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## G = D4×Dic15order 480 = 25·3·5

### Direct product of D4 and Dic15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D4×Dic15
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C2×Dic15 — C22×Dic15 — D4×Dic15
 Lower central C15 — C30 — D4×Dic15
 Upper central C1 — C22 — C2×D4

Generators and relations for D4×Dic15
G = < a,b,c,d | a4=b2=c30=1, d2=c15, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 788 in 188 conjugacy classes, 89 normal (33 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C5, C6, C6, C2×C4, C2×C4, D4, C23, C10, C10, Dic3, C12, C2×C6, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, C2×C10, C2×C10, C2×C10, C2×Dic3, C2×C12, C3×D4, C22×C6, C30, C30, C4×D4, C2×Dic5, C2×C20, C5×D4, C22×C10, C4×Dic3, C4⋊Dic3, C6.D4, C22×Dic3, C6×D4, Dic15, Dic15, C60, C2×C30, C2×C30, C2×C30, C4×Dic5, C4⋊Dic5, C23.D5, C22×Dic5, D4×C10, D4×Dic3, C2×Dic15, C2×Dic15, C2×Dic15, C2×C60, D4×C15, C22×C30, D4×Dic5, C4×Dic15, C605C4, C30.38D4, C22×Dic15, D4×C30, D4×Dic15
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, Dic3, D6, C22×C4, C2×D4, C4○D4, Dic5, D10, C2×Dic3, C22×S3, D15, C4×D4, C2×Dic5, C22×D5, S3×D4, D42S3, C22×Dic3, Dic15, D30, D4×D5, D42D5, C22×Dic5, D4×Dic3, C2×Dic15, C22×D15, D4×Dic5, D4×D15, D42D15, C22×Dic15, D4×Dic15

Smallest permutation representation of D4×Dic15
On 240 points
Generators in S240
(1 34 100 127)(2 35 101 128)(3 36 102 129)(4 37 103 130)(5 38 104 131)(6 39 105 132)(7 40 106 133)(8 41 107 134)(9 42 108 135)(10 43 109 136)(11 44 110 137)(12 45 111 138)(13 46 112 139)(14 47 113 140)(15 48 114 141)(16 49 115 142)(17 50 116 143)(18 51 117 144)(19 52 118 145)(20 53 119 146)(21 54 120 147)(22 55 91 148)(23 56 92 149)(24 57 93 150)(25 58 94 121)(26 59 95 122)(27 60 96 123)(28 31 97 124)(29 32 98 125)(30 33 99 126)(61 152 227 204)(62 153 228 205)(63 154 229 206)(64 155 230 207)(65 156 231 208)(66 157 232 209)(67 158 233 210)(68 159 234 181)(69 160 235 182)(70 161 236 183)(71 162 237 184)(72 163 238 185)(73 164 239 186)(74 165 240 187)(75 166 211 188)(76 167 212 189)(77 168 213 190)(78 169 214 191)(79 170 215 192)(80 171 216 193)(81 172 217 194)(82 173 218 195)(83 174 219 196)(84 175 220 197)(85 176 221 198)(86 177 222 199)(87 178 223 200)(88 179 224 201)(89 180 225 202)(90 151 226 203)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(31 139)(32 140)(33 141)(34 142)(35 143)(36 144)(37 145)(38 146)(39 147)(40 148)(41 149)(42 150)(43 121)(44 122)(45 123)(46 124)(47 125)(48 126)(49 127)(50 128)(51 129)(52 130)(53 131)(54 132)(55 133)(56 134)(57 135)(58 136)(59 137)(60 138)(61 76)(62 77)(63 78)(64 79)(65 80)(66 81)(67 82)(68 83)(69 84)(70 85)(71 86)(72 87)(73 88)(74 89)(75 90)(91 106)(92 107)(93 108)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)(105 120)(151 188)(152 189)(153 190)(154 191)(155 192)(156 193)(157 194)(158 195)(159 196)(160 197)(161 198)(162 199)(163 200)(164 201)(165 202)(166 203)(167 204)(168 205)(169 206)(170 207)(171 208)(172 209)(173 210)(174 181)(175 182)(176 183)(177 184)(178 185)(179 186)(180 187)(211 226)(212 227)(213 228)(214 229)(215 230)(216 231)(217 232)(218 233)(219 234)(220 235)(221 236)(222 237)(223 238)(224 239)(225 240)
(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)
(1 215 16 230)(2 214 17 229)(3 213 18 228)(4 212 19 227)(5 211 20 226)(6 240 21 225)(7 239 22 224)(8 238 23 223)(9 237 24 222)(10 236 25 221)(11 235 26 220)(12 234 27 219)(13 233 28 218)(14 232 29 217)(15 231 30 216)(31 195 46 210)(32 194 47 209)(33 193 48 208)(34 192 49 207)(35 191 50 206)(36 190 51 205)(37 189 52 204)(38 188 53 203)(39 187 54 202)(40 186 55 201)(41 185 56 200)(42 184 57 199)(43 183 58 198)(44 182 59 197)(45 181 60 196)(61 103 76 118)(62 102 77 117)(63 101 78 116)(64 100 79 115)(65 99 80 114)(66 98 81 113)(67 97 82 112)(68 96 83 111)(69 95 84 110)(70 94 85 109)(71 93 86 108)(72 92 87 107)(73 91 88 106)(74 120 89 105)(75 119 90 104)(121 176 136 161)(122 175 137 160)(123 174 138 159)(124 173 139 158)(125 172 140 157)(126 171 141 156)(127 170 142 155)(128 169 143 154)(129 168 144 153)(130 167 145 152)(131 166 146 151)(132 165 147 180)(133 164 148 179)(134 163 149 178)(135 162 150 177)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G ··· 4L 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A ··· 10F 10G ··· 10N 12A 12B 15A 15B 15C 15D 20A 20B 20C 20D 30A ··· 30L 30M ··· 30AB 60A ··· 60H order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 ··· 4 5 5 6 6 6 6 6 6 6 10 ··· 10 10 ··· 10 12 12 15 15 15 15 20 20 20 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 2 2 2 2 2 15 15 15 15 30 ··· 30 2 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 4 4 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

90 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + - + + - + + + - + + - + - + - image C1 C2 C2 C2 C2 C2 C4 S3 D4 D5 D6 Dic3 D6 C4○D4 D10 Dic5 D10 D15 D30 Dic15 D30 S3×D4 D4⋊2S3 D4×D5 D4⋊2D5 D4×D15 D4⋊2D15 kernel D4×Dic15 C4×Dic15 C60⋊5C4 C30.38D4 C22×Dic15 D4×C30 D4×C15 D4×C10 Dic15 C6×D4 C2×C20 C5×D4 C22×C10 C30 C2×C12 C3×D4 C22×C6 C2×D4 C2×C4 D4 C23 C10 C10 C6 C6 C2 C2 # reps 1 1 1 2 2 1 8 1 2 2 1 4 2 2 2 8 4 4 4 16 8 1 1 2 2 4 4

Matrix representation of D4×Dic15 in GL4(𝔽61) generated by

 60 0 0 0 0 60 0 0 0 0 1 2 0 0 60 60
,
 60 0 0 0 0 60 0 0 0 0 1 0 0 0 60 60
,
 52 37 0 0 29 23 0 0 0 0 1 0 0 0 0 1
,
 30 45 0 0 22 31 0 0 0 0 60 0 0 0 0 60
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,1,60,0,0,2,60],[60,0,0,0,0,60,0,0,0,0,1,60,0,0,0,60],[52,29,0,0,37,23,0,0,0,0,1,0,0,0,0,1],[30,22,0,0,45,31,0,0,0,0,60,0,0,0,0,60] >;

D4×Dic15 in GAP, Magma, Sage, TeX

D_4\times {\rm Dic}_{15}
% in TeX

G:=Group("D4xDic15");
// GroupNames label

G:=SmallGroup(480,899);
// by ID

G=gap.SmallGroup(480,899);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,219,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^30=1,d^2=c^15,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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