direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C8⋊D15, C8⋊8D30, C40⋊30D6, C4.6D60, C24⋊30D10, C30⋊7SD16, C20.31D12, C12.31D20, C60.162D4, C120⋊37C22, C22.12D60, C60.244C23, D60.35C22, Dic30⋊18C22, (C2×C40)⋊7S3, (C2×C8)⋊5D15, (C2×C24)⋊7D5, (C2×C120)⋊11C2, C6⋊1(C40⋊C2), (C2×D60).5C2, (C2×C6).18D20, (C2×C4).80D30, C2.11(C2×D60), C6.37(C2×D20), C15⋊16(C2×SD16), C10⋊1(C24⋊C2), (C2×Dic30)⋊8C2, (C2×C30).103D4, C30.265(C2×D4), (C2×C10).18D12, C10.38(C2×D12), (C2×C20).390D6, (C2×C12).396D10, C4.25(C22×D15), C20.215(C22×S3), (C2×C60).477C22, C12.217(C22×D5), C5⋊2(C2×C24⋊C2), C3⋊2(C2×C40⋊C2), SmallGroup(480,867)
Series: Derived ►Chief ►Lower central ►Upper central
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, dbd=b3, dcd=c-1 >
Subgroups: 1076 in 136 conjugacy classes, 55 normal (31 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, S3 [×2], C6, C6 [×2], C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×3], C23, D5 [×2], C10, C10 [×2], Dic3 [×2], C12 [×2], D6 [×4], C2×C6, C15, C2×C8, SD16 [×4], C2×D4, C2×Q8, Dic5 [×2], C20 [×2], D10 [×4], C2×C10, C24 [×2], Dic6 [×3], D12 [×3], C2×Dic3, C2×C12, C22×S3, D15 [×2], C30, C30 [×2], C2×SD16, C40 [×2], Dic10 [×3], D20 [×3], C2×Dic5, C2×C20, C22×D5, C24⋊C2 [×4], C2×C24, C2×Dic6, C2×D12, Dic15 [×2], C60 [×2], D30 [×4], C2×C30, C40⋊C2 [×4], C2×C40, C2×Dic10, C2×D20, C2×C24⋊C2, C120 [×2], Dic30 [×2], Dic30, D60 [×2], D60, C2×Dic15, C2×C60, C22×D15, C2×C40⋊C2, C8⋊D15 [×4], C2×C120, C2×Dic30, C2×D60, C2×C8⋊D15
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], SD16 [×2], C2×D4, D10 [×3], D12 [×2], C22×S3, D15, C2×SD16, D20 [×2], C22×D5, C24⋊C2 [×2], C2×D12, D30 [×3], C40⋊C2 [×2], C2×D20, C2×C24⋊C2, D60 [×2], C22×D15, C2×C40⋊C2, C8⋊D15 [×2], C2×D60, C2×C8⋊D15
►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)(16 143)(17 144)(18 145)(19 146)(20 147)(21 148)(22 149)(23 150)(24 136)(25 137)(26 138)(27 139)(28 140)(29 141)(30 142)(31 158)(32 159)(33 160)(34 161)(35 162)(36 163)(37 164)(38 165)(39 151)(40 152)(41 153)(42 154)(43 155)(44 156)(45 157)(46 168)(47 169)(48 170)(49 171)(50 172)(51 173)(52 174)(53 175)(54 176)(55 177)(56 178)(57 179)(58 180)(59 166)(60 167)(61 182)(62 183)(63 184)(64 185)(65 186)(66 187)(67 188)(68 189)(69 190)(70 191)(71 192)(72 193)(73 194)(74 195)(75 181)(76 196)(77 197)(78 198)(79 199)(80 200)(81 201)(82 202)(83 203)(84 204)(85 205)(86 206)(87 207)(88 208)(89 209)(90 210)(91 221)(92 222)(93 223)(94 224)(95 225)(96 211)(97 212)(98 213)(99 214)(100 215)(101 216)(102 217)(103 218)(104 219)(105 220)(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)
(1 240 54 208 22 225 41 187)(2 226 55 209 23 211 42 188)(3 227 56 210 24 212 43 189)(4 228 57 196 25 213 44 190)(5 229 58 197 26 214 45 191)(6 230 59 198 27 215 31 192)(7 231 60 199 28 216 32 193)(8 232 46 200 29 217 33 194)(9 233 47 201 30 218 34 195)(10 234 48 202 16 219 35 181)(11 235 49 203 17 220 36 182)(12 236 50 204 18 221 37 183)(13 237 51 205 19 222 38 184)(14 238 52 206 20 223 39 185)(15 239 53 207 21 224 40 186)(61 132 117 171 83 144 105 163)(62 133 118 172 84 145 91 164)(63 134 119 173 85 146 92 165)(64 135 120 174 86 147 93 151)(65 121 106 175 87 148 94 152)(66 122 107 176 88 149 95 153)(67 123 108 177 89 150 96 154)(68 124 109 178 90 136 97 155)(69 125 110 179 76 137 98 156)(70 126 111 180 77 138 99 157)(71 127 112 166 78 139 100 158)(72 128 113 167 79 140 101 159)(73 129 114 168 80 141 102 160)(74 130 115 169 81 142 103 161)(75 131 116 170 82 143 104 162)
(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 27)(17 26)(18 25)(19 24)(20 23)(21 22)(28 30)(31 48)(32 47)(33 46)(34 60)(35 59)(36 58)(37 57)(38 56)(39 55)(40 54)(41 53)(42 52)(43 51)(44 50)(45 49)(61 99)(62 98)(63 97)(64 96)(65 95)(66 94)(67 93)(68 92)(69 91)(70 105)(71 104)(72 103)(73 102)(74 101)(75 100)(76 118)(77 117)(78 116)(79 115)(80 114)(81 113)(82 112)(83 111)(84 110)(85 109)(86 108)(87 107)(88 106)(89 120)(90 119)(121 122)(123 135)(124 134)(125 133)(126 132)(127 131)(128 130)(136 146)(137 145)(138 144)(139 143)(140 142)(147 150)(148 149)(151 177)(152 176)(153 175)(154 174)(155 173)(156 172)(157 171)(158 170)(159 169)(160 168)(161 167)(162 166)(163 180)(164 179)(165 178)(181 215)(182 214)(183 213)(184 212)(185 211)(186 225)(187 224)(188 223)(189 222)(190 221)(191 220)(192 219)(193 218)(194 217)(195 216)(196 236)(197 235)(198 234)(199 233)(200 232)(201 231)(202 230)(203 229)(204 228)(205 227)(206 226)(207 240)(208 239)(209 238)(210 237)
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)(16,143)(17,144)(18,145)(19,146)(20,147)(21,148)(22,149)(23,150)(24,136)(25,137)(26,138)(27,139)(28,140)(29,141)(30,142)(31,158)(32,159)(33,160)(34,161)(35,162)(36,163)(37,164)(38,165)(39,151)(40,152)(41,153)(42,154)(43,155)(44,156)(45,157)(46,168)(47,169)(48,170)(49,171)(50,172)(51,173)(52,174)(53,175)(54,176)(55,177)(56,178)(57,179)(58,180)(59,166)(60,167)(61,182)(62,183)(63,184)(64,185)(65,186)(66,187)(67,188)(68,189)(69,190)(70,191)(71,192)(72,193)(73,194)(74,195)(75,181)(76,196)(77,197)(78,198)(79,199)(80,200)(81,201)(82,202)(83,203)(84,204)(85,205)(86,206)(87,207)(88,208)(89,209)(90,210)(91,221)(92,222)(93,223)(94,224)(95,225)(96,211)(97,212)(98,213)(99,214)(100,215)(101,216)(102,217)(103,218)(104,219)(105,220)(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), (1,240,54,208,22,225,41,187)(2,226,55,209,23,211,42,188)(3,227,56,210,24,212,43,189)(4,228,57,196,25,213,44,190)(5,229,58,197,26,214,45,191)(6,230,59,198,27,215,31,192)(7,231,60,199,28,216,32,193)(8,232,46,200,29,217,33,194)(9,233,47,201,30,218,34,195)(10,234,48,202,16,219,35,181)(11,235,49,203,17,220,36,182)(12,236,50,204,18,221,37,183)(13,237,51,205,19,222,38,184)(14,238,52,206,20,223,39,185)(15,239,53,207,21,224,40,186)(61,132,117,171,83,144,105,163)(62,133,118,172,84,145,91,164)(63,134,119,173,85,146,92,165)(64,135,120,174,86,147,93,151)(65,121,106,175,87,148,94,152)(66,122,107,176,88,149,95,153)(67,123,108,177,89,150,96,154)(68,124,109,178,90,136,97,155)(69,125,110,179,76,137,98,156)(70,126,111,180,77,138,99,157)(71,127,112,166,78,139,100,158)(72,128,113,167,79,140,101,159)(73,129,114,168,80,141,102,160)(74,130,115,169,81,142,103,161)(75,131,116,170,82,143,104,162), (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)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(28,30)(31,48)(32,47)(33,46)(34,60)(35,59)(36,58)(37,57)(38,56)(39,55)(40,54)(41,53)(42,52)(43,51)(44,50)(45,49)(61,99)(62,98)(63,97)(64,96)(65,95)(66,94)(67,93)(68,92)(69,91)(70,105)(71,104)(72,103)(73,102)(74,101)(75,100)(76,118)(77,117)(78,116)(79,115)(80,114)(81,113)(82,112)(83,111)(84,110)(85,109)(86,108)(87,107)(88,106)(89,120)(90,119)(121,122)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130)(136,146)(137,145)(138,144)(139,143)(140,142)(147,150)(148,149)(151,177)(152,176)(153,175)(154,174)(155,173)(156,172)(157,171)(158,170)(159,169)(160,168)(161,167)(162,166)(163,180)(164,179)(165,178)(181,215)(182,214)(183,213)(184,212)(185,211)(186,225)(187,224)(188,223)(189,222)(190,221)(191,220)(192,219)(193,218)(194,217)(195,216)(196,236)(197,235)(198,234)(199,233)(200,232)(201,231)(202,230)(203,229)(204,228)(205,227)(206,226)(207,240)(208,239)(209,238)(210,237)>;
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)(16,143)(17,144)(18,145)(19,146)(20,147)(21,148)(22,149)(23,150)(24,136)(25,137)(26,138)(27,139)(28,140)(29,141)(30,142)(31,158)(32,159)(33,160)(34,161)(35,162)(36,163)(37,164)(38,165)(39,151)(40,152)(41,153)(42,154)(43,155)(44,156)(45,157)(46,168)(47,169)(48,170)(49,171)(50,172)(51,173)(52,174)(53,175)(54,176)(55,177)(56,178)(57,179)(58,180)(59,166)(60,167)(61,182)(62,183)(63,184)(64,185)(65,186)(66,187)(67,188)(68,189)(69,190)(70,191)(71,192)(72,193)(73,194)(74,195)(75,181)(76,196)(77,197)(78,198)(79,199)(80,200)(81,201)(82,202)(83,203)(84,204)(85,205)(86,206)(87,207)(88,208)(89,209)(90,210)(91,221)(92,222)(93,223)(94,224)(95,225)(96,211)(97,212)(98,213)(99,214)(100,215)(101,216)(102,217)(103,218)(104,219)(105,220)(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), (1,240,54,208,22,225,41,187)(2,226,55,209,23,211,42,188)(3,227,56,210,24,212,43,189)(4,228,57,196,25,213,44,190)(5,229,58,197,26,214,45,191)(6,230,59,198,27,215,31,192)(7,231,60,199,28,216,32,193)(8,232,46,200,29,217,33,194)(9,233,47,201,30,218,34,195)(10,234,48,202,16,219,35,181)(11,235,49,203,17,220,36,182)(12,236,50,204,18,221,37,183)(13,237,51,205,19,222,38,184)(14,238,52,206,20,223,39,185)(15,239,53,207,21,224,40,186)(61,132,117,171,83,144,105,163)(62,133,118,172,84,145,91,164)(63,134,119,173,85,146,92,165)(64,135,120,174,86,147,93,151)(65,121,106,175,87,148,94,152)(66,122,107,176,88,149,95,153)(67,123,108,177,89,150,96,154)(68,124,109,178,90,136,97,155)(69,125,110,179,76,137,98,156)(70,126,111,180,77,138,99,157)(71,127,112,166,78,139,100,158)(72,128,113,167,79,140,101,159)(73,129,114,168,80,141,102,160)(74,130,115,169,81,142,103,161)(75,131,116,170,82,143,104,162), (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)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(28,30)(31,48)(32,47)(33,46)(34,60)(35,59)(36,58)(37,57)(38,56)(39,55)(40,54)(41,53)(42,52)(43,51)(44,50)(45,49)(61,99)(62,98)(63,97)(64,96)(65,95)(66,94)(67,93)(68,92)(69,91)(70,105)(71,104)(72,103)(73,102)(74,101)(75,100)(76,118)(77,117)(78,116)(79,115)(80,114)(81,113)(82,112)(83,111)(84,110)(85,109)(86,108)(87,107)(88,106)(89,120)(90,119)(121,122)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130)(136,146)(137,145)(138,144)(139,143)(140,142)(147,150)(148,149)(151,177)(152,176)(153,175)(154,174)(155,173)(156,172)(157,171)(158,170)(159,169)(160,168)(161,167)(162,166)(163,180)(164,179)(165,178)(181,215)(182,214)(183,213)(184,212)(185,211)(186,225)(187,224)(188,223)(189,222)(190,221)(191,220)(192,219)(193,218)(194,217)(195,216)(196,236)(197,235)(198,234)(199,233)(200,232)(201,231)(202,230)(203,229)(204,228)(205,227)(206,226)(207,240)(208,239)(209,238)(210,237) );
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),(16,143),(17,144),(18,145),(19,146),(20,147),(21,148),(22,149),(23,150),(24,136),(25,137),(26,138),(27,139),(28,140),(29,141),(30,142),(31,158),(32,159),(33,160),(34,161),(35,162),(36,163),(37,164),(38,165),(39,151),(40,152),(41,153),(42,154),(43,155),(44,156),(45,157),(46,168),(47,169),(48,170),(49,171),(50,172),(51,173),(52,174),(53,175),(54,176),(55,177),(56,178),(57,179),(58,180),(59,166),(60,167),(61,182),(62,183),(63,184),(64,185),(65,186),(66,187),(67,188),(68,189),(69,190),(70,191),(71,192),(72,193),(73,194),(74,195),(75,181),(76,196),(77,197),(78,198),(79,199),(80,200),(81,201),(82,202),(83,203),(84,204),(85,205),(86,206),(87,207),(88,208),(89,209),(90,210),(91,221),(92,222),(93,223),(94,224),(95,225),(96,211),(97,212),(98,213),(99,214),(100,215),(101,216),(102,217),(103,218),(104,219),(105,220),(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)], [(1,240,54,208,22,225,41,187),(2,226,55,209,23,211,42,188),(3,227,56,210,24,212,43,189),(4,228,57,196,25,213,44,190),(5,229,58,197,26,214,45,191),(6,230,59,198,27,215,31,192),(7,231,60,199,28,216,32,193),(8,232,46,200,29,217,33,194),(9,233,47,201,30,218,34,195),(10,234,48,202,16,219,35,181),(11,235,49,203,17,220,36,182),(12,236,50,204,18,221,37,183),(13,237,51,205,19,222,38,184),(14,238,52,206,20,223,39,185),(15,239,53,207,21,224,40,186),(61,132,117,171,83,144,105,163),(62,133,118,172,84,145,91,164),(63,134,119,173,85,146,92,165),(64,135,120,174,86,147,93,151),(65,121,106,175,87,148,94,152),(66,122,107,176,88,149,95,153),(67,123,108,177,89,150,96,154),(68,124,109,178,90,136,97,155),(69,125,110,179,76,137,98,156),(70,126,111,180,77,138,99,157),(71,127,112,166,78,139,100,158),(72,128,113,167,79,140,101,159),(73,129,114,168,80,141,102,160),(74,130,115,169,81,142,103,161),(75,131,116,170,82,143,104,162)], [(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),(16,27),(17,26),(18,25),(19,24),(20,23),(21,22),(28,30),(31,48),(32,47),(33,46),(34,60),(35,59),(36,58),(37,57),(38,56),(39,55),(40,54),(41,53),(42,52),(43,51),(44,50),(45,49),(61,99),(62,98),(63,97),(64,96),(65,95),(66,94),(67,93),(68,92),(69,91),(70,105),(71,104),(72,103),(73,102),(74,101),(75,100),(76,118),(77,117),(78,116),(79,115),(80,114),(81,113),(82,112),(83,111),(84,110),(85,109),(86,108),(87,107),(88,106),(89,120),(90,119),(121,122),(123,135),(124,134),(125,133),(126,132),(127,131),(128,130),(136,146),(137,145),(138,144),(139,143),(140,142),(147,150),(148,149),(151,177),(152,176),(153,175),(154,174),(155,173),(156,172),(157,171),(158,170),(159,169),(160,168),(161,167),(162,166),(163,180),(164,179),(165,178),(181,215),(182,214),(183,213),(184,212),(185,211),(186,225),(187,224),(188,223),(189,222),(190,221),(191,220),(192,219),(193,218),(194,217),(195,216),(196,236),(197,235),(198,234),(199,233),(200,232),(201,231),(202,230),(203,229),(204,228),(205,227),(206,226),(207,240),(208,239),(209,238),(210,237)])
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 | C8⋊D15 |
| kernel | C2×C8⋊D15 | C8⋊D15 | 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×C8⋊D15 ►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×C8⋊D15 in GAP, Magma, Sage, TeX
C_2\times C_8\rtimes D_{15} % in TeX
G:=Group("C2xC8:D15"); // 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