direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×D24⋊C2, D24⋊5C10, C40.62D6, C60.226C23, C120.69C22, (S3×C8)⋊3C10, (C5×Q16)⋊7S3, Q16⋊3(C5×S3), D6.3(C5×D4), (S3×C40)⋊12C2, (C5×D24)⋊13C2, C15⋊36(C4○D8), C8.10(S3×C10), C24.8(C2×C10), (C3×Q16)⋊3C10, C6.36(D4×C10), (C5×Q8).48D6, (C15×Q16)⋊10C2, Q8⋊2S3⋊4C10, Q8⋊3S3⋊3C10, (S3×C10).27D4, D12.5(C2×C10), C30.372(C2×D4), C10.190(S3×D4), Q8.10(S3×C10), Dic3.14(C5×D4), (C5×Dic3).51D4, (S3×C20).61C22, C20.199(C22×S3), C12.10(C22×C10), (C5×D12).34C22, (Q8×C15).36C22, C3⋊4(C5×C4○D8), C2.24(C5×S3×D4), C3⋊C8.8(C2×C10), C4.10(S3×C2×C10), (C5×C3⋊C8).44C22, (C3×Q8).5(C2×C10), (C4×S3).12(C2×C10), (C5×Q8⋊2S3)⋊12C2, (C5×Q8⋊3S3)⋊10C2, SmallGroup(480,798)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D24⋊C2
G = < a,b,c,d | a5=b24=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd=b17, dcd=b4c >
Subgroups: 324 in 124 conjugacy classes, 54 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C8, C8, C2×C4, D4, Q8, C10, C10, Dic3, C12, C12, D6, D6, C15, C2×C8, D8, SD16, Q16, C4○D4, C20, C20, C2×C10, C3⋊C8, C24, C4×S3, C4×S3, D12, D12, C3×Q8, C5×S3, C30, C4○D8, C40, C40, C2×C20, C5×D4, C5×Q8, S3×C8, D24, Q8⋊2S3, C3×Q16, Q8⋊3S3, C5×Dic3, C60, C60, S3×C10, S3×C10, C2×C40, C5×D8, C5×SD16, C5×Q16, C5×C4○D4, D24⋊C2, C5×C3⋊C8, C120, S3×C20, S3×C20, C5×D12, C5×D12, Q8×C15, C5×C4○D8, S3×C40, C5×D24, C5×Q8⋊2S3, C15×Q16, C5×Q8⋊3S3, C5×D24⋊C2
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C2×C10, C22×S3, C5×S3, C4○D8, C5×D4, C22×C10, S3×D4, S3×C10, D4×C10, D24⋊C2, S3×C2×C10, C5×C4○D8, C5×S3×D4, C5×D24⋊C2
►On 240 points
Generators in S240
(1 159 114 53 130)(2 160 115 54 131)(3 161 116 55 132)(4 162 117 56 133)(5 163 118 57 134)(6 164 119 58 135)(7 165 120 59 136)(8 166 97 60 137)(9 167 98 61 138)(10 168 99 62 139)(11 145 100 63 140)(12 146 101 64 141)(13 147 102 65 142)(14 148 103 66 143)(15 149 104 67 144)(16 150 105 68 121)(17 151 106 69 122)(18 152 107 70 123)(19 153 108 71 124)(20 154 109 72 125)(21 155 110 49 126)(22 156 111 50 127)(23 157 112 51 128)(24 158 113 52 129)(25 213 76 229 176)(26 214 77 230 177)(27 215 78 231 178)(28 216 79 232 179)(29 193 80 233 180)(30 194 81 234 181)(31 195 82 235 182)(32 196 83 236 183)(33 197 84 237 184)(34 198 85 238 185)(35 199 86 239 186)(36 200 87 240 187)(37 201 88 217 188)(38 202 89 218 189)(39 203 90 219 190)(40 204 91 220 191)(41 205 92 221 192)(42 206 93 222 169)(43 207 94 223 170)(44 208 95 224 171)(45 209 96 225 172)(46 210 73 226 173)(47 211 74 227 174)(48 212 75 228 175)
(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)
(1 210)(2 209)(3 208)(4 207)(5 206)(6 205)(7 204)(8 203)(9 202)(10 201)(11 200)(12 199)(13 198)(14 197)(15 196)(16 195)(17 194)(18 193)(19 216)(20 215)(21 214)(22 213)(23 212)(24 211)(25 127)(26 126)(27 125)(28 124)(29 123)(30 122)(31 121)(32 144)(33 143)(34 142)(35 141)(36 140)(37 139)(38 138)(39 137)(40 136)(41 135)(42 134)(43 133)(44 132)(45 131)(46 130)(47 129)(48 128)(49 177)(50 176)(51 175)(52 174)(53 173)(54 172)(55 171)(56 170)(57 169)(58 192)(59 191)(60 190)(61 189)(62 188)(63 187)(64 186)(65 185)(66 184)(67 183)(68 182)(69 181)(70 180)(71 179)(72 178)(73 159)(74 158)(75 157)(76 156)(77 155)(78 154)(79 153)(80 152)(81 151)(82 150)(83 149)(84 148)(85 147)(86 146)(87 145)(88 168)(89 167)(90 166)(91 165)(92 164)(93 163)(94 162)(95 161)(96 160)(97 219)(98 218)(99 217)(100 240)(101 239)(102 238)(103 237)(104 236)(105 235)(106 234)(107 233)(108 232)(109 231)(110 230)(111 229)(112 228)(113 227)(114 226)(115 225)(116 224)(117 223)(118 222)(119 221)(120 220)
(2 18)(3 11)(5 21)(6 14)(8 24)(9 17)(12 20)(15 23)(25 45)(26 38)(27 31)(28 48)(29 41)(30 34)(32 44)(33 37)(35 47)(36 40)(39 43)(42 46)(49 57)(51 67)(52 60)(54 70)(55 63)(58 66)(61 69)(64 72)(73 93)(74 86)(75 79)(76 96)(77 89)(78 82)(80 92)(81 85)(83 95)(84 88)(87 91)(90 94)(97 113)(98 106)(100 116)(101 109)(103 119)(104 112)(107 115)(110 118)(122 138)(123 131)(125 141)(126 134)(128 144)(129 137)(132 140)(135 143)(145 161)(146 154)(148 164)(149 157)(151 167)(152 160)(155 163)(158 166)(169 173)(170 190)(171 183)(172 176)(174 186)(175 179)(177 189)(178 182)(180 192)(181 185)(184 188)(187 191)(193 205)(194 198)(195 215)(196 208)(197 201)(199 211)(200 204)(202 214)(203 207)(206 210)(209 213)(212 216)(217 237)(218 230)(219 223)(220 240)(221 233)(222 226)(224 236)(225 229)(227 239)(228 232)(231 235)(234 238)
G:=sub<Sym(240)| (1,159,114,53,130)(2,160,115,54,131)(3,161,116,55,132)(4,162,117,56,133)(5,163,118,57,134)(6,164,119,58,135)(7,165,120,59,136)(8,166,97,60,137)(9,167,98,61,138)(10,168,99,62,139)(11,145,100,63,140)(12,146,101,64,141)(13,147,102,65,142)(14,148,103,66,143)(15,149,104,67,144)(16,150,105,68,121)(17,151,106,69,122)(18,152,107,70,123)(19,153,108,71,124)(20,154,109,72,125)(21,155,110,49,126)(22,156,111,50,127)(23,157,112,51,128)(24,158,113,52,129)(25,213,76,229,176)(26,214,77,230,177)(27,215,78,231,178)(28,216,79,232,179)(29,193,80,233,180)(30,194,81,234,181)(31,195,82,235,182)(32,196,83,236,183)(33,197,84,237,184)(34,198,85,238,185)(35,199,86,239,186)(36,200,87,240,187)(37,201,88,217,188)(38,202,89,218,189)(39,203,90,219,190)(40,204,91,220,191)(41,205,92,221,192)(42,206,93,222,169)(43,207,94,223,170)(44,208,95,224,171)(45,209,96,225,172)(46,210,73,226,173)(47,211,74,227,174)(48,212,75,228,175), (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), (1,210)(2,209)(3,208)(4,207)(5,206)(6,205)(7,204)(8,203)(9,202)(10,201)(11,200)(12,199)(13,198)(14,197)(15,196)(16,195)(17,194)(18,193)(19,216)(20,215)(21,214)(22,213)(23,212)(24,211)(25,127)(26,126)(27,125)(28,124)(29,123)(30,122)(31,121)(32,144)(33,143)(34,142)(35,141)(36,140)(37,139)(38,138)(39,137)(40,136)(41,135)(42,134)(43,133)(44,132)(45,131)(46,130)(47,129)(48,128)(49,177)(50,176)(51,175)(52,174)(53,173)(54,172)(55,171)(56,170)(57,169)(58,192)(59,191)(60,190)(61,189)(62,188)(63,187)(64,186)(65,185)(66,184)(67,183)(68,182)(69,181)(70,180)(71,179)(72,178)(73,159)(74,158)(75,157)(76,156)(77,155)(78,154)(79,153)(80,152)(81,151)(82,150)(83,149)(84,148)(85,147)(86,146)(87,145)(88,168)(89,167)(90,166)(91,165)(92,164)(93,163)(94,162)(95,161)(96,160)(97,219)(98,218)(99,217)(100,240)(101,239)(102,238)(103,237)(104,236)(105,235)(106,234)(107,233)(108,232)(109,231)(110,230)(111,229)(112,228)(113,227)(114,226)(115,225)(116,224)(117,223)(118,222)(119,221)(120,220), (2,18)(3,11)(5,21)(6,14)(8,24)(9,17)(12,20)(15,23)(25,45)(26,38)(27,31)(28,48)(29,41)(30,34)(32,44)(33,37)(35,47)(36,40)(39,43)(42,46)(49,57)(51,67)(52,60)(54,70)(55,63)(58,66)(61,69)(64,72)(73,93)(74,86)(75,79)(76,96)(77,89)(78,82)(80,92)(81,85)(83,95)(84,88)(87,91)(90,94)(97,113)(98,106)(100,116)(101,109)(103,119)(104,112)(107,115)(110,118)(122,138)(123,131)(125,141)(126,134)(128,144)(129,137)(132,140)(135,143)(145,161)(146,154)(148,164)(149,157)(151,167)(152,160)(155,163)(158,166)(169,173)(170,190)(171,183)(172,176)(174,186)(175,179)(177,189)(178,182)(180,192)(181,185)(184,188)(187,191)(193,205)(194,198)(195,215)(196,208)(197,201)(199,211)(200,204)(202,214)(203,207)(206,210)(209,213)(212,216)(217,237)(218,230)(219,223)(220,240)(221,233)(222,226)(224,236)(225,229)(227,239)(228,232)(231,235)(234,238)>;
G:=Group( (1,159,114,53,130)(2,160,115,54,131)(3,161,116,55,132)(4,162,117,56,133)(5,163,118,57,134)(6,164,119,58,135)(7,165,120,59,136)(8,166,97,60,137)(9,167,98,61,138)(10,168,99,62,139)(11,145,100,63,140)(12,146,101,64,141)(13,147,102,65,142)(14,148,103,66,143)(15,149,104,67,144)(16,150,105,68,121)(17,151,106,69,122)(18,152,107,70,123)(19,153,108,71,124)(20,154,109,72,125)(21,155,110,49,126)(22,156,111,50,127)(23,157,112,51,128)(24,158,113,52,129)(25,213,76,229,176)(26,214,77,230,177)(27,215,78,231,178)(28,216,79,232,179)(29,193,80,233,180)(30,194,81,234,181)(31,195,82,235,182)(32,196,83,236,183)(33,197,84,237,184)(34,198,85,238,185)(35,199,86,239,186)(36,200,87,240,187)(37,201,88,217,188)(38,202,89,218,189)(39,203,90,219,190)(40,204,91,220,191)(41,205,92,221,192)(42,206,93,222,169)(43,207,94,223,170)(44,208,95,224,171)(45,209,96,225,172)(46,210,73,226,173)(47,211,74,227,174)(48,212,75,228,175), (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), (1,210)(2,209)(3,208)(4,207)(5,206)(6,205)(7,204)(8,203)(9,202)(10,201)(11,200)(12,199)(13,198)(14,197)(15,196)(16,195)(17,194)(18,193)(19,216)(20,215)(21,214)(22,213)(23,212)(24,211)(25,127)(26,126)(27,125)(28,124)(29,123)(30,122)(31,121)(32,144)(33,143)(34,142)(35,141)(36,140)(37,139)(38,138)(39,137)(40,136)(41,135)(42,134)(43,133)(44,132)(45,131)(46,130)(47,129)(48,128)(49,177)(50,176)(51,175)(52,174)(53,173)(54,172)(55,171)(56,170)(57,169)(58,192)(59,191)(60,190)(61,189)(62,188)(63,187)(64,186)(65,185)(66,184)(67,183)(68,182)(69,181)(70,180)(71,179)(72,178)(73,159)(74,158)(75,157)(76,156)(77,155)(78,154)(79,153)(80,152)(81,151)(82,150)(83,149)(84,148)(85,147)(86,146)(87,145)(88,168)(89,167)(90,166)(91,165)(92,164)(93,163)(94,162)(95,161)(96,160)(97,219)(98,218)(99,217)(100,240)(101,239)(102,238)(103,237)(104,236)(105,235)(106,234)(107,233)(108,232)(109,231)(110,230)(111,229)(112,228)(113,227)(114,226)(115,225)(116,224)(117,223)(118,222)(119,221)(120,220), (2,18)(3,11)(5,21)(6,14)(8,24)(9,17)(12,20)(15,23)(25,45)(26,38)(27,31)(28,48)(29,41)(30,34)(32,44)(33,37)(35,47)(36,40)(39,43)(42,46)(49,57)(51,67)(52,60)(54,70)(55,63)(58,66)(61,69)(64,72)(73,93)(74,86)(75,79)(76,96)(77,89)(78,82)(80,92)(81,85)(83,95)(84,88)(87,91)(90,94)(97,113)(98,106)(100,116)(101,109)(103,119)(104,112)(107,115)(110,118)(122,138)(123,131)(125,141)(126,134)(128,144)(129,137)(132,140)(135,143)(145,161)(146,154)(148,164)(149,157)(151,167)(152,160)(155,163)(158,166)(169,173)(170,190)(171,183)(172,176)(174,186)(175,179)(177,189)(178,182)(180,192)(181,185)(184,188)(187,191)(193,205)(194,198)(195,215)(196,208)(197,201)(199,211)(200,204)(202,214)(203,207)(206,210)(209,213)(212,216)(217,237)(218,230)(219,223)(220,240)(221,233)(222,226)(224,236)(225,229)(227,239)(228,232)(231,235)(234,238) );
G=PermutationGroup([[(1,159,114,53,130),(2,160,115,54,131),(3,161,116,55,132),(4,162,117,56,133),(5,163,118,57,134),(6,164,119,58,135),(7,165,120,59,136),(8,166,97,60,137),(9,167,98,61,138),(10,168,99,62,139),(11,145,100,63,140),(12,146,101,64,141),(13,147,102,65,142),(14,148,103,66,143),(15,149,104,67,144),(16,150,105,68,121),(17,151,106,69,122),(18,152,107,70,123),(19,153,108,71,124),(20,154,109,72,125),(21,155,110,49,126),(22,156,111,50,127),(23,157,112,51,128),(24,158,113,52,129),(25,213,76,229,176),(26,214,77,230,177),(27,215,78,231,178),(28,216,79,232,179),(29,193,80,233,180),(30,194,81,234,181),(31,195,82,235,182),(32,196,83,236,183),(33,197,84,237,184),(34,198,85,238,185),(35,199,86,239,186),(36,200,87,240,187),(37,201,88,217,188),(38,202,89,218,189),(39,203,90,219,190),(40,204,91,220,191),(41,205,92,221,192),(42,206,93,222,169),(43,207,94,223,170),(44,208,95,224,171),(45,209,96,225,172),(46,210,73,226,173),(47,211,74,227,174),(48,212,75,228,175)], [(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)], [(1,210),(2,209),(3,208),(4,207),(5,206),(6,205),(7,204),(8,203),(9,202),(10,201),(11,200),(12,199),(13,198),(14,197),(15,196),(16,195),(17,194),(18,193),(19,216),(20,215),(21,214),(22,213),(23,212),(24,211),(25,127),(26,126),(27,125),(28,124),(29,123),(30,122),(31,121),(32,144),(33,143),(34,142),(35,141),(36,140),(37,139),(38,138),(39,137),(40,136),(41,135),(42,134),(43,133),(44,132),(45,131),(46,130),(47,129),(48,128),(49,177),(50,176),(51,175),(52,174),(53,173),(54,172),(55,171),(56,170),(57,169),(58,192),(59,191),(60,190),(61,189),(62,188),(63,187),(64,186),(65,185),(66,184),(67,183),(68,182),(69,181),(70,180),(71,179),(72,178),(73,159),(74,158),(75,157),(76,156),(77,155),(78,154),(79,153),(80,152),(81,151),(82,150),(83,149),(84,148),(85,147),(86,146),(87,145),(88,168),(89,167),(90,166),(91,165),(92,164),(93,163),(94,162),(95,161),(96,160),(97,219),(98,218),(99,217),(100,240),(101,239),(102,238),(103,237),(104,236),(105,235),(106,234),(107,233),(108,232),(109,231),(110,230),(111,229),(112,228),(113,227),(114,226),(115,225),(116,224),(117,223),(118,222),(119,221),(120,220)], [(2,18),(3,11),(5,21),(6,14),(8,24),(9,17),(12,20),(15,23),(25,45),(26,38),(27,31),(28,48),(29,41),(30,34),(32,44),(33,37),(35,47),(36,40),(39,43),(42,46),(49,57),(51,67),(52,60),(54,70),(55,63),(58,66),(61,69),(64,72),(73,93),(74,86),(75,79),(76,96),(77,89),(78,82),(80,92),(81,85),(83,95),(84,88),(87,91),(90,94),(97,113),(98,106),(100,116),(101,109),(103,119),(104,112),(107,115),(110,118),(122,138),(123,131),(125,141),(126,134),(128,144),(129,137),(132,140),(135,143),(145,161),(146,154),(148,164),(149,157),(151,167),(152,160),(155,163),(158,166),(169,173),(170,190),(171,183),(172,176),(174,186),(175,179),(177,189),(178,182),(180,192),(181,185),(184,188),(187,191),(193,205),(194,198),(195,215),(196,208),(197,201),(199,211),(200,204),(202,214),(203,207),(206,210),(209,213),(212,216),(217,237),(218,230),(219,223),(220,240),(221,233),(222,226),(224,236),(225,229),(227,239),(228,232),(231,235),(234,238)]])
105 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 6 | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10P | 12A | 12B | 12C | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 20M | ··· | 20T | 24A | 24B | 30A | 30B | 30C | 30D | 40A | ··· | 40H | 40I | ··· | 40P | 60A | 60B | 60C | 60D | 60E | ··· | 60L | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 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 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 40 | ··· | 40 | 60 | 60 | 60 | 60 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 6 | 12 | 12 | 2 | 2 | 3 | 3 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | ··· | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
105 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 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | S3 | D4 | D4 | D6 | D6 | C5×S3 | C4○D8 | C5×D4 | C5×D4 | S3×C10 | S3×C10 | C5×C4○D8 | S3×D4 | D24⋊C2 | C5×S3×D4 | C5×D24⋊C2 |
| kernel | C5×D24⋊C2 | S3×C40 | C5×D24 | C5×Q8⋊2S3 | C15×Q16 | C5×Q8⋊3S3 | D24⋊C2 | S3×C8 | D24 | Q8⋊2S3 | C3×Q16 | Q8⋊3S3 | C5×Q16 | C5×Dic3 | S3×C10 | C40 | C5×Q8 | Q16 | C15 | Dic3 | D6 | C8 | Q8 | C3 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 8 | 4 | 8 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 16 | 1 | 2 | 4 | 8 |
Matrix representation of C5×D24⋊C2 ►in GL4(𝔽241) generated by
| 87 | 0 | 0 | 0 |
| 0 | 87 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 |
| 0 | 0 | 8 | 19 |
| 0 | 0 | 0 | 211 |
| 1 | 1 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 233 | 222 |
| 0 | 0 | 16 | 8 |
| 1 | 0 | 0 | 0 |
| 240 | 240 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 240 |
G:=sub<GL(4,GF(241))| [87,0,0,0,0,87,0,0,0,0,1,0,0,0,0,1],[1,240,0,0,1,0,0,0,0,0,8,0,0,0,19,211],[1,0,0,0,1,240,0,0,0,0,233,16,0,0,222,8],[1,240,0,0,0,240,0,0,0,0,1,0,0,0,1,240] >;
C5×D24⋊C2 in GAP, Magma, Sage, TeX
C_5\times D_{24}\rtimes C_2 % in TeX
G:=Group("C5xD24:C2"); // GroupNames label
G:=SmallGroup(480,798);
// by ID
G=gap.SmallGroup(480,798);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,568,1766,471,436,2111,1068,102,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^24=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d=b^17,d*c*d=b^4*c>;
// generators/relations