direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C2.D24, D12⋊2C20, C30.36D8, C60.223D4, C10.17D24, C30.24SD16, (C2×C40)⋊2S3, C6.5(C5×D8), (C2×C24)⋊2C10, (C2×C120)⋊4C2, C4.8(S3×C20), C2.2(C5×D24), (C5×D12)⋊14C4, C20.97(C4×S3), C4⋊Dic3⋊1C10, C12.45(C5×D4), C6.3(C5×SD16), C12.18(C2×C20), C60.205(C2×C4), (C2×D12).1C10, (C2×C10).51D12, (C2×C20).424D6, (C2×C30).121D4, C15⋊16(D4⋊C4), C10.54(D6⋊C4), (C10×D12).10C2, C22.10(C5×D12), C10.11(C24⋊C2), C20.113(C3⋊D4), C30.96(C22⋊C4), (C2×C60).524C22, (C2×C8)⋊2(C5×S3), C3⋊2(C5×D4⋊C4), C2.8(C5×D6⋊C4), C2.3(C5×C24⋊C2), (C2×C6).15(C5×D4), C4.20(C5×C3⋊D4), C6.7(C5×C22⋊C4), (C2×C4).74(S3×C10), (C5×C4⋊Dic3)⋊13C2, (C2×C12).88(C2×C10), SmallGroup(480,140)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C2.D24
G = < a,b,c,d | a5=b2=c24=1, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=bc-1 >
Subgroups: 324 in 100 conjugacy classes, 46 normal (42 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C8, C2×C4, C2×C4, D4, C23, C10, C10, Dic3, C12, D6, C2×C6, C15, C4⋊C4, C2×C8, C2×D4, C20, C20, C2×C10, C2×C10, C24, D12, D12, C2×Dic3, C2×C12, C22×S3, C5×S3, C30, D4⋊C4, C40, C2×C20, C2×C20, C5×D4, C22×C10, C4⋊Dic3, C2×C24, C2×D12, C5×Dic3, C60, S3×C10, C2×C30, C5×C4⋊C4, C2×C40, D4×C10, C2.D24, C120, C5×D12, C5×D12, C10×Dic3, C2×C60, S3×C2×C10, C5×D4⋊C4, C5×C4⋊Dic3, C2×C120, C10×D12, C5×C2.D24
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, C10, D6, C22⋊C4, D8, SD16, C20, C2×C10, C4×S3, D12, C3⋊D4, C5×S3, D4⋊C4, C2×C20, C5×D4, C24⋊C2, D24, D6⋊C4, S3×C10, C5×C22⋊C4, C5×D8, C5×SD16, C2.D24, S3×C20, C5×D12, C5×C3⋊D4, C5×D4⋊C4, C5×C24⋊C2, C5×D24, C5×D6⋊C4, C5×C2.D24
►On 240 points
Generators in S240
(1 96 97 71 155)(2 73 98 72 156)(3 74 99 49 157)(4 75 100 50 158)(5 76 101 51 159)(6 77 102 52 160)(7 78 103 53 161)(8 79 104 54 162)(9 80 105 55 163)(10 81 106 56 164)(11 82 107 57 165)(12 83 108 58 166)(13 84 109 59 167)(14 85 110 60 168)(15 86 111 61 145)(16 87 112 62 146)(17 88 113 63 147)(18 89 114 64 148)(19 90 115 65 149)(20 91 116 66 150)(21 92 117 67 151)(22 93 118 68 152)(23 94 119 69 153)(24 95 120 70 154)(25 130 174 222 213)(26 131 175 223 214)(27 132 176 224 215)(28 133 177 225 216)(29 134 178 226 193)(30 135 179 227 194)(31 136 180 228 195)(32 137 181 229 196)(33 138 182 230 197)(34 139 183 231 198)(35 140 184 232 199)(36 141 185 233 200)(37 142 186 234 201)(38 143 187 235 202)(39 144 188 236 203)(40 121 189 237 204)(41 122 190 238 205)(42 123 191 239 206)(43 124 192 240 207)(44 125 169 217 208)(45 126 170 218 209)(46 127 171 219 210)(47 128 172 220 211)(48 129 173 221 212)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 31)(21 32)(22 33)(23 34)(24 35)(49 235)(50 236)(51 237)(52 238)(53 239)(54 240)(55 217)(56 218)(57 219)(58 220)(59 221)(60 222)(61 223)(62 224)(63 225)(64 226)(65 227)(66 228)(67 229)(68 230)(69 231)(70 232)(71 233)(72 234)(73 142)(74 143)(75 144)(76 121)(77 122)(78 123)(79 124)(80 125)(81 126)(82 127)(83 128)(84 129)(85 130)(86 131)(87 132)(88 133)(89 134)(90 135)(91 136)(92 137)(93 138)(94 139)(95 140)(96 141)(97 185)(98 186)(99 187)(100 188)(101 189)(102 190)(103 191)(104 192)(105 169)(106 170)(107 171)(108 172)(109 173)(110 174)(111 175)(112 176)(113 177)(114 178)(115 179)(116 180)(117 181)(118 182)(119 183)(120 184)(145 214)(146 215)(147 216)(148 193)(149 194)(150 195)(151 196)(152 197)(153 198)(154 199)(155 200)(156 201)(157 202)(158 203)(159 204)(160 205)(161 206)(162 207)(163 208)(164 209)(165 210)(166 211)(167 212)(168 213)
(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 35 36 24)(2 23 37 34)(3 33 38 22)(4 21 39 32)(5 31 40 20)(6 19 41 30)(7 29 42 18)(8 17 43 28)(9 27 44 16)(10 15 45 26)(11 25 46 14)(12 13 47 48)(49 230 235 68)(50 67 236 229)(51 228 237 66)(52 65 238 227)(53 226 239 64)(54 63 240 225)(55 224 217 62)(56 61 218 223)(57 222 219 60)(58 59 220 221)(69 234 231 72)(70 71 232 233)(73 94 142 139)(74 138 143 93)(75 92 144 137)(76 136 121 91)(77 90 122 135)(78 134 123 89)(79 88 124 133)(80 132 125 87)(81 86 126 131)(82 130 127 85)(83 84 128 129)(95 96 140 141)(97 184 185 120)(98 119 186 183)(99 182 187 118)(100 117 188 181)(101 180 189 116)(102 115 190 179)(103 178 191 114)(104 113 192 177)(105 176 169 112)(106 111 170 175)(107 174 171 110)(108 109 172 173)(145 209 214 164)(146 163 215 208)(147 207 216 162)(148 161 193 206)(149 205 194 160)(150 159 195 204)(151 203 196 158)(152 157 197 202)(153 201 198 156)(154 155 199 200)(165 213 210 168)(166 167 211 212)
G:=sub<Sym(240)| (1,96,97,71,155)(2,73,98,72,156)(3,74,99,49,157)(4,75,100,50,158)(5,76,101,51,159)(6,77,102,52,160)(7,78,103,53,161)(8,79,104,54,162)(9,80,105,55,163)(10,81,106,56,164)(11,82,107,57,165)(12,83,108,58,166)(13,84,109,59,167)(14,85,110,60,168)(15,86,111,61,145)(16,87,112,62,146)(17,88,113,63,147)(18,89,114,64,148)(19,90,115,65,149)(20,91,116,66,150)(21,92,117,67,151)(22,93,118,68,152)(23,94,119,69,153)(24,95,120,70,154)(25,130,174,222,213)(26,131,175,223,214)(27,132,176,224,215)(28,133,177,225,216)(29,134,178,226,193)(30,135,179,227,194)(31,136,180,228,195)(32,137,181,229,196)(33,138,182,230,197)(34,139,183,231,198)(35,140,184,232,199)(36,141,185,233,200)(37,142,186,234,201)(38,143,187,235,202)(39,144,188,236,203)(40,121,189,237,204)(41,122,190,238,205)(42,123,191,239,206)(43,124,192,240,207)(44,125,169,217,208)(45,126,170,218,209)(46,127,171,219,210)(47,128,172,220,211)(48,129,173,221,212), (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33)(23,34)(24,35)(49,235)(50,236)(51,237)(52,238)(53,239)(54,240)(55,217)(56,218)(57,219)(58,220)(59,221)(60,222)(61,223)(62,224)(63,225)(64,226)(65,227)(66,228)(67,229)(68,230)(69,231)(70,232)(71,233)(72,234)(73,142)(74,143)(75,144)(76,121)(77,122)(78,123)(79,124)(80,125)(81,126)(82,127)(83,128)(84,129)(85,130)(86,131)(87,132)(88,133)(89,134)(90,135)(91,136)(92,137)(93,138)(94,139)(95,140)(96,141)(97,185)(98,186)(99,187)(100,188)(101,189)(102,190)(103,191)(104,192)(105,169)(106,170)(107,171)(108,172)(109,173)(110,174)(111,175)(112,176)(113,177)(114,178)(115,179)(116,180)(117,181)(118,182)(119,183)(120,184)(145,214)(146,215)(147,216)(148,193)(149,194)(150,195)(151,196)(152,197)(153,198)(154,199)(155,200)(156,201)(157,202)(158,203)(159,204)(160,205)(161,206)(162,207)(163,208)(164,209)(165,210)(166,211)(167,212)(168,213), (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), (1,35,36,24)(2,23,37,34)(3,33,38,22)(4,21,39,32)(5,31,40,20)(6,19,41,30)(7,29,42,18)(8,17,43,28)(9,27,44,16)(10,15,45,26)(11,25,46,14)(12,13,47,48)(49,230,235,68)(50,67,236,229)(51,228,237,66)(52,65,238,227)(53,226,239,64)(54,63,240,225)(55,224,217,62)(56,61,218,223)(57,222,219,60)(58,59,220,221)(69,234,231,72)(70,71,232,233)(73,94,142,139)(74,138,143,93)(75,92,144,137)(76,136,121,91)(77,90,122,135)(78,134,123,89)(79,88,124,133)(80,132,125,87)(81,86,126,131)(82,130,127,85)(83,84,128,129)(95,96,140,141)(97,184,185,120)(98,119,186,183)(99,182,187,118)(100,117,188,181)(101,180,189,116)(102,115,190,179)(103,178,191,114)(104,113,192,177)(105,176,169,112)(106,111,170,175)(107,174,171,110)(108,109,172,173)(145,209,214,164)(146,163,215,208)(147,207,216,162)(148,161,193,206)(149,205,194,160)(150,159,195,204)(151,203,196,158)(152,157,197,202)(153,201,198,156)(154,155,199,200)(165,213,210,168)(166,167,211,212)>;
G:=Group( (1,96,97,71,155)(2,73,98,72,156)(3,74,99,49,157)(4,75,100,50,158)(5,76,101,51,159)(6,77,102,52,160)(7,78,103,53,161)(8,79,104,54,162)(9,80,105,55,163)(10,81,106,56,164)(11,82,107,57,165)(12,83,108,58,166)(13,84,109,59,167)(14,85,110,60,168)(15,86,111,61,145)(16,87,112,62,146)(17,88,113,63,147)(18,89,114,64,148)(19,90,115,65,149)(20,91,116,66,150)(21,92,117,67,151)(22,93,118,68,152)(23,94,119,69,153)(24,95,120,70,154)(25,130,174,222,213)(26,131,175,223,214)(27,132,176,224,215)(28,133,177,225,216)(29,134,178,226,193)(30,135,179,227,194)(31,136,180,228,195)(32,137,181,229,196)(33,138,182,230,197)(34,139,183,231,198)(35,140,184,232,199)(36,141,185,233,200)(37,142,186,234,201)(38,143,187,235,202)(39,144,188,236,203)(40,121,189,237,204)(41,122,190,238,205)(42,123,191,239,206)(43,124,192,240,207)(44,125,169,217,208)(45,126,170,218,209)(46,127,171,219,210)(47,128,172,220,211)(48,129,173,221,212), (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33)(23,34)(24,35)(49,235)(50,236)(51,237)(52,238)(53,239)(54,240)(55,217)(56,218)(57,219)(58,220)(59,221)(60,222)(61,223)(62,224)(63,225)(64,226)(65,227)(66,228)(67,229)(68,230)(69,231)(70,232)(71,233)(72,234)(73,142)(74,143)(75,144)(76,121)(77,122)(78,123)(79,124)(80,125)(81,126)(82,127)(83,128)(84,129)(85,130)(86,131)(87,132)(88,133)(89,134)(90,135)(91,136)(92,137)(93,138)(94,139)(95,140)(96,141)(97,185)(98,186)(99,187)(100,188)(101,189)(102,190)(103,191)(104,192)(105,169)(106,170)(107,171)(108,172)(109,173)(110,174)(111,175)(112,176)(113,177)(114,178)(115,179)(116,180)(117,181)(118,182)(119,183)(120,184)(145,214)(146,215)(147,216)(148,193)(149,194)(150,195)(151,196)(152,197)(153,198)(154,199)(155,200)(156,201)(157,202)(158,203)(159,204)(160,205)(161,206)(162,207)(163,208)(164,209)(165,210)(166,211)(167,212)(168,213), (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), (1,35,36,24)(2,23,37,34)(3,33,38,22)(4,21,39,32)(5,31,40,20)(6,19,41,30)(7,29,42,18)(8,17,43,28)(9,27,44,16)(10,15,45,26)(11,25,46,14)(12,13,47,48)(49,230,235,68)(50,67,236,229)(51,228,237,66)(52,65,238,227)(53,226,239,64)(54,63,240,225)(55,224,217,62)(56,61,218,223)(57,222,219,60)(58,59,220,221)(69,234,231,72)(70,71,232,233)(73,94,142,139)(74,138,143,93)(75,92,144,137)(76,136,121,91)(77,90,122,135)(78,134,123,89)(79,88,124,133)(80,132,125,87)(81,86,126,131)(82,130,127,85)(83,84,128,129)(95,96,140,141)(97,184,185,120)(98,119,186,183)(99,182,187,118)(100,117,188,181)(101,180,189,116)(102,115,190,179)(103,178,191,114)(104,113,192,177)(105,176,169,112)(106,111,170,175)(107,174,171,110)(108,109,172,173)(145,209,214,164)(146,163,215,208)(147,207,216,162)(148,161,193,206)(149,205,194,160)(150,159,195,204)(151,203,196,158)(152,157,197,202)(153,201,198,156)(154,155,199,200)(165,213,210,168)(166,167,211,212) );
G=PermutationGroup([[(1,96,97,71,155),(2,73,98,72,156),(3,74,99,49,157),(4,75,100,50,158),(5,76,101,51,159),(6,77,102,52,160),(7,78,103,53,161),(8,79,104,54,162),(9,80,105,55,163),(10,81,106,56,164),(11,82,107,57,165),(12,83,108,58,166),(13,84,109,59,167),(14,85,110,60,168),(15,86,111,61,145),(16,87,112,62,146),(17,88,113,63,147),(18,89,114,64,148),(19,90,115,65,149),(20,91,116,66,150),(21,92,117,67,151),(22,93,118,68,152),(23,94,119,69,153),(24,95,120,70,154),(25,130,174,222,213),(26,131,175,223,214),(27,132,176,224,215),(28,133,177,225,216),(29,134,178,226,193),(30,135,179,227,194),(31,136,180,228,195),(32,137,181,229,196),(33,138,182,230,197),(34,139,183,231,198),(35,140,184,232,199),(36,141,185,233,200),(37,142,186,234,201),(38,143,187,235,202),(39,144,188,236,203),(40,121,189,237,204),(41,122,190,238,205),(42,123,191,239,206),(43,124,192,240,207),(44,125,169,217,208),(45,126,170,218,209),(46,127,171,219,210),(47,128,172,220,211),(48,129,173,221,212)], [(1,36),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,43),(9,44),(10,45),(11,46),(12,47),(13,48),(14,25),(15,26),(16,27),(17,28),(18,29),(19,30),(20,31),(21,32),(22,33),(23,34),(24,35),(49,235),(50,236),(51,237),(52,238),(53,239),(54,240),(55,217),(56,218),(57,219),(58,220),(59,221),(60,222),(61,223),(62,224),(63,225),(64,226),(65,227),(66,228),(67,229),(68,230),(69,231),(70,232),(71,233),(72,234),(73,142),(74,143),(75,144),(76,121),(77,122),(78,123),(79,124),(80,125),(81,126),(82,127),(83,128),(84,129),(85,130),(86,131),(87,132),(88,133),(89,134),(90,135),(91,136),(92,137),(93,138),(94,139),(95,140),(96,141),(97,185),(98,186),(99,187),(100,188),(101,189),(102,190),(103,191),(104,192),(105,169),(106,170),(107,171),(108,172),(109,173),(110,174),(111,175),(112,176),(113,177),(114,178),(115,179),(116,180),(117,181),(118,182),(119,183),(120,184),(145,214),(146,215),(147,216),(148,193),(149,194),(150,195),(151,196),(152,197),(153,198),(154,199),(155,200),(156,201),(157,202),(158,203),(159,204),(160,205),(161,206),(162,207),(163,208),(164,209),(165,210),(166,211),(167,212),(168,213)], [(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)], [(1,35,36,24),(2,23,37,34),(3,33,38,22),(4,21,39,32),(5,31,40,20),(6,19,41,30),(7,29,42,18),(8,17,43,28),(9,27,44,16),(10,15,45,26),(11,25,46,14),(12,13,47,48),(49,230,235,68),(50,67,236,229),(51,228,237,66),(52,65,238,227),(53,226,239,64),(54,63,240,225),(55,224,217,62),(56,61,218,223),(57,222,219,60),(58,59,220,221),(69,234,231,72),(70,71,232,233),(73,94,142,139),(74,138,143,93),(75,92,144,137),(76,136,121,91),(77,90,122,135),(78,134,123,89),(79,88,124,133),(80,132,125,87),(81,86,126,131),(82,130,127,85),(83,84,128,129),(95,96,140,141),(97,184,185,120),(98,119,186,183),(99,182,187,118),(100,117,188,181),(101,180,189,116),(102,115,190,179),(103,178,191,114),(104,113,192,177),(105,176,169,112),(106,111,170,175),(107,174,171,110),(108,109,172,173),(145,209,214,164),(146,163,215,208),(147,207,216,162),(148,161,193,206),(149,205,194,160),(150,159,195,204),(151,203,196,158),(152,157,197,202),(153,201,198,156),(154,155,199,200),(165,213,210,168),(166,167,211,212)]])
150 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | ··· | 20P | 24A | ··· | 24H | 30A | ··· | 30L | 40A | ··· | 40P | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | 2 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C20 | S3 | D4 | D4 | D6 | D8 | SD16 | C4×S3 | C3⋊D4 | D12 | C5×S3 | C5×D4 | C5×D4 | C24⋊C2 | D24 | S3×C10 | C5×D8 | C5×SD16 | S3×C20 | C5×C3⋊D4 | C5×D12 | C5×C24⋊C2 | C5×D24 |
| kernel | C5×C2.D24 | C5×C4⋊Dic3 | C2×C120 | C10×D12 | C5×D12 | C2.D24 | C4⋊Dic3 | C2×C24 | C2×D12 | D12 | C2×C40 | C60 | C2×C30 | C2×C20 | C30 | C30 | C20 | C20 | C2×C10 | C2×C8 | C12 | C2×C6 | C10 | C10 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 16 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 16 | 16 |
Matrix representation of C5×C2.D24 ►in GL3(𝔽241) generated by
| 1 | 0 | 0 |
| 0 | 91 | 0 |
| 0 | 0 | 91 |
| 240 | 0 | 0 |
| 0 | 240 | 0 |
| 0 | 0 | 240 |
| 177 | 0 | 0 |
| 0 | 94 | 213 |
| 0 | 28 | 66 |
| 64 | 0 | 0 |
| 0 | 175 | 94 |
| 0 | 28 | 66 |
G:=sub<GL(3,GF(241))| [1,0,0,0,91,0,0,0,91],[240,0,0,0,240,0,0,0,240],[177,0,0,0,94,28,0,213,66],[64,0,0,0,175,28,0,94,66] >;
C5×C2.D24 in GAP, Magma, Sage, TeX
C_5\times C_2.D_{24} % in TeX
G:=Group("C5xC2.D24"); // GroupNames label
G:=SmallGroup(480,140);
// by ID
G=gap.SmallGroup(480,140);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,309,428,2803,136,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^24=1,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations