(69,170)(70,169)(71,192)(72,191)(97,155)(98,154)(99,153)(100,152)(101,151)(102,150)(103,149)(104,148)(105,147)(106,146)(107,145)(108,168)(109,167)(110,166)(111,165)(112,164)(113,163)(114,162)(115,161)(116,160)(117,159)(118,158)(119,157)(120,156)(193,235)(194,234)(195,233)(196,232)(197,231)(198,230)(199,229)(200,228)(201,227)(202,226)(203,225)(204,224)(205,223)(206,222)(207,221)(208,220)(209,219)(210,218)(211,217)(212,240)(213,239)(214,238)(215,237)(216,236) );
G=PermutationGroup([[(1,50,93,214,160),(2,51,94,215,161),(3,52,95,216,162),(4,53,96,193,163),(5,54,73,194,164),(6,55,74,195,165),(7,56,75,196,166),(8,57,76,197,167),(9,58,77,198,168),(10,59,78,199,145),(11,60,79,200,146),(12,61,80,201,147),(13,62,81,202,148),(14,63,82,203,149),(15,64,83,204,150),(16,65,84,205,151),(17,66,85,206,152),(18,67,86,207,153),(19,68,87,208,154),(20,69,88,209,155),(21,70,89,210,156),(22,71,90,211,157),(23,72,91,212,158),(24,49,92,213,159),(25,238,116,125,189),(26,239,117,126,190),(27,240,118,127,191),(28,217,119,128,192),(29,218,120,129,169),(30,219,97,130,170),(31,220,98,131,171),(32,221,99,132,172),(33,222,100,133,173),(34,223,101,134,174),(35,224,102,135,175),(36,225,103,136,176),(37,226,104,137,177),(38,227,105,138,178),(39,228,106,139,179),(40,229,107,140,180),(41,230,108,141,181),(42,231,109,142,182),(43,232,110,143,183),(44,233,111,144,184),(45,234,112,121,185),(46,235,113,122,186),(47,236,114,123,187),(48,237,115,124,188)], [(1,19,13,7),(2,20,14,8),(3,21,15,9),(4,22,16,10),(5,23,17,11),(6,24,18,12),(25,31,37,43),(26,32,38,44),(27,33,39,45),(28,34,40,46),(29,35,41,47),(30,36,42,48),(49,67,61,55),(50,68,62,56),(51,69,63,57),(52,70,64,58),(53,71,65,59),(54,72,66,60),(73,91,85,79),(74,92,86,80),(75,93,87,81),(76,94,88,82),(77,95,89,83),(78,96,90,84),(97,103,109,115),(98,104,110,116),(99,105,111,117),(100,106,112,118),(101,107,113,119),(102,108,114,120),(121,127,133,139),(122,128,134,140),(123,129,135,141),(124,130,136,142),(125,131,137,143),(126,132,138,144),(145,163,157,151),(146,164,158,152),(147,165,159,153),(148,166,160,154),(149,167,161,155),(150,168,162,156),(169,175,181,187),(170,176,182,188),(171,177,183,189),(172,178,184,190),(173,179,185,191),(174,180,186,192),(193,211,205,199),(194,212,206,200),(195,213,207,201),(196,214,208,202),(197,215,209,203),(198,216,210,204),(217,223,229,235),(218,224,230,236),(219,225,231,237),(220,226,232,238),(221,227,233,239),(222,228,234,240)], [(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)], [(1,125),(2,124),(3,123),(4,122),(5,121),(6,144),(7,143),(8,142),(9,141),(10,140),(11,139),(12,138),(13,137),(14,136),(15,135),(16,134),(17,133),(18,132),(19,131),(20,130),(21,129),(22,128),(23,127),(24,126),(25,93),(26,92),(27,91),(28,90),(29,89),(30,88),(31,87),(32,86),(33,85),(34,84),(35,83),(36,82),(37,81),(38,80),(39,79),(40,78),(41,77),(42,76),(43,75),(44,74),(45,73),(46,96),(47,95),(48,94),(49,190),(50,189),(51,188),(52,187),(53,186),(54,185),(55,184),(56,183),(57,182),(58,181),(59,180),(60,179),(61,178),(62,177),(63,176),(64,175),(65,174),(66,173),(67,172),(68,171),(69,170),(70,169),(71,192),(72,191),(97,155),(98,154),(99,153),(100,152),(101,151),(102,150),(103,149),(104,148),(105,147),(106,146),(107,145),(108,168),(109,167),(110,166),(111,165),(112,164),(113,163),(114,162),(115,161),(116,160),(117,159),(118,158),(119,157),(120,156),(193,235),(194,234),(195,233),(196,232),(197,231),(198,230),(199,229),(200,228),(201,227),(202,226),(203,225),(204,224),(205,223),(206,222),(207,221),(208,220),(209,219),(210,218),(211,217),(212,240),(213,239),(214,238),(215,237),(216,236)]])
150 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 | 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 | 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 | 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 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 12 | 12 | 2 | 1 | 1 | 2 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
150 irreducible representations
| dim | 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 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | S3 | D4 | D4 | D6 | D6 | D12 | D12 | C5×S3 | C4○D8 | C5×D4 | C5×D4 | S3×C10 | S3×C10 | C4○D24 | C5×D12 | C5×D12 | C5×C4○D8 | C5×C4○D24 |
| kernel | C5×C4○D24 | C5×C24⋊C2 | C5×D24 | C5×Dic12 | C2×C120 | C5×C4○D12 | C4○D24 | C24⋊C2 | D24 | Dic12 | C2×C24 | C4○D12 | C2×C40 | C60 | C2×C30 | C40 | C2×C20 | C20 | C2×C10 | C2×C8 | C15 | C12 | C2×C6 | C8 | C2×C4 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 8 | 4 | 4 | 4 | 8 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 4 | 8 | 8 | 8 | 16 | 32 |
Matrix representation of C5×C4○D24 ►in GL2(𝔽241) generated by
| 91 | 0 |
| 0 | 91 |
| 177 | 0 |
| 0 | 177 |
| 209 | 0 |
| 23 | 128 |
| 113 | 105 |
| 23 | 128 |
G:=sub<GL(2,GF(241))| [91,0,0,91],[177,0,0,177],[209,23,0,128],[113,23,105,128] >;
C5×C4○D24 in GAP, Magma, Sage, TeX
C_5\times C_4\circ D_{24} % in TeX
G:=Group("C5xC4oD24"); // GroupNames label
G:=SmallGroup(480,783);
// by ID
G=gap.SmallGroup(480,783);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,568,926,226,4204,102,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=d^2=1,c^12=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^11>;
// generators/relations