direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C5⋊D24, C30⋊3D8, C10⋊2D24, C60.61D4, D12⋊18D10, C20.13D12, D60⋊35C22, C60.126C23, C15⋊6(C2×D8), C5⋊3(C2×D24), C6⋊1(D4⋊D5), (C2×D12)⋊1D5, C5⋊2C8⋊25D6, (C10×D12)⋊4C2, (C2×D60)⋊25C2, C30.77(C2×D4), (C2×C20).88D6, (C2×C30).45D4, (C2×C10).38D12, C10.47(C2×D12), C4.5(C5⋊D12), (C2×C12).289D10, (C5×D12)⋊20C22, C12.55(C5⋊D4), C20.88(C22×S3), (C2×C60).133C22, C12.149(C22×D5), C22.19(C5⋊D12), C3⋊1(C2×D4⋊D5), C4.74(C2×S3×D5), (C6×C5⋊2C8)⋊6C2, (C2×C5⋊2C8)⋊5S3, C6.1(C2×C5⋊D4), C2.5(C2×C5⋊D12), (C2×C4).145(S3×D5), (C3×C5⋊2C8)⋊29C22, (C2×C6).30(C5⋊D4), SmallGroup(480,378)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C5⋊D24
G = < a,b,c,d | a2=b5=c24=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 1052 in 152 conjugacy classes, 52 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, D4, C23, D5, C10, C10, C10, C12, D6, C2×C6, C15, C2×C8, D8, C2×D4, C20, D10, C2×C10, C2×C10, C24, D12, D12, C2×C12, C22×S3, C5×S3, D15, C30, C30, C2×D8, C5⋊2C8, D20, C2×C20, C5×D4, C22×D5, C22×C10, D24, C2×C24, C2×D12, C2×D12, C60, S3×C10, D30, C2×C30, C2×C5⋊2C8, D4⋊D5, C2×D20, D4×C10, C2×D24, C3×C5⋊2C8, C5×D12, C5×D12, D60, D60, C2×C60, S3×C2×C10, C22×D15, C2×D4⋊D5, C5⋊D24, C6×C5⋊2C8, C10×D12, C2×D60, C2×C5⋊D24
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, D8, C2×D4, D10, D12, C22×S3, C2×D8, C5⋊D4, C22×D5, D24, C2×D12, S3×D5, D4⋊D5, C2×C5⋊D4, C2×D24, C5⋊D12, C2×S3×D5, C2×D4⋊D5, C5⋊D24, C2×C5⋊D12, C2×C5⋊D24
►On 240 points
Generators in S240
(1 161)(2 162)(3 163)(4 164)(5 165)(6 166)(7 167)(8 168)(9 145)(10 146)(11 147)(12 148)(13 149)(14 150)(15 151)(16 152)(17 153)(18 154)(19 155)(20 156)(21 157)(22 158)(23 159)(24 160)(25 70)(26 71)(27 72)(28 49)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 57)(37 58)(38 59)(39 60)(40 61)(41 62)(42 63)(43 64)(44 65)(45 66)(46 67)(47 68)(48 69)(73 118)(74 119)(75 120)(76 97)(77 98)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)(85 106)(86 107)(87 108)(88 109)(89 110)(90 111)(91 112)(92 113)(93 114)(94 115)(95 116)(96 117)(121 173)(122 174)(123 175)(124 176)(125 177)(126 178)(127 179)(128 180)(129 181)(130 182)(131 183)(132 184)(133 185)(134 186)(135 187)(136 188)(137 189)(138 190)(139 191)(140 192)(141 169)(142 170)(143 171)(144 172)(193 226)(194 227)(195 228)(196 229)(197 230)(198 231)(199 232)(200 233)(201 234)(202 235)(203 236)(204 237)(205 238)(206 239)(207 240)(208 217)(209 218)(210 219)(211 220)(212 221)(213 222)(214 223)(215 224)(216 225)
(1 177 60 109 202)(2 203 110 61 178)(3 179 62 111 204)(4 205 112 63 180)(5 181 64 113 206)(6 207 114 65 182)(7 183 66 115 208)(8 209 116 67 184)(9 185 68 117 210)(10 211 118 69 186)(11 187 70 119 212)(12 213 120 71 188)(13 189 72 97 214)(14 215 98 49 190)(15 191 50 99 216)(16 193 100 51 192)(17 169 52 101 194)(18 195 102 53 170)(19 171 54 103 196)(20 197 104 55 172)(21 173 56 105 198)(22 199 106 57 174)(23 175 58 107 200)(24 201 108 59 176)(25 74 221 147 135)(26 136 148 222 75)(27 76 223 149 137)(28 138 150 224 77)(29 78 225 151 139)(30 140 152 226 79)(31 80 227 153 141)(32 142 154 228 81)(33 82 229 155 143)(34 144 156 230 83)(35 84 231 157 121)(36 122 158 232 85)(37 86 233 159 123)(38 124 160 234 87)(39 88 235 161 125)(40 126 162 236 89)(41 90 237 163 127)(42 128 164 238 91)(43 92 239 165 129)(44 130 166 240 93)(45 94 217 167 131)(46 132 168 218 95)(47 96 219 145 133)(48 134 146 220 73)
(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 9)(2 8)(3 7)(4 6)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(25 86)(26 85)(27 84)(28 83)(29 82)(30 81)(31 80)(32 79)(33 78)(34 77)(35 76)(36 75)(37 74)(38 73)(39 96)(40 95)(41 94)(42 93)(43 92)(44 91)(45 90)(46 89)(47 88)(48 87)(49 104)(50 103)(51 102)(52 101)(53 100)(54 99)(55 98)(56 97)(57 120)(58 119)(59 118)(60 117)(61 116)(62 115)(63 114)(64 113)(65 112)(66 111)(67 110)(68 109)(69 108)(70 107)(71 106)(72 105)(121 223)(122 222)(123 221)(124 220)(125 219)(126 218)(127 217)(128 240)(129 239)(130 238)(131 237)(132 236)(133 235)(134 234)(135 233)(136 232)(137 231)(138 230)(139 229)(140 228)(141 227)(142 226)(143 225)(144 224)(145 161)(146 160)(147 159)(148 158)(149 157)(150 156)(151 155)(152 154)(162 168)(163 167)(164 166)(169 194)(170 193)(171 216)(172 215)(173 214)(174 213)(175 212)(176 211)(177 210)(178 209)(179 208)(180 207)(181 206)(182 205)(183 204)(184 203)(185 202)(186 201)(187 200)(188 199)(189 198)(190 197)(191 196)(192 195)
G:=sub<Sym(240)| (1,161)(2,162)(3,163)(4,164)(5,165)(6,166)(7,167)(8,168)(9,145)(10,146)(11,147)(12,148)(13,149)(14,150)(15,151)(16,152)(17,153)(18,154)(19,155)(20,156)(21,157)(22,158)(23,159)(24,160)(25,70)(26,71)(27,72)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63)(43,64)(44,65)(45,66)(46,67)(47,68)(48,69)(73,118)(74,119)(75,120)(76,97)(77,98)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105)(85,106)(86,107)(87,108)(88,109)(89,110)(90,111)(91,112)(92,113)(93,114)(94,115)(95,116)(96,117)(121,173)(122,174)(123,175)(124,176)(125,177)(126,178)(127,179)(128,180)(129,181)(130,182)(131,183)(132,184)(133,185)(134,186)(135,187)(136,188)(137,189)(138,190)(139,191)(140,192)(141,169)(142,170)(143,171)(144,172)(193,226)(194,227)(195,228)(196,229)(197,230)(198,231)(199,232)(200,233)(201,234)(202,235)(203,236)(204,237)(205,238)(206,239)(207,240)(208,217)(209,218)(210,219)(211,220)(212,221)(213,222)(214,223)(215,224)(216,225), (1,177,60,109,202)(2,203,110,61,178)(3,179,62,111,204)(4,205,112,63,180)(5,181,64,113,206)(6,207,114,65,182)(7,183,66,115,208)(8,209,116,67,184)(9,185,68,117,210)(10,211,118,69,186)(11,187,70,119,212)(12,213,120,71,188)(13,189,72,97,214)(14,215,98,49,190)(15,191,50,99,216)(16,193,100,51,192)(17,169,52,101,194)(18,195,102,53,170)(19,171,54,103,196)(20,197,104,55,172)(21,173,56,105,198)(22,199,106,57,174)(23,175,58,107,200)(24,201,108,59,176)(25,74,221,147,135)(26,136,148,222,75)(27,76,223,149,137)(28,138,150,224,77)(29,78,225,151,139)(30,140,152,226,79)(31,80,227,153,141)(32,142,154,228,81)(33,82,229,155,143)(34,144,156,230,83)(35,84,231,157,121)(36,122,158,232,85)(37,86,233,159,123)(38,124,160,234,87)(39,88,235,161,125)(40,126,162,236,89)(41,90,237,163,127)(42,128,164,238,91)(43,92,239,165,129)(44,130,166,240,93)(45,94,217,167,131)(46,132,168,218,95)(47,96,219,145,133)(48,134,146,220,73), (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), (1,9)(2,8)(3,7)(4,6)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(25,86)(26,85)(27,84)(28,83)(29,82)(30,81)(31,80)(32,79)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,96)(40,95)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,104)(50,103)(51,102)(52,101)(53,100)(54,99)(55,98)(56,97)(57,120)(58,119)(59,118)(60,117)(61,116)(62,115)(63,114)(64,113)(65,112)(66,111)(67,110)(68,109)(69,108)(70,107)(71,106)(72,105)(121,223)(122,222)(123,221)(124,220)(125,219)(126,218)(127,217)(128,240)(129,239)(130,238)(131,237)(132,236)(133,235)(134,234)(135,233)(136,232)(137,231)(138,230)(139,229)(140,228)(141,227)(142,226)(143,225)(144,224)(145,161)(146,160)(147,159)(148,158)(149,157)(150,156)(151,155)(152,154)(162,168)(163,167)(164,166)(169,194)(170,193)(171,216)(172,215)(173,214)(174,213)(175,212)(176,211)(177,210)(178,209)(179,208)(180,207)(181,206)(182,205)(183,204)(184,203)(185,202)(186,201)(187,200)(188,199)(189,198)(190,197)(191,196)(192,195)>;
G:=Group( (1,161)(2,162)(3,163)(4,164)(5,165)(6,166)(7,167)(8,168)(9,145)(10,146)(11,147)(12,148)(13,149)(14,150)(15,151)(16,152)(17,153)(18,154)(19,155)(20,156)(21,157)(22,158)(23,159)(24,160)(25,70)(26,71)(27,72)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63)(43,64)(44,65)(45,66)(46,67)(47,68)(48,69)(73,118)(74,119)(75,120)(76,97)(77,98)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105)(85,106)(86,107)(87,108)(88,109)(89,110)(90,111)(91,112)(92,113)(93,114)(94,115)(95,116)(96,117)(121,173)(122,174)(123,175)(124,176)(125,177)(126,178)(127,179)(128,180)(129,181)(130,182)(131,183)(132,184)(133,185)(134,186)(135,187)(136,188)(137,189)(138,190)(139,191)(140,192)(141,169)(142,170)(143,171)(144,172)(193,226)(194,227)(195,228)(196,229)(197,230)(198,231)(199,232)(200,233)(201,234)(202,235)(203,236)(204,237)(205,238)(206,239)(207,240)(208,217)(209,218)(210,219)(211,220)(212,221)(213,222)(214,223)(215,224)(216,225), (1,177,60,109,202)(2,203,110,61,178)(3,179,62,111,204)(4,205,112,63,180)(5,181,64,113,206)(6,207,114,65,182)(7,183,66,115,208)(8,209,116,67,184)(9,185,68,117,210)(10,211,118,69,186)(11,187,70,119,212)(12,213,120,71,188)(13,189,72,97,214)(14,215,98,49,190)(15,191,50,99,216)(16,193,100,51,192)(17,169,52,101,194)(18,195,102,53,170)(19,171,54,103,196)(20,197,104,55,172)(21,173,56,105,198)(22,199,106,57,174)(23,175,58,107,200)(24,201,108,59,176)(25,74,221,147,135)(26,136,148,222,75)(27,76,223,149,137)(28,138,150,224,77)(29,78,225,151,139)(30,140,152,226,79)(31,80,227,153,141)(32,142,154,228,81)(33,82,229,155,143)(34,144,156,230,83)(35,84,231,157,121)(36,122,158,232,85)(37,86,233,159,123)(38,124,160,234,87)(39,88,235,161,125)(40,126,162,236,89)(41,90,237,163,127)(42,128,164,238,91)(43,92,239,165,129)(44,130,166,240,93)(45,94,217,167,131)(46,132,168,218,95)(47,96,219,145,133)(48,134,146,220,73), (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), (1,9)(2,8)(3,7)(4,6)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(25,86)(26,85)(27,84)(28,83)(29,82)(30,81)(31,80)(32,79)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,96)(40,95)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,104)(50,103)(51,102)(52,101)(53,100)(54,99)(55,98)(56,97)(57,120)(58,119)(59,118)(60,117)(61,116)(62,115)(63,114)(64,113)(65,112)(66,111)(67,110)(68,109)(69,108)(70,107)(71,106)(72,105)(121,223)(122,222)(123,221)(124,220)(125,219)(126,218)(127,217)(128,240)(129,239)(130,238)(131,237)(132,236)(133,235)(134,234)(135,233)(136,232)(137,231)(138,230)(139,229)(140,228)(141,227)(142,226)(143,225)(144,224)(145,161)(146,160)(147,159)(148,158)(149,157)(150,156)(151,155)(152,154)(162,168)(163,167)(164,166)(169,194)(170,193)(171,216)(172,215)(173,214)(174,213)(175,212)(176,211)(177,210)(178,209)(179,208)(180,207)(181,206)(182,205)(183,204)(184,203)(185,202)(186,201)(187,200)(188,199)(189,198)(190,197)(191,196)(192,195) );
G=PermutationGroup([[(1,161),(2,162),(3,163),(4,164),(5,165),(6,166),(7,167),(8,168),(9,145),(10,146),(11,147),(12,148),(13,149),(14,150),(15,151),(16,152),(17,153),(18,154),(19,155),(20,156),(21,157),(22,158),(23,159),(24,160),(25,70),(26,71),(27,72),(28,49),(29,50),(30,51),(31,52),(32,53),(33,54),(34,55),(35,56),(36,57),(37,58),(38,59),(39,60),(40,61),(41,62),(42,63),(43,64),(44,65),(45,66),(46,67),(47,68),(48,69),(73,118),(74,119),(75,120),(76,97),(77,98),(78,99),(79,100),(80,101),(81,102),(82,103),(83,104),(84,105),(85,106),(86,107),(87,108),(88,109),(89,110),(90,111),(91,112),(92,113),(93,114),(94,115),(95,116),(96,117),(121,173),(122,174),(123,175),(124,176),(125,177),(126,178),(127,179),(128,180),(129,181),(130,182),(131,183),(132,184),(133,185),(134,186),(135,187),(136,188),(137,189),(138,190),(139,191),(140,192),(141,169),(142,170),(143,171),(144,172),(193,226),(194,227),(195,228),(196,229),(197,230),(198,231),(199,232),(200,233),(201,234),(202,235),(203,236),(204,237),(205,238),(206,239),(207,240),(208,217),(209,218),(210,219),(211,220),(212,221),(213,222),(214,223),(215,224),(216,225)], [(1,177,60,109,202),(2,203,110,61,178),(3,179,62,111,204),(4,205,112,63,180),(5,181,64,113,206),(6,207,114,65,182),(7,183,66,115,208),(8,209,116,67,184),(9,185,68,117,210),(10,211,118,69,186),(11,187,70,119,212),(12,213,120,71,188),(13,189,72,97,214),(14,215,98,49,190),(15,191,50,99,216),(16,193,100,51,192),(17,169,52,101,194),(18,195,102,53,170),(19,171,54,103,196),(20,197,104,55,172),(21,173,56,105,198),(22,199,106,57,174),(23,175,58,107,200),(24,201,108,59,176),(25,74,221,147,135),(26,136,148,222,75),(27,76,223,149,137),(28,138,150,224,77),(29,78,225,151,139),(30,140,152,226,79),(31,80,227,153,141),(32,142,154,228,81),(33,82,229,155,143),(34,144,156,230,83),(35,84,231,157,121),(36,122,158,232,85),(37,86,233,159,123),(38,124,160,234,87),(39,88,235,161,125),(40,126,162,236,89),(41,90,237,163,127),(42,128,164,238,91),(43,92,239,165,129),(44,130,166,240,93),(45,94,217,167,131),(46,132,168,218,95),(47,96,219,145,133),(48,134,146,220,73)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,24),(11,23),(12,22),(13,21),(14,20),(15,19),(16,18),(25,86),(26,85),(27,84),(28,83),(29,82),(30,81),(31,80),(32,79),(33,78),(34,77),(35,76),(36,75),(37,74),(38,73),(39,96),(40,95),(41,94),(42,93),(43,92),(44,91),(45,90),(46,89),(47,88),(48,87),(49,104),(50,103),(51,102),(52,101),(53,100),(54,99),(55,98),(56,97),(57,120),(58,119),(59,118),(60,117),(61,116),(62,115),(63,114),(64,113),(65,112),(66,111),(67,110),(68,109),(69,108),(70,107),(71,106),(72,105),(121,223),(122,222),(123,221),(124,220),(125,219),(126,218),(127,217),(128,240),(129,239),(130,238),(131,237),(132,236),(133,235),(134,234),(135,233),(136,232),(137,231),(138,230),(139,229),(140,228),(141,227),(142,226),(143,225),(144,224),(145,161),(146,160),(147,159),(148,158),(149,157),(150,156),(151,155),(152,154),(162,168),(163,167),(164,166),(169,194),(170,193),(171,216),(172,215),(173,214),(174,213),(175,212),(176,211),(177,210),(178,209),(179,208),(180,207),(181,206),(182,205),(183,204),(184,203),(185,202),(186,201),(187,200),(188,199),(189,198),(190,197),(191,196),(192,195)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 24A | ··· | 24H | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D8 | D10 | D10 | D12 | D12 | C5⋊D4 | C5⋊D4 | D24 | S3×D5 | D4⋊D5 | C5⋊D12 | C2×S3×D5 | C5⋊D12 | C5⋊D24 |
| kernel | C2×C5⋊D24 | C5⋊D24 | C6×C5⋊2C8 | C10×D12 | C2×D60 | C2×C5⋊2C8 | C60 | C2×C30 | C2×D12 | C5⋊2C8 | C2×C20 | C30 | D12 | C2×C12 | C20 | C2×C10 | C12 | C2×C6 | C10 | C2×C4 | C6 | C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 8 | 2 | 4 | 2 | 2 | 2 | 8 |
Matrix representation of C2×C5⋊D24 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 190 | 240 | 0 | 0 |
| 0 | 0 | 191 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 172 | 214 | 0 | 0 |
| 0 | 0 | 221 | 69 | 0 | 0 |
| 0 | 0 | 0 | 0 | 219 | 146 |
| 0 | 0 | 0 | 0 | 137 | 0 |
| 240 | 240 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 1 | 0 | 0 |
| 0 | 0 | 51 | 190 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 119 | 240 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,190,191,0,0,0,0,240,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,240,0,0,0,0,1,0,0,0,0,0,0,0,172,221,0,0,0,0,214,69,0,0,0,0,0,0,219,137,0,0,0,0,146,0],[240,0,0,0,0,0,240,1,0,0,0,0,0,0,51,51,0,0,0,0,1,190,0,0,0,0,0,0,1,119,0,0,0,0,0,240] >;
C2×C5⋊D24 in GAP, Magma, Sage, TeX
C_2\times C_5\rtimes D_{24} % in TeX
G:=Group("C2xC5:D24"); // GroupNames label
G:=SmallGroup(480,378);
// by ID
G=gap.SmallGroup(480,378);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^5=c^24=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations