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## G = Dic15⋊D4order 480 = 25·3·5

### 1st semidirect product of Dic15 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — Dic15⋊D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×D5×Dic3 — Dic15⋊D4
 Lower central C15 — C2×C30 — Dic15⋊D4
 Upper central C1 — C22 — C2×C4

Generators and relations for Dic15⋊D4
G = < a,b,c,d | a30=c4=d2=1, b2=a15, bab-1=a-1, ac=ca, dad=a19, cbc-1=dbd=a15b, dcd=c-1 >

Subgroups: 1052 in 188 conjugacy classes, 50 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C5×S3, C3×D5, C30, C4⋊D4, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, Dic3⋊C4, D6⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4, C6×D4, C5×Dic3, Dic15, Dic15, C60, C6×D5, C6×D5, S3×C10, C2×C30, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C23.14D6, D5×Dic3, C15⋊D4, C3×D20, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, S3×C2×C10, D10⋊D4, D10⋊Dic3, C5×D6⋊C4, C30.4Q8, C2×D5×Dic3, C2×C15⋊D4, C6×D20, Dic15⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C3⋊D4, C22×S3, C4⋊D4, C22×D5, S3×D4, D42S3, C2×C3⋊D4, S3×D5, C4○D20, D4×D5, C23.14D6, C2×S3×D5, D10⋊D4, D205S3, C20⋊D6, D5×C3⋊D4, Dic15⋊D4

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

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + - + + + - image C1 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 C4○D4 D10 D10 D10 C3⋊D4 C4○D20 S3×D4 D4⋊2S3 S3×D5 D4×D5 C2×S3×D5 D20⋊5S3 C20⋊D6 D5×C3⋊D4 kernel Dic15⋊D4 D10⋊Dic3 C5×D6⋊C4 C30.4Q8 C2×D5×Dic3 C2×C15⋊D4 C6×D20 C2×D20 Dic15 C6×D5 D6⋊C4 C2×C20 C22×D5 C30 C2×Dic3 C2×C12 C22×S3 D10 C6 C10 C10 C2×C4 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 2 1 1 2 2 2 1 2 2 2 2 2 4 8 1 1 2 4 2 4 4 4

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

 0 60 0 0 1 18 0 0 0 0 2 15 0 0 12 60
,
 11 0 0 0 46 50 0 0 0 0 46 52 0 0 59 15
,
 25 57 0 0 4 36 0 0 0 0 27 26 0 0 33 34
,
 25 57 0 0 34 36 0 0 0 0 34 35 0 0 28 27
G:=sub<GL(4,GF(61))| [0,1,0,0,60,18,0,0,0,0,2,12,0,0,15,60],[11,46,0,0,0,50,0,0,0,0,46,59,0,0,52,15],[25,4,0,0,57,36,0,0,0,0,27,33,0,0,26,34],[25,34,0,0,57,36,0,0,0,0,34,28,0,0,35,27] >;

Dic15⋊D4 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(480,484);
// by ID

G=gap.SmallGroup(480,484);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,590,219,142,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=c^4=d^2=1,b^2=a^15,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^19,c*b*c^-1=d*b*d=a^15*b,d*c*d=c^-1>;
// generators/relations

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