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## G = Dic15⋊12D4order 480 = 25·3·5

### 2nd semidirect product of Dic15 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — Dic15⋊12D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C22×D15 — C2×C15⋊7D4 — Dic15⋊12D4
 Lower central C15 — C2×C30 — Dic15⋊12D4
 Upper central C1 — C22 — C2×D4

Generators and relations for Dic1512D4
G = < a,b,c,d | a30=c4=d2=1, b2=a15, bab-1=a-1, ac=ca, ad=da, cbc-1=a15b, bd=db, dcd=c-1 >

Subgroups: 1076 in 188 conjugacy classes, 57 normal (47 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, D15, C30, C30, C4⋊D4, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, Dic3⋊C4, D6⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4, C6×D4, Dic15, Dic15, C60, D30, C2×C30, C2×C30, C2×C30, C10.D4, D10⋊C4, C23.D5, C22×Dic5, C2×C5⋊D4, D4×C10, C23.14D6, C2×Dic15, C2×Dic15, C157D4, C2×C60, D4×C15, C22×D15, C22×C30, Dic5⋊D4, C30.4Q8, D303C4, C30.38D4, C22×Dic15, C2×C157D4, D4×C30, Dic1512D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C3⋊D4, C22×S3, D15, C4⋊D4, C5⋊D4, C22×D5, S3×D4, D42S3, C2×C3⋊D4, D30, D4×D5, D42D5, C2×C5⋊D4, C23.14D6, C157D4, C22×D15, Dic5⋊D4, D4×D15, D42D15, C2×C157D4, Dic1512D4

Smallest permutation representation of Dic1512D4
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)
(1 143 16 128)(2 142 17 127)(3 141 18 126)(4 140 19 125)(5 139 20 124)(6 138 21 123)(7 137 22 122)(8 136 23 121)(9 135 24 150)(10 134 25 149)(11 133 26 148)(12 132 27 147)(13 131 28 146)(14 130 29 145)(15 129 30 144)(31 117 46 102)(32 116 47 101)(33 115 48 100)(34 114 49 99)(35 113 50 98)(36 112 51 97)(37 111 52 96)(38 110 53 95)(39 109 54 94)(40 108 55 93)(41 107 56 92)(42 106 57 91)(43 105 58 120)(44 104 59 119)(45 103 60 118)(61 234 76 219)(62 233 77 218)(63 232 78 217)(64 231 79 216)(65 230 80 215)(66 229 81 214)(67 228 82 213)(68 227 83 212)(69 226 84 211)(70 225 85 240)(71 224 86 239)(72 223 87 238)(73 222 88 237)(74 221 89 236)(75 220 90 235)(151 204 166 189)(152 203 167 188)(153 202 168 187)(154 201 169 186)(155 200 170 185)(156 199 171 184)(157 198 172 183)(158 197 173 182)(159 196 174 181)(160 195 175 210)(161 194 176 209)(162 193 177 208)(163 192 178 207)(164 191 179 206)(165 190 180 205)
(1 93 85 197)(2 94 86 198)(3 95 87 199)(4 96 88 200)(5 97 89 201)(6 98 90 202)(7 99 61 203)(8 100 62 204)(9 101 63 205)(10 102 64 206)(11 103 65 207)(12 104 66 208)(13 105 67 209)(14 106 68 210)(15 107 69 181)(16 108 70 182)(17 109 71 183)(18 110 72 184)(19 111 73 185)(20 112 74 186)(21 113 75 187)(22 114 76 188)(23 115 77 189)(24 116 78 190)(25 117 79 191)(26 118 80 192)(27 119 81 193)(28 120 82 194)(29 91 83 195)(30 92 84 196)(31 216 164 149)(32 217 165 150)(33 218 166 121)(34 219 167 122)(35 220 168 123)(36 221 169 124)(37 222 170 125)(38 223 171 126)(39 224 172 127)(40 225 173 128)(41 226 174 129)(42 227 175 130)(43 228 176 131)(44 229 177 132)(45 230 178 133)(46 231 179 134)(47 232 180 135)(48 233 151 136)(49 234 152 137)(50 235 153 138)(51 236 154 139)(52 237 155 140)(53 238 156 141)(54 239 157 142)(55 240 158 143)(56 211 159 144)(57 212 160 145)(58 213 161 146)(59 214 162 147)(60 215 163 148)
(31 164)(32 165)(33 166)(34 167)(35 168)(36 169)(37 170)(38 171)(39 172)(40 173)(41 174)(42 175)(43 176)(44 177)(45 178)(46 179)(47 180)(48 151)(49 152)(50 153)(51 154)(52 155)(53 156)(54 157)(55 158)(56 159)(57 160)(58 161)(59 162)(60 163)(91 195)(92 196)(93 197)(94 198)(95 199)(96 200)(97 201)(98 202)(99 203)(100 204)(101 205)(102 206)(103 207)(104 208)(105 209)(106 210)(107 181)(108 182)(109 183)(110 184)(111 185)(112 186)(113 187)(114 188)(115 189)(116 190)(117 191)(118 192)(119 193)(120 194)

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

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

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

84 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 ··· 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 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 4 60 2 4 30 30 30 30 60 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

84 irreducible representations

 dim 1 1 1 1 1 1 1 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 C2 C2 C2 S3 D4 D4 D5 D6 D6 C4○D4 D10 D10 C3⋊D4 D15 C5⋊D4 D30 D30 C15⋊7D4 S3×D4 D4⋊2S3 D4×D5 D4⋊2D5 D4×D15 D4⋊2D15 kernel Dic15⋊12D4 C30.4Q8 D30⋊3C4 C30.38D4 C22×Dic15 C2×C15⋊7D4 D4×C30 D4×C10 Dic15 C2×C30 C6×D4 C2×C20 C22×C10 C30 C2×C12 C22×C6 C2×C10 C2×D4 C2×C6 C2×C4 C23 C22 C10 C10 C6 C6 C2 C2 # reps 1 1 1 1 1 2 1 1 2 2 2 1 2 2 2 4 4 4 8 4 8 16 1 1 2 2 4 4

Matrix representation of Dic1512D4 in GL4(𝔽61) generated by

 28 37 0 0 24 47 0 0 0 0 1 0 0 0 0 1
,
 11 57 0 0 0 50 0 0 0 0 60 0 0 0 0 60
,
 14 45 0 0 16 47 0 0 0 0 1 21 0 0 58 60
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 58 60
G:=sub<GL(4,GF(61))| [28,24,0,0,37,47,0,0,0,0,1,0,0,0,0,1],[11,0,0,0,57,50,0,0,0,0,60,0,0,0,0,60],[14,16,0,0,45,47,0,0,0,0,1,58,0,0,21,60],[1,0,0,0,0,1,0,0,0,0,1,58,0,0,0,60] >;

Dic1512D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}\rtimes_{12}D_4
% in TeX

G:=Group("Dic15:12D4");
// GroupNames label

G:=SmallGroup(480,904);
// by ID

G=gap.SmallGroup(480,904);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,254,219,2693,18822]);
// 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,a*d=d*a,c*b*c^-1=a^15*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽