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## G = C2×C8×D15order 480 = 25·3·5

### Direct product of C2×C8 and D15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C2×C8×D15
 Chief series C1 — C5 — C15 — C30 — C60 — C4×D15 — C2×C4×D15 — C2×C8×D15
 Lower central C15 — C2×C8×D15
 Upper central C1 — C2×C8

Generators and relations for C2×C8×D15
G = < a,b,c,d | a2=b8=c15=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 692 in 152 conjugacy classes, 71 normal (33 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C8, C2×C4, C2×C4, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, C2×C8, C22×C4, Dic5, C20, D10, C2×C10, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, D15, C30, C30, C22×C8, C52C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, S3×C8, C2×C3⋊C8, C2×C24, S3×C2×C4, Dic15, C60, D30, C2×C30, C8×D5, C2×C52C8, C2×C40, C2×C4×D5, S3×C2×C8, C153C8, C120, C4×D15, C2×Dic15, C2×C60, C22×D15, D5×C2×C8, C8×D15, C2×C153C8, C2×C120, C2×C4×D15, C2×C8×D15
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, D5, D6, C2×C8, C22×C4, D10, C4×S3, C22×S3, D15, C22×C8, C4×D5, C22×D5, S3×C8, S3×C2×C4, D30, C8×D5, C2×C4×D5, S3×C2×C8, C4×D15, C22×D15, D5×C2×C8, C8×D15, C2×C4×D15, C2×C8×D15

Smallest permutation representation of C2×C8×D15
On 240 points
Generators in S240
(1 137)(2 138)(3 139)(4 140)(5 141)(6 142)(7 143)(8 144)(9 145)(10 146)(11 147)(12 148)(13 149)(14 150)(15 136)(16 121)(17 122)(18 123)(19 124)(20 125)(21 126)(22 127)(23 128)(24 129)(25 130)(26 131)(27 132)(28 133)(29 134)(30 135)(31 174)(32 175)(33 176)(34 177)(35 178)(36 179)(37 180)(38 166)(39 167)(40 168)(41 169)(42 170)(43 171)(44 172)(45 173)(46 163)(47 164)(48 165)(49 151)(50 152)(51 153)(52 154)(53 155)(54 156)(55 157)(56 158)(57 159)(58 160)(59 161)(60 162)(61 207)(62 208)(63 209)(64 210)(65 196)(66 197)(67 198)(68 199)(69 200)(70 201)(71 202)(72 203)(73 204)(74 205)(75 206)(76 192)(77 193)(78 194)(79 195)(80 181)(81 182)(82 183)(83 184)(84 185)(85 186)(86 187)(87 188)(88 189)(89 190)(90 191)(91 230)(92 231)(93 232)(94 233)(95 234)(96 235)(97 236)(98 237)(99 238)(100 239)(101 240)(102 226)(103 227)(104 228)(105 229)(106 211)(107 212)(108 213)(109 214)(110 215)(111 216)(112 217)(113 218)(114 219)(115 220)(116 221)(117 222)(118 223)(119 224)(120 225)
(1 106 50 84 16 95 35 65)(2 107 51 85 17 96 36 66)(3 108 52 86 18 97 37 67)(4 109 53 87 19 98 38 68)(5 110 54 88 20 99 39 69)(6 111 55 89 21 100 40 70)(7 112 56 90 22 101 41 71)(8 113 57 76 23 102 42 72)(9 114 58 77 24 103 43 73)(10 115 59 78 25 104 44 74)(11 116 60 79 26 105 45 75)(12 117 46 80 27 91 31 61)(13 118 47 81 28 92 32 62)(14 119 48 82 29 93 33 63)(15 120 49 83 30 94 34 64)(121 234 178 196 137 211 152 185)(122 235 179 197 138 212 153 186)(123 236 180 198 139 213 154 187)(124 237 166 199 140 214 155 188)(125 238 167 200 141 215 156 189)(126 239 168 201 142 216 157 190)(127 240 169 202 143 217 158 191)(128 226 170 203 144 218 159 192)(129 227 171 204 145 219 160 193)(130 228 172 205 146 220 161 194)(131 229 173 206 147 221 162 195)(132 230 174 207 148 222 163 181)(133 231 175 208 149 223 164 182)(134 232 176 209 150 224 165 183)(135 233 177 210 136 225 151 184)
(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 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(31 38)(32 37)(33 36)(34 35)(39 45)(40 44)(41 43)(46 53)(47 52)(48 51)(49 50)(54 60)(55 59)(56 58)(61 68)(62 67)(63 66)(64 65)(69 75)(70 74)(71 73)(77 90)(78 89)(79 88)(80 87)(81 86)(82 85)(83 84)(91 98)(92 97)(93 96)(94 95)(99 105)(100 104)(101 103)(106 120)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)(121 135)(122 134)(123 133)(124 132)(125 131)(126 130)(127 129)(136 137)(138 150)(139 149)(140 148)(141 147)(142 146)(143 145)(151 152)(153 165)(154 164)(155 163)(156 162)(157 161)(158 160)(166 174)(167 173)(168 172)(169 171)(175 180)(176 179)(177 178)(181 188)(182 187)(183 186)(184 185)(189 195)(190 194)(191 193)(196 210)(197 209)(198 208)(199 207)(200 206)(201 205)(202 204)(211 225)(212 224)(213 223)(214 222)(215 221)(216 220)(217 219)(227 240)(228 239)(229 238)(230 237)(231 236)(232 235)(233 234)

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

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

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

144 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 6A 6B 6C 8A ··· 8H 8I ··· 8P 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 2 2 3 4 4 4 4 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 15 15 15 15 2 1 1 1 1 15 15 15 15 2 2 2 2 2 1 ··· 1 15 ··· 15 2 ··· 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

144 irreducible representations

 dim 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 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C8 S3 D5 D6 D6 D10 D10 C4×S3 C4×S3 D15 C4×D5 C4×D5 S3×C8 D30 D30 C8×D5 C4×D15 C4×D15 C8×D15 kernel C2×C8×D15 C8×D15 C2×C15⋊3C8 C2×C120 C2×C4×D15 C4×D15 C2×Dic15 C22×D15 D30 C2×C40 C2×C24 C40 C2×C20 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 4 2 2 16 1 2 2 1 4 2 2 2 4 4 4 8 8 4 16 8 8 32

Matrix representation of C2×C8×D15 in GL4(𝔽241) generated by

 240 0 0 0 0 240 0 0 0 0 1 0 0 0 0 1
,
 233 0 0 0 0 233 0 0 0 0 8 0 0 0 0 8
,
 240 189 0 0 52 52 0 0 0 0 211 143 0 0 161 68
,
 177 16 0 0 211 64 0 0 0 0 177 114 0 0 148 64
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[233,0,0,0,0,233,0,0,0,0,8,0,0,0,0,8],[240,52,0,0,189,52,0,0,0,0,211,161,0,0,143,68],[177,211,0,0,16,64,0,0,0,0,177,148,0,0,114,64] >;

C2×C8×D15 in GAP, Magma, Sage, TeX

C_2\times C_8\times D_{15}
% in TeX

G:=Group("C2xC8xD15");
// GroupNames label

G:=SmallGroup(480,864);
// by ID

G=gap.SmallGroup(480,864);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,58,80,2693,18822]);
// 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,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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