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## G = C15×D4⋊C4order 480 = 25·3·5

### Direct product of C15 and D4⋊C4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C15×D4⋊C4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C20 — C2×C60 — C15×C4⋊C4 — C15×D4⋊C4
 Lower central C1 — C2 — C4 — C15×D4⋊C4
 Upper central C1 — C2×C30 — C2×C60 — C15×D4⋊C4

Generators and relations for C15×D4⋊C4
G = < a,b,c,d | a15=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >

Subgroups: 184 in 100 conjugacy classes, 56 normal (48 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×4], C5, C6 [×3], C6 [×2], C8, C2×C4, C2×C4, D4 [×2], D4, C23, C10 [×3], C10 [×2], C12 [×2], C12, C2×C6, C2×C6 [×4], C15, C4⋊C4, C2×C8, C2×D4, C20 [×2], C20, C2×C10, C2×C10 [×4], C24, C2×C12, C2×C12, C3×D4 [×2], C3×D4, C22×C6, C30 [×3], C30 [×2], D4⋊C4, C40, C2×C20, C2×C20, C5×D4 [×2], C5×D4, C22×C10, C3×C4⋊C4, C2×C24, C6×D4, C60 [×2], C60, C2×C30, C2×C30 [×4], C5×C4⋊C4, C2×C40, D4×C10, C3×D4⋊C4, C120, C2×C60, C2×C60, D4×C15 [×2], D4×C15, C22×C30, C5×D4⋊C4, C15×C4⋊C4, C2×C120, D4×C30, C15×D4⋊C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C5, C6 [×3], C2×C4, D4 [×2], C10 [×3], C12 [×2], C2×C6, C15, C22⋊C4, D8, SD16, C20 [×2], C2×C10, C2×C12, C3×D4 [×2], C30 [×3], D4⋊C4, C2×C20, C5×D4 [×2], C3×C22⋊C4, C3×D8, C3×SD16, C60 [×2], C2×C30, C5×C22⋊C4, C5×D8, C5×SD16, C3×D4⋊C4, C2×C60, D4×C15 [×2], C5×D4⋊C4, C15×C22⋊C4, C15×D8, C15×SD16, C15×D4⋊C4

Smallest permutation representation of C15×D4⋊C4
On 240 points
Generators in S240
(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)
(1 196 213 64)(2 197 214 65)(3 198 215 66)(4 199 216 67)(5 200 217 68)(6 201 218 69)(7 202 219 70)(8 203 220 71)(9 204 221 72)(10 205 222 73)(11 206 223 74)(12 207 224 75)(13 208 225 61)(14 209 211 62)(15 210 212 63)(16 173 195 164)(17 174 181 165)(18 175 182 151)(19 176 183 152)(20 177 184 153)(21 178 185 154)(22 179 186 155)(23 180 187 156)(24 166 188 157)(25 167 189 158)(26 168 190 159)(27 169 191 160)(28 170 192 161)(29 171 193 162)(30 172 194 163)(31 81 231 93)(32 82 232 94)(33 83 233 95)(34 84 234 96)(35 85 235 97)(36 86 236 98)(37 87 237 99)(38 88 238 100)(39 89 239 101)(40 90 240 102)(41 76 226 103)(42 77 227 104)(43 78 228 105)(44 79 229 91)(45 80 230 92)(46 111 130 144)(47 112 131 145)(48 113 132 146)(49 114 133 147)(50 115 134 148)(51 116 135 149)(52 117 121 150)(53 118 122 136)(54 119 123 137)(55 120 124 138)(56 106 125 139)(57 107 126 140)(58 108 127 141)(59 109 128 142)(60 110 129 143)
(1 187)(2 188)(3 189)(4 190)(5 191)(6 192)(7 193)(8 194)(9 195)(10 181)(11 182)(12 183)(13 184)(14 185)(15 186)(16 221)(17 222)(18 223)(19 224)(20 225)(21 211)(22 212)(23 213)(24 214)(25 215)(26 216)(27 217)(28 218)(29 219)(30 220)(31 132)(32 133)(33 134)(34 135)(35 121)(36 122)(37 123)(38 124)(39 125)(40 126)(41 127)(42 128)(43 129)(44 130)(45 131)(46 229)(47 230)(48 231)(49 232)(50 233)(51 234)(52 235)(53 236)(54 237)(55 238)(56 239)(57 240)(58 226)(59 227)(60 228)(61 153)(62 154)(63 155)(64 156)(65 157)(66 158)(67 159)(68 160)(69 161)(70 162)(71 163)(72 164)(73 165)(74 151)(75 152)(76 108)(77 109)(78 110)(79 111)(80 112)(81 113)(82 114)(83 115)(84 116)(85 117)(86 118)(87 119)(88 120)(89 106)(90 107)(91 144)(92 145)(93 146)(94 147)(95 148)(96 149)(97 150)(98 136)(99 137)(100 138)(101 139)(102 140)(103 141)(104 142)(105 143)(166 197)(167 198)(168 199)(169 200)(170 201)(171 202)(172 203)(173 204)(174 205)(175 206)(176 207)(177 208)(178 209)(179 210)(180 196)
(1 41 156 58)(2 42 157 59)(3 43 158 60)(4 44 159 46)(5 45 160 47)(6 31 161 48)(7 32 162 49)(8 33 163 50)(9 34 164 51)(10 35 165 52)(11 36 151 53)(12 37 152 54)(13 38 153 55)(14 39 154 56)(15 40 155 57)(16 149 204 96)(17 150 205 97)(18 136 206 98)(19 137 207 99)(20 138 208 100)(21 139 209 101)(22 140 210 102)(23 141 196 103)(24 142 197 104)(25 143 198 105)(26 144 199 91)(27 145 200 92)(28 146 201 93)(29 147 202 94)(30 148 203 95)(61 88 184 120)(62 89 185 106)(63 90 186 107)(64 76 187 108)(65 77 188 109)(66 78 189 110)(67 79 190 111)(68 80 191 112)(69 81 192 113)(70 82 193 114)(71 83 194 115)(72 84 195 116)(73 85 181 117)(74 86 182 118)(75 87 183 119)(121 222 235 174)(122 223 236 175)(123 224 237 176)(124 225 238 177)(125 211 239 178)(126 212 240 179)(127 213 226 180)(128 214 227 166)(129 215 228 167)(130 216 229 168)(131 217 230 169)(132 218 231 170)(133 219 232 171)(134 220 233 172)(135 221 234 173)

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

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

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

210 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 5A 5B 5C 5D 6A ··· 6F 6G 6H 6I 6J 8A 8B 8C 8D 10A ··· 10L 10M ··· 10T 12A 12B 12C 12D 12E 12F 12G 12H 15A ··· 15H 20A ··· 20H 20I ··· 20P 24A ··· 24H 30A ··· 30X 30Y ··· 30AN 40A ··· 40P 60A ··· 60P 60Q ··· 60AF 120A ··· 120AF order 1 2 2 2 2 2 3 3 4 4 4 4 5 5 5 5 6 ··· 6 6 6 6 6 8 8 8 8 10 ··· 10 10 ··· 10 12 12 12 12 12 12 12 12 15 ··· 15 20 ··· 20 20 ··· 20 24 ··· 24 30 ··· 30 30 ··· 30 40 ··· 40 60 ··· 60 60 ··· 60 120 ··· 120 size 1 1 1 1 4 4 1 1 2 2 4 4 1 1 1 1 1 ··· 1 4 4 4 4 2 2 2 2 1 ··· 1 4 ··· 4 2 2 2 2 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 1 ··· 1 4 ··· 4 2 ··· 2 2 ··· 2 4 ··· 4 2 ··· 2

210 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 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 type + + + + + + + image C1 C2 C2 C2 C3 C4 C5 C6 C6 C6 C10 C10 C10 C12 C15 C20 C30 C30 C30 C60 D4 D4 D8 SD16 C3×D4 C3×D4 C5×D4 C5×D4 C3×D8 C3×SD16 C5×D8 C5×SD16 D4×C15 D4×C15 C15×D8 C15×SD16 kernel C15×D4⋊C4 C15×C4⋊C4 C2×C120 D4×C30 C5×D4⋊C4 D4×C15 C3×D4⋊C4 C5×C4⋊C4 C2×C40 D4×C10 C3×C4⋊C4 C2×C24 C6×D4 C5×D4 D4⋊C4 C3×D4 C4⋊C4 C2×C8 C2×D4 D4 C60 C2×C30 C30 C30 C20 C2×C10 C12 C2×C6 C10 C10 C6 C6 C4 C22 C2 C2 # reps 1 1 1 1 2 4 4 2 2 2 4 4 4 8 8 16 8 8 8 32 1 1 2 2 2 2 4 4 4 4 8 8 8 8 16 16

Matrix representation of C15×D4⋊C4 in GL3(𝔽241) generated by

 98 0 0 0 94 0 0 0 94
,
 1 0 0 0 0 1 0 240 0
,
 1 0 0 0 0 240 0 240 0
,
 64 0 0 0 19 222 0 222 222
G:=sub<GL(3,GF(241))| [98,0,0,0,94,0,0,0,94],[1,0,0,0,0,240,0,1,0],[1,0,0,0,0,240,0,240,0],[64,0,0,0,19,222,0,222,222] >;

C15×D4⋊C4 in GAP, Magma, Sage, TeX

C_{15}\times D_4\rtimes C_4
% in TeX

G:=Group("C15xD4:C4");
// GroupNames label

G:=SmallGroup(480,205);
// by ID

G=gap.SmallGroup(480,205);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,840,869,10504,5261,172]);
// Polycyclic

G:=Group<a,b,c,d|a^15=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b*c>;
// generators/relations

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