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## G = C2×C24⋊D5order 480 = 25·3·5

### Direct product of C2 and C24⋊D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C2×C24⋊D5
 Chief series C1 — C5 — C15 — C30 — C60 — D60 — C2×D60 — C2×C24⋊D5
 Lower central C15 — C30 — C60 — C2×C24⋊D5
 Upper central C1 — C22 — C2×C4 — C2×C8

Generators and relations for C2×C24⋊D5
G = < a,b,c,d | a2=b8=c15=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b3, dcd=c-1 >

Subgroups: 1076 in 136 conjugacy classes, 55 normal (31 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, SD16, C2×D4, C2×Q8, Dic5, C20, D10, C2×C10, C24, Dic6, D12, C2×Dic3, C2×C12, C22×S3, D15, C30, C30, C2×SD16, C40, Dic10, D20, C2×Dic5, C2×C20, C22×D5, C24⋊C2, C2×C24, C2×Dic6, C2×D12, Dic15, C60, D30, C2×C30, C40⋊C2, C2×C40, C2×Dic10, C2×D20, C2×C24⋊C2, C120, Dic30, Dic30, D60, D60, C2×Dic15, C2×C60, C22×D15, C2×C40⋊C2, C24⋊D5, C2×C120, C2×Dic30, C2×D60, C2×C24⋊D5
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, SD16, C2×D4, D10, D12, C22×S3, D15, C2×SD16, D20, C22×D5, C24⋊C2, C2×D12, D30, C40⋊C2, C2×D20, C2×C24⋊C2, D60, C22×D15, C2×C40⋊C2, C24⋊D5, C2×D60, C2×C24⋊D5

Smallest permutation representation of C2×C24⋊D5
On 240 points
Generators in S240
(1 122)(2 123)(3 124)(4 125)(5 126)(6 127)(7 128)(8 129)(9 130)(10 131)(11 132)(12 133)(13 134)(14 135)(15 121)(16 142)(17 143)(18 144)(19 145)(20 146)(21 147)(22 148)(23 149)(24 150)(25 136)(26 137)(27 138)(28 139)(29 140)(30 141)(31 156)(32 157)(33 158)(34 159)(35 160)(36 161)(37 162)(38 163)(39 164)(40 165)(41 151)(42 152)(43 153)(44 154)(45 155)(46 167)(47 168)(48 169)(49 170)(50 171)(51 172)(52 173)(53 174)(54 175)(55 176)(56 177)(57 178)(58 179)(59 180)(60 166)(61 181)(62 182)(63 183)(64 184)(65 185)(66 186)(67 187)(68 188)(69 189)(70 190)(71 191)(72 192)(73 193)(74 194)(75 195)(76 205)(77 206)(78 207)(79 208)(80 209)(81 210)(82 196)(83 197)(84 198)(85 199)(86 200)(87 201)(88 202)(89 203)(90 204)(91 219)(92 220)(93 221)(94 222)(95 223)(96 224)(97 225)(98 211)(99 212)(100 213)(101 214)(102 215)(103 216)(104 217)(105 218)(106 239)(107 240)(108 226)(109 227)(110 228)(111 229)(112 230)(113 231)(114 232)(115 233)(116 234)(117 235)(118 236)(119 237)(120 238)
(1 238 55 206 18 222 45 182)(2 239 56 207 19 223 31 183)(3 240 57 208 20 224 32 184)(4 226 58 209 21 225 33 185)(5 227 59 210 22 211 34 186)(6 228 60 196 23 212 35 187)(7 229 46 197 24 213 36 188)(8 230 47 198 25 214 37 189)(9 231 48 199 26 215 38 190)(10 232 49 200 27 216 39 191)(11 233 50 201 28 217 40 192)(12 234 51 202 29 218 41 193)(13 235 52 203 30 219 42 194)(14 236 53 204 16 220 43 195)(15 237 54 205 17 221 44 181)(61 121 119 175 76 143 93 154)(62 122 120 176 77 144 94 155)(63 123 106 177 78 145 95 156)(64 124 107 178 79 146 96 157)(65 125 108 179 80 147 97 158)(66 126 109 180 81 148 98 159)(67 127 110 166 82 149 99 160)(68 128 111 167 83 150 100 161)(69 129 112 168 84 136 101 162)(70 130 113 169 85 137 102 163)(71 131 114 170 86 138 103 164)(72 132 115 171 87 139 104 165)(73 133 116 172 88 140 105 151)(74 134 117 173 89 141 91 152)(75 135 118 174 90 142 92 153)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 19)(17 18)(20 30)(21 29)(22 28)(23 27)(24 26)(31 53)(32 52)(33 51)(34 50)(35 49)(36 48)(37 47)(38 46)(39 60)(40 59)(41 58)(42 57)(43 56)(44 55)(45 54)(61 94)(62 93)(63 92)(64 91)(65 105)(66 104)(67 103)(68 102)(69 101)(70 100)(71 99)(72 98)(73 97)(74 96)(75 95)(76 120)(77 119)(78 118)(79 117)(80 116)(81 115)(82 114)(83 113)(84 112)(85 111)(86 110)(87 109)(88 108)(89 107)(90 106)(121 122)(123 135)(124 134)(125 133)(126 132)(127 131)(128 130)(137 150)(138 149)(139 148)(140 147)(141 146)(142 145)(143 144)(151 179)(152 178)(153 177)(154 176)(155 175)(156 174)(157 173)(158 172)(159 171)(160 170)(161 169)(162 168)(163 167)(164 166)(165 180)(181 222)(182 221)(183 220)(184 219)(185 218)(186 217)(187 216)(188 215)(189 214)(190 213)(191 212)(192 211)(193 225)(194 224)(195 223)(196 232)(197 231)(198 230)(199 229)(200 228)(201 227)(202 226)(203 240)(204 239)(205 238)(206 237)(207 236)(208 235)(209 234)(210 233)

G:=sub<Sym(240)| (1,122)(2,123)(3,124)(4,125)(5,126)(6,127)(7,128)(8,129)(9,130)(10,131)(11,132)(12,133)(13,134)(14,135)(15,121)(16,142)(17,143)(18,144)(19,145)(20,146)(21,147)(22,148)(23,149)(24,150)(25,136)(26,137)(27,138)(28,139)(29,140)(30,141)(31,156)(32,157)(33,158)(34,159)(35,160)(36,161)(37,162)(38,163)(39,164)(40,165)(41,151)(42,152)(43,153)(44,154)(45,155)(46,167)(47,168)(48,169)(49,170)(50,171)(51,172)(52,173)(53,174)(54,175)(55,176)(56,177)(57,178)(58,179)(59,180)(60,166)(61,181)(62,182)(63,183)(64,184)(65,185)(66,186)(67,187)(68,188)(69,189)(70,190)(71,191)(72,192)(73,193)(74,194)(75,195)(76,205)(77,206)(78,207)(79,208)(80,209)(81,210)(82,196)(83,197)(84,198)(85,199)(86,200)(87,201)(88,202)(89,203)(90,204)(91,219)(92,220)(93,221)(94,222)(95,223)(96,224)(97,225)(98,211)(99,212)(100,213)(101,214)(102,215)(103,216)(104,217)(105,218)(106,239)(107,240)(108,226)(109,227)(110,228)(111,229)(112,230)(113,231)(114,232)(115,233)(116,234)(117,235)(118,236)(119,237)(120,238), (1,238,55,206,18,222,45,182)(2,239,56,207,19,223,31,183)(3,240,57,208,20,224,32,184)(4,226,58,209,21,225,33,185)(5,227,59,210,22,211,34,186)(6,228,60,196,23,212,35,187)(7,229,46,197,24,213,36,188)(8,230,47,198,25,214,37,189)(9,231,48,199,26,215,38,190)(10,232,49,200,27,216,39,191)(11,233,50,201,28,217,40,192)(12,234,51,202,29,218,41,193)(13,235,52,203,30,219,42,194)(14,236,53,204,16,220,43,195)(15,237,54,205,17,221,44,181)(61,121,119,175,76,143,93,154)(62,122,120,176,77,144,94,155)(63,123,106,177,78,145,95,156)(64,124,107,178,79,146,96,157)(65,125,108,179,80,147,97,158)(66,126,109,180,81,148,98,159)(67,127,110,166,82,149,99,160)(68,128,111,167,83,150,100,161)(69,129,112,168,84,136,101,162)(70,130,113,169,85,137,102,163)(71,131,114,170,86,138,103,164)(72,132,115,171,87,139,104,165)(73,133,116,172,88,140,105,151)(74,134,117,173,89,141,91,152)(75,135,118,174,90,142,92,153), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,19)(17,18)(20,30)(21,29)(22,28)(23,27)(24,26)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(61,94)(62,93)(63,92)(64,91)(65,105)(66,104)(67,103)(68,102)(69,101)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,120)(77,119)(78,118)(79,117)(80,116)(81,115)(82,114)(83,113)(84,112)(85,111)(86,110)(87,109)(88,108)(89,107)(90,106)(121,122)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130)(137,150)(138,149)(139,148)(140,147)(141,146)(142,145)(143,144)(151,179)(152,178)(153,177)(154,176)(155,175)(156,174)(157,173)(158,172)(159,171)(160,170)(161,169)(162,168)(163,167)(164,166)(165,180)(181,222)(182,221)(183,220)(184,219)(185,218)(186,217)(187,216)(188,215)(189,214)(190,213)(191,212)(192,211)(193,225)(194,224)(195,223)(196,232)(197,231)(198,230)(199,229)(200,228)(201,227)(202,226)(203,240)(204,239)(205,238)(206,237)(207,236)(208,235)(209,234)(210,233)>;

G:=Group( (1,122)(2,123)(3,124)(4,125)(5,126)(6,127)(7,128)(8,129)(9,130)(10,131)(11,132)(12,133)(13,134)(14,135)(15,121)(16,142)(17,143)(18,144)(19,145)(20,146)(21,147)(22,148)(23,149)(24,150)(25,136)(26,137)(27,138)(28,139)(29,140)(30,141)(31,156)(32,157)(33,158)(34,159)(35,160)(36,161)(37,162)(38,163)(39,164)(40,165)(41,151)(42,152)(43,153)(44,154)(45,155)(46,167)(47,168)(48,169)(49,170)(50,171)(51,172)(52,173)(53,174)(54,175)(55,176)(56,177)(57,178)(58,179)(59,180)(60,166)(61,181)(62,182)(63,183)(64,184)(65,185)(66,186)(67,187)(68,188)(69,189)(70,190)(71,191)(72,192)(73,193)(74,194)(75,195)(76,205)(77,206)(78,207)(79,208)(80,209)(81,210)(82,196)(83,197)(84,198)(85,199)(86,200)(87,201)(88,202)(89,203)(90,204)(91,219)(92,220)(93,221)(94,222)(95,223)(96,224)(97,225)(98,211)(99,212)(100,213)(101,214)(102,215)(103,216)(104,217)(105,218)(106,239)(107,240)(108,226)(109,227)(110,228)(111,229)(112,230)(113,231)(114,232)(115,233)(116,234)(117,235)(118,236)(119,237)(120,238), (1,238,55,206,18,222,45,182)(2,239,56,207,19,223,31,183)(3,240,57,208,20,224,32,184)(4,226,58,209,21,225,33,185)(5,227,59,210,22,211,34,186)(6,228,60,196,23,212,35,187)(7,229,46,197,24,213,36,188)(8,230,47,198,25,214,37,189)(9,231,48,199,26,215,38,190)(10,232,49,200,27,216,39,191)(11,233,50,201,28,217,40,192)(12,234,51,202,29,218,41,193)(13,235,52,203,30,219,42,194)(14,236,53,204,16,220,43,195)(15,237,54,205,17,221,44,181)(61,121,119,175,76,143,93,154)(62,122,120,176,77,144,94,155)(63,123,106,177,78,145,95,156)(64,124,107,178,79,146,96,157)(65,125,108,179,80,147,97,158)(66,126,109,180,81,148,98,159)(67,127,110,166,82,149,99,160)(68,128,111,167,83,150,100,161)(69,129,112,168,84,136,101,162)(70,130,113,169,85,137,102,163)(71,131,114,170,86,138,103,164)(72,132,115,171,87,139,104,165)(73,133,116,172,88,140,105,151)(74,134,117,173,89,141,91,152)(75,135,118,174,90,142,92,153), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,19)(17,18)(20,30)(21,29)(22,28)(23,27)(24,26)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(61,94)(62,93)(63,92)(64,91)(65,105)(66,104)(67,103)(68,102)(69,101)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,120)(77,119)(78,118)(79,117)(80,116)(81,115)(82,114)(83,113)(84,112)(85,111)(86,110)(87,109)(88,108)(89,107)(90,106)(121,122)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130)(137,150)(138,149)(139,148)(140,147)(141,146)(142,145)(143,144)(151,179)(152,178)(153,177)(154,176)(155,175)(156,174)(157,173)(158,172)(159,171)(160,170)(161,169)(162,168)(163,167)(164,166)(165,180)(181,222)(182,221)(183,220)(184,219)(185,218)(186,217)(187,216)(188,215)(189,214)(190,213)(191,212)(192,211)(193,225)(194,224)(195,223)(196,232)(197,231)(198,230)(199,229)(200,228)(201,227)(202,226)(203,240)(204,239)(205,238)(206,237)(207,236)(208,235)(209,234)(210,233) );

G=PermutationGroup([[(1,122),(2,123),(3,124),(4,125),(5,126),(6,127),(7,128),(8,129),(9,130),(10,131),(11,132),(12,133),(13,134),(14,135),(15,121),(16,142),(17,143),(18,144),(19,145),(20,146),(21,147),(22,148),(23,149),(24,150),(25,136),(26,137),(27,138),(28,139),(29,140),(30,141),(31,156),(32,157),(33,158),(34,159),(35,160),(36,161),(37,162),(38,163),(39,164),(40,165),(41,151),(42,152),(43,153),(44,154),(45,155),(46,167),(47,168),(48,169),(49,170),(50,171),(51,172),(52,173),(53,174),(54,175),(55,176),(56,177),(57,178),(58,179),(59,180),(60,166),(61,181),(62,182),(63,183),(64,184),(65,185),(66,186),(67,187),(68,188),(69,189),(70,190),(71,191),(72,192),(73,193),(74,194),(75,195),(76,205),(77,206),(78,207),(79,208),(80,209),(81,210),(82,196),(83,197),(84,198),(85,199),(86,200),(87,201),(88,202),(89,203),(90,204),(91,219),(92,220),(93,221),(94,222),(95,223),(96,224),(97,225),(98,211),(99,212),(100,213),(101,214),(102,215),(103,216),(104,217),(105,218),(106,239),(107,240),(108,226),(109,227),(110,228),(111,229),(112,230),(113,231),(114,232),(115,233),(116,234),(117,235),(118,236),(119,237),(120,238)], [(1,238,55,206,18,222,45,182),(2,239,56,207,19,223,31,183),(3,240,57,208,20,224,32,184),(4,226,58,209,21,225,33,185),(5,227,59,210,22,211,34,186),(6,228,60,196,23,212,35,187),(7,229,46,197,24,213,36,188),(8,230,47,198,25,214,37,189),(9,231,48,199,26,215,38,190),(10,232,49,200,27,216,39,191),(11,233,50,201,28,217,40,192),(12,234,51,202,29,218,41,193),(13,235,52,203,30,219,42,194),(14,236,53,204,16,220,43,195),(15,237,54,205,17,221,44,181),(61,121,119,175,76,143,93,154),(62,122,120,176,77,144,94,155),(63,123,106,177,78,145,95,156),(64,124,107,178,79,146,96,157),(65,125,108,179,80,147,97,158),(66,126,109,180,81,148,98,159),(67,127,110,166,82,149,99,160),(68,128,111,167,83,150,100,161),(69,129,112,168,84,136,101,162),(70,130,113,169,85,137,102,163),(71,131,114,170,86,138,103,164),(72,132,115,171,87,139,104,165),(73,133,116,172,88,140,105,151),(74,134,117,173,89,141,91,152),(75,135,118,174,90,142,92,153)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,19),(17,18),(20,30),(21,29),(22,28),(23,27),(24,26),(31,53),(32,52),(33,51),(34,50),(35,49),(36,48),(37,47),(38,46),(39,60),(40,59),(41,58),(42,57),(43,56),(44,55),(45,54),(61,94),(62,93),(63,92),(64,91),(65,105),(66,104),(67,103),(68,102),(69,101),(70,100),(71,99),(72,98),(73,97),(74,96),(75,95),(76,120),(77,119),(78,118),(79,117),(80,116),(81,115),(82,114),(83,113),(84,112),(85,111),(86,110),(87,109),(88,108),(89,107),(90,106),(121,122),(123,135),(124,134),(125,133),(126,132),(127,131),(128,130),(137,150),(138,149),(139,148),(140,147),(141,146),(142,145),(143,144),(151,179),(152,178),(153,177),(154,176),(155,175),(156,174),(157,173),(158,172),(159,171),(160,170),(161,169),(162,168),(163,167),(164,166),(165,180),(181,222),(182,221),(183,220),(184,219),(185,218),(186,217),(187,216),(188,215),(189,214),(190,213),(191,212),(192,211),(193,225),(194,224),(195,223),(196,232),(197,231),(198,230),(199,229),(200,228),(201,227),(202,226),(203,240),(204,239),(205,238),(206,237),(207,236),(208,235),(209,234),(210,233)]])

126 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 5A 5B 6A 6B 6C 8A 8B 8C 8D 10A ··· 10F 12A 12B 12C 12D 15A 15B 15C 15D 20A ··· 20H 24A ··· 24H 30A ··· 30L 40A ··· 40P 60A ··· 60P 120A ··· 120AF order 1 2 2 2 2 2 3 4 4 4 4 5 5 6 6 6 8 8 8 8 10 ··· 10 12 12 12 12 15 15 15 15 20 ··· 20 24 ··· 24 30 ··· 30 40 ··· 40 60 ··· 60 120 ··· 120 size 1 1 1 1 60 60 2 2 2 60 60 2 2 2 2 2 2 2 2 2 2 ··· 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

126 irreducible representations

 dim 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 type + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 SD16 D10 D10 D12 D12 D15 D20 D20 C24⋊C2 D30 D30 C40⋊C2 D60 D60 C24⋊D5 kernel C2×C24⋊D5 C24⋊D5 C2×C120 C2×Dic30 C2×D60 C2×C40 C60 C2×C30 C2×C24 C40 C2×C20 C30 C24 C2×C12 C20 C2×C10 C2×C8 C12 C2×C6 C10 C8 C2×C4 C6 C4 C22 C2 # reps 1 4 1 1 1 1 1 1 2 2 1 4 4 2 2 2 4 4 4 8 8 4 16 8 8 32

Matrix representation of C2×C24⋊D5 in GL5(𝔽241)

 240 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 240 0 0 0 0 0 240
,
 240 0 0 0 0 0 240 0 0 0 0 0 240 0 0 0 0 0 94 169 0 0 0 15 109
,
 1 0 0 0 0 0 131 161 0 0 0 80 148 0 0 0 0 0 51 52 0 0 0 190 0
,
 240 0 0 0 0 0 131 161 0 0 0 94 110 0 0 0 0 0 190 1 0 0 0 51 51

G:=sub<GL(5,GF(241))| [240,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,240,0,0,0,0,0,240],[240,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,94,15,0,0,0,169,109],[1,0,0,0,0,0,131,80,0,0,0,161,148,0,0,0,0,0,51,190,0,0,0,52,0],[240,0,0,0,0,0,131,94,0,0,0,161,110,0,0,0,0,0,190,51,0,0,0,1,51] >;

C2×C24⋊D5 in GAP, Magma, Sage, TeX

C_2\times C_{24}\rtimes D_5
% in TeX

G:=Group("C2xC24:D5");
// GroupNames label

G:=SmallGroup(480,867);
// by ID

G=gap.SmallGroup(480,867);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,58,675,80,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^15=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^3,d*c*d=c^-1>;
// generators/relations

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