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## G = Dic15⋊3D4order 480 = 25·3·5

### 3rd semidirect product of Dic15 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — Dic15⋊3D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×D5×Dic3 — Dic15⋊3D4
 Lower central C15 — C2×C30 — Dic15⋊3D4
 Upper central C1 — C22 — C23

Generators and relations for Dic153D4
G = < a,b,c,d | a30=c4=d2=1, b2=a15, bab-1=a-1, cac-1=dad=a19, cbc-1=a15b, bd=db, dcd=c-1 >

Subgroups: 1116 in 188 conjugacy classes, 50 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×5], C22, C22 [×10], C5, S3, C6 [×3], C6 [×3], C2×C4 [×6], D4 [×6], C23, C23 [×2], D5 [×3], C10 [×3], C10, Dic3 [×4], C12, D6 [×3], C2×C6, C2×C6 [×7], C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5 [×3], C20 [×2], D10 [×2], D10 [×5], C2×C10, C2×C10 [×3], C2×Dic3 [×2], C2×Dic3 [×3], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3, C22×C6, C22×C6, C3×D5 [×2], D15, C30 [×3], C30, C4⋊D4, C4×D5 [×2], D20 [×2], C2×Dic5, C2×Dic5, C5⋊D4 [×4], C2×C20 [×2], C22×D5, C22×D5, C22×C10, Dic3⋊C4, D6⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4 [×2], C6×D4, C5×Dic3 [×2], C3×Dic5, Dic15 [×2], C6×D5 [×2], C6×D5 [×2], D30 [×3], C2×C30, C2×C30 [×3], C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C2×C5⋊D4, C23.14D6, D5×Dic3 [×2], C3⋊D20 [×2], C6×Dic5, C3×C5⋊D4 [×2], C10×Dic3 [×2], C2×Dic15, C157D4 [×2], D5×C2×C6, C22×D15, C22×C30, D10⋊D4, D304C4, Dic155C4, C5×C6.D4, C2×D5×Dic3, C2×C3⋊D20, C6×C5⋊D4, C2×C157D4, Dic153D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D5, D6 [×3], C2×D4 [×2], C4○D4, D10 [×3], C3⋊D4 [×2], C22×S3, C4⋊D4, C22×D5, S3×D4, D42S3, C2×C3⋊D4, S3×D5, C4○D20, D4×D5 [×2], C23.14D6, C2×S3×D5, D10⋊D4, Dic5.D6, D5×C3⋊D4, D10⋊D6, Dic153D4

Smallest permutation representation of Dic153D4
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 66 16 81)(2 65 17 80)(3 64 18 79)(4 63 19 78)(5 62 20 77)(6 61 21 76)(7 90 22 75)(8 89 23 74)(9 88 24 73)(10 87 25 72)(11 86 26 71)(12 85 27 70)(13 84 28 69)(14 83 29 68)(15 82 30 67)(31 232 46 217)(32 231 47 216)(33 230 48 215)(34 229 49 214)(35 228 50 213)(36 227 51 212)(37 226 52 211)(38 225 53 240)(39 224 54 239)(40 223 55 238)(41 222 56 237)(42 221 57 236)(43 220 58 235)(44 219 59 234)(45 218 60 233)(91 189 106 204)(92 188 107 203)(93 187 108 202)(94 186 109 201)(95 185 110 200)(96 184 111 199)(97 183 112 198)(98 182 113 197)(99 181 114 196)(100 210 115 195)(101 209 116 194)(102 208 117 193)(103 207 118 192)(104 206 119 191)(105 205 120 190)(121 171 136 156)(122 170 137 155)(123 169 138 154)(124 168 139 153)(125 167 140 152)(126 166 141 151)(127 165 142 180)(128 164 143 179)(129 163 144 178)(130 162 145 177)(131 161 146 176)(132 160 147 175)(133 159 148 174)(134 158 149 173)(135 157 150 172)
(1 226 118 132)(2 215 119 121)(3 234 120 140)(4 223 91 129)(5 212 92 148)(6 231 93 137)(7 220 94 126)(8 239 95 145)(9 228 96 134)(10 217 97 123)(11 236 98 142)(12 225 99 131)(13 214 100 150)(14 233 101 139)(15 222 102 128)(16 211 103 147)(17 230 104 136)(18 219 105 125)(19 238 106 144)(20 227 107 133)(21 216 108 122)(22 235 109 141)(23 224 110 130)(24 213 111 149)(25 232 112 138)(26 221 113 127)(27 240 114 146)(28 229 115 135)(29 218 116 124)(30 237 117 143)(31 198 169 72)(32 187 170 61)(33 206 171 80)(34 195 172 69)(35 184 173 88)(36 203 174 77)(37 192 175 66)(38 181 176 85)(39 200 177 74)(40 189 178 63)(41 208 179 82)(42 197 180 71)(43 186 151 90)(44 205 152 79)(45 194 153 68)(46 183 154 87)(47 202 155 76)(48 191 156 65)(49 210 157 84)(50 199 158 73)(51 188 159 62)(52 207 160 81)(53 196 161 70)(54 185 162 89)(55 204 163 78)(56 193 164 67)(57 182 165 86)(58 201 166 75)(59 190 167 64)(60 209 168 83)
(1 103)(2 92)(3 111)(4 100)(5 119)(6 108)(7 97)(8 116)(9 105)(10 94)(11 113)(12 102)(13 91)(14 110)(15 99)(16 118)(17 107)(18 96)(19 115)(20 104)(21 93)(22 112)(23 101)(24 120)(25 109)(26 98)(27 117)(28 106)(29 95)(30 114)(31 58)(32 47)(33 36)(34 55)(35 44)(37 52)(38 41)(39 60)(40 49)(42 57)(43 46)(45 54)(48 51)(50 59)(53 56)(61 202)(62 191)(63 210)(64 199)(65 188)(66 207)(67 196)(68 185)(69 204)(70 193)(71 182)(72 201)(73 190)(74 209)(75 198)(76 187)(77 206)(78 195)(79 184)(80 203)(81 192)(82 181)(83 200)(84 189)(85 208)(86 197)(87 186)(88 205)(89 194)(90 183)(121 148)(122 137)(123 126)(124 145)(125 134)(127 142)(128 131)(129 150)(130 139)(132 147)(133 136)(135 144)(138 141)(140 149)(143 146)(151 154)(152 173)(153 162)(155 170)(156 159)(157 178)(158 167)(160 175)(161 164)(163 172)(165 180)(166 169)(168 177)(171 174)(176 179)(211 226)(212 215)(213 234)(214 223)(216 231)(217 220)(218 239)(219 228)(221 236)(222 225)(224 233)(227 230)(229 238)(232 235)(237 240)

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

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

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

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 ··· 20H 30A ··· 30N 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 30 ··· 30 size 1 1 1 1 4 10 10 60 2 6 6 12 20 30 30 2 2 2 2 2 4 4 20 20 2 ··· 2 4 4 4 4 20 20 4 4 12 ··· 12 4 ··· 4

60 irreducible representations

 dim 1 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 C2 S3 D4 D4 D5 D6 D6 D6 C4○D4 D10 D10 C3⋊D4 C4○D20 S3×D4 D4⋊2S3 S3×D5 D4×D5 C2×S3×D5 Dic5.D6 D5×C3⋊D4 D10⋊D6 kernel Dic15⋊3D4 D30⋊4C4 Dic15⋊5C4 C5×C6.D4 C2×D5×Dic3 C2×C3⋊D20 C6×C5⋊D4 C2×C15⋊7D4 C2×C5⋊D4 Dic15 C6×D5 C6.D4 C2×Dic5 C22×D5 C22×C10 C30 C2×Dic3 C22×C6 D10 C6 C10 C10 C23 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 2 4 2 4 8 1 1 2 4 2 4 4 4

Matrix representation of Dic153D4 in GL4(𝔽61) generated by

 60 56 0 0 25 2 0 0 0 0 17 60 0 0 45 1
,
 26 28 0 0 39 35 0 0 0 0 4 15 0 0 7 57
,
 34 32 0 0 23 27 0 0 0 0 39 8 0 0 8 22
,
 60 0 0 0 0 60 0 0 0 0 17 18 0 0 45 44
G:=sub<GL(4,GF(61))| [60,25,0,0,56,2,0,0,0,0,17,45,0,0,60,1],[26,39,0,0,28,35,0,0,0,0,4,7,0,0,15,57],[34,23,0,0,32,27,0,0,0,0,39,8,0,0,8,22],[60,0,0,0,0,60,0,0,0,0,17,45,0,0,18,44] >;

Dic153D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}\rtimes_3D_4
% in TeX

G:=Group("Dic15:3D4");
// GroupNames label

G:=SmallGroup(480,626);
// by ID

G=gap.SmallGroup(480,626);
# by ID

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

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

׿
×
𝔽