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## G = C5×C2.D24order 480 = 25·3·5

### Direct product of C5 and C2.D24

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×C2.D24
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×C60 — C10×D12 — C5×C2.D24
 Lower central C3 — C6 — C12 — C5×C2.D24
 Upper central C1 — C2×C10 — C2×C20 — C2×C40

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 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×4], C5, S3 [×2], C6 [×3], C8, C2×C4, C2×C4, D4 [×3], C23, C10 [×3], C10 [×2], Dic3, C12 [×2], D6 [×4], C2×C6, C15, C4⋊C4, C2×C8, C2×D4, C20 [×2], C20, C2×C10, C2×C10 [×4], C24, D12 [×2], D12, C2×Dic3, C2×C12, C22×S3, C5×S3 [×2], C30 [×3], D4⋊C4, C40, C2×C20, C2×C20, C5×D4 [×3], C22×C10, C4⋊Dic3, C2×C24, C2×D12, C5×Dic3, C60 [×2], S3×C10 [×4], C2×C30, C5×C4⋊C4, C2×C40, D4×C10, C2.D24, C120, C5×D12 [×2], 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 [×3], C4 [×2], C22, C5, S3, C2×C4, D4 [×2], C10 [×3], D6, C22⋊C4, D8, SD16, C20 [×2], C2×C10, C4×S3, D12, C3⋊D4, C5×S3, D4⋊C4, C2×C20, C5×D4 [×2], 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

Smallest permutation representation of C5×C2.D24
On 240 points
Generators in S240
(1 220 199 169 141)(2 221 200 170 142)(3 222 201 171 143)(4 223 202 172 144)(5 224 203 173 121)(6 225 204 174 122)(7 226 205 175 123)(8 227 206 176 124)(9 228 207 177 125)(10 229 208 178 126)(11 230 209 179 127)(12 231 210 180 128)(13 232 211 181 129)(14 233 212 182 130)(15 234 213 183 131)(16 235 214 184 132)(17 236 215 185 133)(18 237 216 186 134)(19 238 193 187 135)(20 239 194 188 136)(21 240 195 189 137)(22 217 196 190 138)(23 218 197 191 139)(24 219 198 192 140)(25 107 80 72 147)(26 108 81 49 148)(27 109 82 50 149)(28 110 83 51 150)(29 111 84 52 151)(30 112 85 53 152)(31 113 86 54 153)(32 114 87 55 154)(33 115 88 56 155)(34 116 89 57 156)(35 117 90 58 157)(36 118 91 59 158)(37 119 92 60 159)(38 120 93 61 160)(39 97 94 62 161)(40 98 95 63 162)(41 99 96 64 163)(42 100 73 65 164)(43 101 74 66 165)(44 102 75 67 166)(45 103 76 68 167)(46 104 77 69 168)(47 105 78 70 145)(48 106 79 71 146)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 43)(7 44)(8 45)(9 46)(10 47)(11 48)(12 25)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 37)(49 181)(50 182)(51 183)(52 184)(53 185)(54 186)(55 187)(56 188)(57 189)(58 190)(59 191)(60 192)(61 169)(62 170)(63 171)(64 172)(65 173)(66 174)(67 175)(68 176)(69 177)(70 178)(71 179)(72 180)(73 203)(74 204)(75 205)(76 206)(77 207)(78 208)(79 209)(80 210)(81 211)(82 212)(83 213)(84 214)(85 215)(86 216)(87 193)(88 194)(89 195)(90 196)(91 197)(92 198)(93 199)(94 200)(95 201)(96 202)(97 221)(98 222)(99 223)(100 224)(101 225)(102 226)(103 227)(104 228)(105 229)(106 230)(107 231)(108 232)(109 233)(110 234)(111 235)(112 236)(113 237)(114 238)(115 239)(116 240)(117 217)(118 218)(119 219)(120 220)(121 164)(122 165)(123 166)(124 167)(125 168)(126 145)(127 146)(128 147)(129 148)(130 149)(131 150)(132 151)(133 152)(134 153)(135 154)(136 155)(137 156)(138 157)(139 158)(140 159)(141 160)(142 161)(143 162)(144 163)
(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 37 38 24)(2 23 39 36)(3 35 40 22)(4 21 41 34)(5 33 42 20)(6 19 43 32)(7 31 44 18)(8 17 45 30)(9 29 46 16)(10 15 47 28)(11 27 48 14)(12 13 25 26)(49 180 181 72)(50 71 182 179)(51 178 183 70)(52 69 184 177)(53 176 185 68)(54 67 186 175)(55 174 187 66)(56 65 188 173)(57 172 189 64)(58 63 190 171)(59 170 191 62)(60 61 192 169)(73 194 203 88)(74 87 204 193)(75 216 205 86)(76 85 206 215)(77 214 207 84)(78 83 208 213)(79 212 209 82)(80 81 210 211)(89 202 195 96)(90 95 196 201)(91 200 197 94)(92 93 198 199)(97 118 221 218)(98 217 222 117)(99 116 223 240)(100 239 224 115)(101 114 225 238)(102 237 226 113)(103 112 227 236)(104 235 228 111)(105 110 229 234)(106 233 230 109)(107 108 231 232)(119 120 219 220)(121 155 164 136)(122 135 165 154)(123 153 166 134)(124 133 167 152)(125 151 168 132)(126 131 145 150)(127 149 146 130)(128 129 147 148)(137 163 156 144)(138 143 157 162)(139 161 158 142)(140 141 159 160)

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

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

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

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

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