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## G = D4⋊Dic15order 480 = 25·3·5

### 1st semidirect product of D4 and Dic15 acting via Dic15/C30=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D4⋊Dic15
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C2×C60 — C60⋊5C4 — D4⋊Dic15
 Lower central C15 — C30 — C60 — D4⋊Dic15
 Upper central C1 — C22 — C2×C4 — C2×D4

Generators and relations for D4⋊Dic15
G = < a,b,c,d | a4=b2=c30=1, d2=c15, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

Subgroups: 404 in 100 conjugacy classes, 47 normal (39 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], Dic3, C12 [×2], C2×C6, C2×C6 [×4], C15, C4⋊C4, C2×C8, C2×D4, Dic5, C20 [×2], C2×C10, C2×C10 [×4], C3⋊C8, C2×Dic3, C2×C12, C3×D4 [×2], C3×D4, C22×C6, C30 [×3], C30 [×2], D4⋊C4, C52C8, C2×Dic5, C2×C20, C5×D4 [×2], C5×D4, C22×C10, C2×C3⋊C8, C4⋊Dic3, C6×D4, Dic15, C60 [×2], C2×C30, C2×C30 [×4], C2×C52C8, C4⋊Dic5, D4×C10, D4⋊Dic3, C153C8, C2×Dic15, C2×C60, D4×C15 [×2], D4×C15, C22×C30, D4⋊Dic5, C2×C153C8, C605C4, D4×C30, D4⋊Dic15
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, Dic3 [×2], D6, C22⋊C4, D8, SD16, Dic5 [×2], D10, C2×Dic3, C3⋊D4 [×2], D15, D4⋊C4, C2×Dic5, C5⋊D4 [×2], D4⋊S3, D4.S3, C6.D4, Dic15 [×2], D30, D4⋊D5, D4.D5, C23.D5, D4⋊Dic3, C2×Dic15, C157D4 [×2], D4⋊Dic5, D4⋊D15, D4.D15, C30.38D4, D4⋊Dic15

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

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 5A 5B 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 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 3 4 4 4 4 5 5 6 6 6 6 6 6 6 8 8 8 8 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 4 4 2 2 2 60 60 2 2 2 2 2 4 4 4 4 30 30 30 30 2 ··· 2 4 ··· 4 4 4 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

84 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 4 4 4 4 4 4 type + + + + + + + + + - + + - + + - + - + - + - image C1 C2 C2 C2 C4 S3 D4 D4 D5 D6 Dic3 D8 SD16 D10 Dic5 C3⋊D4 C3⋊D4 D15 C5⋊D4 C5⋊D4 D30 Dic15 C15⋊7D4 C15⋊7D4 D4⋊S3 D4.S3 D4⋊D5 D4.D5 D4⋊D15 D4.D15 kernel D4⋊Dic15 C2×C15⋊3C8 C60⋊5C4 D4×C30 D4×C15 D4×C10 C60 C2×C30 C6×D4 C2×C20 C5×D4 C30 C30 C2×C12 C3×D4 C20 C2×C10 C2×D4 C12 C2×C6 C2×C4 D4 C4 C22 C10 C10 C6 C6 C2 C2 # reps 1 1 1 1 4 1 1 1 2 1 2 2 2 2 4 2 2 4 4 4 4 8 8 8 1 1 2 2 4 4

Matrix representation of D4⋊Dic15 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 0 1 0 0 0 0 240 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 11 240 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 231 0 0 0 0 0 0 24 0 0 0 0 0 0 143 0 0 0 0 0 82 150 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 55 0 0 0 0 149 0 0 0 0 0 0 0 77 227 0 0 0 0 217 164 0 0 0 0 0 0 230 11 0 0 0 0 11 11

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,240,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,11,0,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[231,0,0,0,0,0,0,24,0,0,0,0,0,0,143,82,0,0,0,0,0,150,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,149,0,0,0,0,55,0,0,0,0,0,0,0,77,217,0,0,0,0,227,164,0,0,0,0,0,0,230,11,0,0,0,0,11,11] >;

D4⋊Dic15 in GAP, Magma, Sage, TeX

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

G:=Group("D4:Dic15");
// GroupNames label

G:=SmallGroup(480,192);
// by ID

G=gap.SmallGroup(480,192);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,675,346,80,2693,18822]);
// Polycyclic

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

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