metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.Dic15, Q8.2Dic15, C60.259C23, (C3×Q8).Dic5, (C3×D4).Dic5, C15⋊27(C8○D4), C60.89(C2×C4), C4○D4.5D15, (D4×C15).3C4, (C2×C4).58D30, (Q8×C15).3C4, (C2×C20).157D6, C5⋊5(D4.Dic3), C3⋊3(D4.Dic5), (C5×Q8).5Dic3, (C5×D4).2Dic3, C60.7C4⋊20C2, C4.5(C2×Dic15), (C2×C12).155D10, (C2×C60).82C22, C12.15(C2×Dic5), C20.36(C2×Dic3), C4.41(C22×D15), C30.217(C22×C4), C20.229(C22×S3), C15⋊3C8.39C22, C12.231(C22×D5), C22.1(C2×Dic15), C6.28(C22×Dic5), C2.8(C22×Dic15), C10.41(C22×Dic3), (C2×C15⋊3C8)⋊7C2, (C5×C4○D4).6S3, (C3×C4○D4).2D5, (C2×C30).70(C2×C4), (C15×C4○D4).2C2, (C2×C6).7(C2×Dic5), (C2×C10).27(C2×Dic3), SmallGroup(480,913)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.Dic15
G = < a,b,c,d | a4=b2=1, c30=a2, d2=a2c15, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c29 >
Subgroups: 356 in 124 conjugacy classes, 79 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, D4, Q8, C10, C10, C12, C12, C2×C6, C15, C2×C8, M4(2), C4○D4, C20, C20, C2×C10, C3⋊C8, C2×C12, C3×D4, C3×Q8, C30, C30, C8○D4, C5⋊2C8, C2×C20, C5×D4, C5×Q8, C2×C3⋊C8, C4.Dic3, C3×C4○D4, C60, C60, C2×C30, C2×C5⋊2C8, C4.Dic5, C5×C4○D4, D4.Dic3, C15⋊3C8, C15⋊3C8, C2×C60, D4×C15, Q8×C15, D4.Dic5, C2×C15⋊3C8, C60.7C4, C15×C4○D4, D4.Dic15
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, Dic3, D6, C22×C4, Dic5, D10, C2×Dic3, C22×S3, D15, C8○D4, C2×Dic5, C22×D5, C22×Dic3, Dic15, D30, C22×Dic5, D4.Dic3, C2×Dic15, C22×D15, D4.Dic5, C22×Dic15, D4.Dic15
►On 240 points
Generators in S240
(1 46 31 16)(2 47 32 17)(3 48 33 18)(4 49 34 19)(5 50 35 20)(6 51 36 21)(7 52 37 22)(8 53 38 23)(9 54 39 24)(10 55 40 25)(11 56 41 26)(12 57 42 27)(13 58 43 28)(14 59 44 29)(15 60 45 30)(61 76 91 106)(62 77 92 107)(63 78 93 108)(64 79 94 109)(65 80 95 110)(66 81 96 111)(67 82 97 112)(68 83 98 113)(69 84 99 114)(70 85 100 115)(71 86 101 116)(72 87 102 117)(73 88 103 118)(74 89 104 119)(75 90 105 120)(121 166 151 136)(122 167 152 137)(123 168 153 138)(124 169 154 139)(125 170 155 140)(126 171 156 141)(127 172 157 142)(128 173 158 143)(129 174 159 144)(130 175 160 145)(131 176 161 146)(132 177 162 147)(133 178 163 148)(134 179 164 149)(135 180 165 150)(181 196 211 226)(182 197 212 227)(183 198 213 228)(184 199 214 229)(185 200 215 230)(186 201 216 231)(187 202 217 232)(188 203 218 233)(189 204 219 234)(190 205 220 235)(191 206 221 236)(192 207 222 237)(193 208 223 238)(194 209 224 239)(195 210 225 240)
(1 98)(2 99)(3 100)(4 101)(5 102)(6 103)(7 104)(8 105)(9 106)(10 107)(11 108)(12 109)(13 110)(14 111)(15 112)(16 113)(17 114)(18 115)(19 116)(20 117)(21 118)(22 119)(23 120)(24 61)(25 62)(26 63)(27 64)(28 65)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 73)(37 74)(38 75)(39 76)(40 77)(41 78)(42 79)(43 80)(44 81)(45 82)(46 83)(47 84)(48 85)(49 86)(50 87)(51 88)(52 89)(53 90)(54 91)(55 92)(56 93)(57 94)(58 95)(59 96)(60 97)(121 196)(122 197)(123 198)(124 199)(125 200)(126 201)(127 202)(128 203)(129 204)(130 205)(131 206)(132 207)(133 208)(134 209)(135 210)(136 211)(137 212)(138 213)(139 214)(140 215)(141 216)(142 217)(143 218)(144 219)(145 220)(146 221)(147 222)(148 223)(149 224)(150 225)(151 226)(152 227)(153 228)(154 229)(155 230)(156 231)(157 232)(158 233)(159 234)(160 235)(161 236)(162 237)(163 238)(164 239)(165 240)(166 181)(167 182)(168 183)(169 184)(170 185)(171 186)(172 187)(173 188)(174 189)(175 190)(176 191)(177 192)(178 193)(179 194)(180 195)
(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)
(1 171 46 156 31 141 16 126)(2 140 47 125 32 170 17 155)(3 169 48 154 33 139 18 124)(4 138 49 123 34 168 19 153)(5 167 50 152 35 137 20 122)(6 136 51 121 36 166 21 151)(7 165 52 150 37 135 22 180)(8 134 53 179 38 164 23 149)(9 163 54 148 39 133 24 178)(10 132 55 177 40 162 25 147)(11 161 56 146 41 131 26 176)(12 130 57 175 42 160 27 145)(13 159 58 144 43 129 28 174)(14 128 59 173 44 158 29 143)(15 157 60 142 45 127 30 172)(61 193 106 238 91 223 76 208)(62 222 107 207 92 192 77 237)(63 191 108 236 93 221 78 206)(64 220 109 205 94 190 79 235)(65 189 110 234 95 219 80 204)(66 218 111 203 96 188 81 233)(67 187 112 232 97 217 82 202)(68 216 113 201 98 186 83 231)(69 185 114 230 99 215 84 200)(70 214 115 199 100 184 85 229)(71 183 116 228 101 213 86 198)(72 212 117 197 102 182 87 227)(73 181 118 226 103 211 88 196)(74 210 119 195 104 240 89 225)(75 239 120 224 105 209 90 194)
G:=sub<Sym(240)| (1,46,31,16)(2,47,32,17)(3,48,33,18)(4,49,34,19)(5,50,35,20)(6,51,36,21)(7,52,37,22)(8,53,38,23)(9,54,39,24)(10,55,40,25)(11,56,41,26)(12,57,42,27)(13,58,43,28)(14,59,44,29)(15,60,45,30)(61,76,91,106)(62,77,92,107)(63,78,93,108)(64,79,94,109)(65,80,95,110)(66,81,96,111)(67,82,97,112)(68,83,98,113)(69,84,99,114)(70,85,100,115)(71,86,101,116)(72,87,102,117)(73,88,103,118)(74,89,104,119)(75,90,105,120)(121,166,151,136)(122,167,152,137)(123,168,153,138)(124,169,154,139)(125,170,155,140)(126,171,156,141)(127,172,157,142)(128,173,158,143)(129,174,159,144)(130,175,160,145)(131,176,161,146)(132,177,162,147)(133,178,163,148)(134,179,164,149)(135,180,165,150)(181,196,211,226)(182,197,212,227)(183,198,213,228)(184,199,214,229)(185,200,215,230)(186,201,216,231)(187,202,217,232)(188,203,218,233)(189,204,219,234)(190,205,220,235)(191,206,221,236)(192,207,222,237)(193,208,223,238)(194,209,224,239)(195,210,225,240), (1,98)(2,99)(3,100)(4,101)(5,102)(6,103)(7,104)(8,105)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,112)(16,113)(17,114)(18,115)(19,116)(20,117)(21,118)(22,119)(23,120)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,73)(37,74)(38,75)(39,76)(40,77)(41,78)(42,79)(43,80)(44,81)(45,82)(46,83)(47,84)(48,85)(49,86)(50,87)(51,88)(52,89)(53,90)(54,91)(55,92)(56,93)(57,94)(58,95)(59,96)(60,97)(121,196)(122,197)(123,198)(124,199)(125,200)(126,201)(127,202)(128,203)(129,204)(130,205)(131,206)(132,207)(133,208)(134,209)(135,210)(136,211)(137,212)(138,213)(139,214)(140,215)(141,216)(142,217)(143,218)(144,219)(145,220)(146,221)(147,222)(148,223)(149,224)(150,225)(151,226)(152,227)(153,228)(154,229)(155,230)(156,231)(157,232)(158,233)(159,234)(160,235)(161,236)(162,237)(163,238)(164,239)(165,240)(166,181)(167,182)(168,183)(169,184)(170,185)(171,186)(172,187)(173,188)(174,189)(175,190)(176,191)(177,192)(178,193)(179,194)(180,195), (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), (1,171,46,156,31,141,16,126)(2,140,47,125,32,170,17,155)(3,169,48,154,33,139,18,124)(4,138,49,123,34,168,19,153)(5,167,50,152,35,137,20,122)(6,136,51,121,36,166,21,151)(7,165,52,150,37,135,22,180)(8,134,53,179,38,164,23,149)(9,163,54,148,39,133,24,178)(10,132,55,177,40,162,25,147)(11,161,56,146,41,131,26,176)(12,130,57,175,42,160,27,145)(13,159,58,144,43,129,28,174)(14,128,59,173,44,158,29,143)(15,157,60,142,45,127,30,172)(61,193,106,238,91,223,76,208)(62,222,107,207,92,192,77,237)(63,191,108,236,93,221,78,206)(64,220,109,205,94,190,79,235)(65,189,110,234,95,219,80,204)(66,218,111,203,96,188,81,233)(67,187,112,232,97,217,82,202)(68,216,113,201,98,186,83,231)(69,185,114,230,99,215,84,200)(70,214,115,199,100,184,85,229)(71,183,116,228,101,213,86,198)(72,212,117,197,102,182,87,227)(73,181,118,226,103,211,88,196)(74,210,119,195,104,240,89,225)(75,239,120,224,105,209,90,194)>;
G:=Group( (1,46,31,16)(2,47,32,17)(3,48,33,18)(4,49,34,19)(5,50,35,20)(6,51,36,21)(7,52,37,22)(8,53,38,23)(9,54,39,24)(10,55,40,25)(11,56,41,26)(12,57,42,27)(13,58,43,28)(14,59,44,29)(15,60,45,30)(61,76,91,106)(62,77,92,107)(63,78,93,108)(64,79,94,109)(65,80,95,110)(66,81,96,111)(67,82,97,112)(68,83,98,113)(69,84,99,114)(70,85,100,115)(71,86,101,116)(72,87,102,117)(73,88,103,118)(74,89,104,119)(75,90,105,120)(121,166,151,136)(122,167,152,137)(123,168,153,138)(124,169,154,139)(125,170,155,140)(126,171,156,141)(127,172,157,142)(128,173,158,143)(129,174,159,144)(130,175,160,145)(131,176,161,146)(132,177,162,147)(133,178,163,148)(134,179,164,149)(135,180,165,150)(181,196,211,226)(182,197,212,227)(183,198,213,228)(184,199,214,229)(185,200,215,230)(186,201,216,231)(187,202,217,232)(188,203,218,233)(189,204,219,234)(190,205,220,235)(191,206,221,236)(192,207,222,237)(193,208,223,238)(194,209,224,239)(195,210,225,240), (1,98)(2,99)(3,100)(4,101)(5,102)(6,103)(7,104)(8,105)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,112)(16,113)(17,114)(18,115)(19,116)(20,117)(21,118)(22,119)(23,120)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,73)(37,74)(38,75)(39,76)(40,77)(41,78)(42,79)(43,80)(44,81)(45,82)(46,83)(47,84)(48,85)(49,86)(50,87)(51,88)(52,89)(53,90)(54,91)(55,92)(56,93)(57,94)(58,95)(59,96)(60,97)(121,196)(122,197)(123,198)(124,199)(125,200)(126,201)(127,202)(128,203)(129,204)(130,205)(131,206)(132,207)(133,208)(134,209)(135,210)(136,211)(137,212)(138,213)(139,214)(140,215)(141,216)(142,217)(143,218)(144,219)(145,220)(146,221)(147,222)(148,223)(149,224)(150,225)(151,226)(152,227)(153,228)(154,229)(155,230)(156,231)(157,232)(158,233)(159,234)(160,235)(161,236)(162,237)(163,238)(164,239)(165,240)(166,181)(167,182)(168,183)(169,184)(170,185)(171,186)(172,187)(173,188)(174,189)(175,190)(176,191)(177,192)(178,193)(179,194)(180,195), (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), (1,171,46,156,31,141,16,126)(2,140,47,125,32,170,17,155)(3,169,48,154,33,139,18,124)(4,138,49,123,34,168,19,153)(5,167,50,152,35,137,20,122)(6,136,51,121,36,166,21,151)(7,165,52,150,37,135,22,180)(8,134,53,179,38,164,23,149)(9,163,54,148,39,133,24,178)(10,132,55,177,40,162,25,147)(11,161,56,146,41,131,26,176)(12,130,57,175,42,160,27,145)(13,159,58,144,43,129,28,174)(14,128,59,173,44,158,29,143)(15,157,60,142,45,127,30,172)(61,193,106,238,91,223,76,208)(62,222,107,207,92,192,77,237)(63,191,108,236,93,221,78,206)(64,220,109,205,94,190,79,235)(65,189,110,234,95,219,80,204)(66,218,111,203,96,188,81,233)(67,187,112,232,97,217,82,202)(68,216,113,201,98,186,83,231)(69,185,114,230,99,215,84,200)(70,214,115,199,100,184,85,229)(71,183,116,228,101,213,86,198)(72,212,117,197,102,182,87,227)(73,181,118,226,103,211,88,196)(74,210,119,195,104,240,89,225)(75,239,120,224,105,209,90,194) );
G=PermutationGroup([[(1,46,31,16),(2,47,32,17),(3,48,33,18),(4,49,34,19),(5,50,35,20),(6,51,36,21),(7,52,37,22),(8,53,38,23),(9,54,39,24),(10,55,40,25),(11,56,41,26),(12,57,42,27),(13,58,43,28),(14,59,44,29),(15,60,45,30),(61,76,91,106),(62,77,92,107),(63,78,93,108),(64,79,94,109),(65,80,95,110),(66,81,96,111),(67,82,97,112),(68,83,98,113),(69,84,99,114),(70,85,100,115),(71,86,101,116),(72,87,102,117),(73,88,103,118),(74,89,104,119),(75,90,105,120),(121,166,151,136),(122,167,152,137),(123,168,153,138),(124,169,154,139),(125,170,155,140),(126,171,156,141),(127,172,157,142),(128,173,158,143),(129,174,159,144),(130,175,160,145),(131,176,161,146),(132,177,162,147),(133,178,163,148),(134,179,164,149),(135,180,165,150),(181,196,211,226),(182,197,212,227),(183,198,213,228),(184,199,214,229),(185,200,215,230),(186,201,216,231),(187,202,217,232),(188,203,218,233),(189,204,219,234),(190,205,220,235),(191,206,221,236),(192,207,222,237),(193,208,223,238),(194,209,224,239),(195,210,225,240)], [(1,98),(2,99),(3,100),(4,101),(5,102),(6,103),(7,104),(8,105),(9,106),(10,107),(11,108),(12,109),(13,110),(14,111),(15,112),(16,113),(17,114),(18,115),(19,116),(20,117),(21,118),(22,119),(23,120),(24,61),(25,62),(26,63),(27,64),(28,65),(29,66),(30,67),(31,68),(32,69),(33,70),(34,71),(35,72),(36,73),(37,74),(38,75),(39,76),(40,77),(41,78),(42,79),(43,80),(44,81),(45,82),(46,83),(47,84),(48,85),(49,86),(50,87),(51,88),(52,89),(53,90),(54,91),(55,92),(56,93),(57,94),(58,95),(59,96),(60,97),(121,196),(122,197),(123,198),(124,199),(125,200),(126,201),(127,202),(128,203),(129,204),(130,205),(131,206),(132,207),(133,208),(134,209),(135,210),(136,211),(137,212),(138,213),(139,214),(140,215),(141,216),(142,217),(143,218),(144,219),(145,220),(146,221),(147,222),(148,223),(149,224),(150,225),(151,226),(152,227),(153,228),(154,229),(155,230),(156,231),(157,232),(158,233),(159,234),(160,235),(161,236),(162,237),(163,238),(164,239),(165,240),(166,181),(167,182),(168,183),(169,184),(170,185),(171,186),(172,187),(173,188),(174,189),(175,190),(176,191),(177,192),(178,193),(179,194),(180,195)], [(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)], [(1,171,46,156,31,141,16,126),(2,140,47,125,32,170,17,155),(3,169,48,154,33,139,18,124),(4,138,49,123,34,168,19,153),(5,167,50,152,35,137,20,122),(6,136,51,121,36,166,21,151),(7,165,52,150,37,135,22,180),(8,134,53,179,38,164,23,149),(9,163,54,148,39,133,24,178),(10,132,55,177,40,162,25,147),(11,161,56,146,41,131,26,176),(12,130,57,175,42,160,27,145),(13,159,58,144,43,129,28,174),(14,128,59,173,44,158,29,143),(15,157,60,142,45,127,30,172),(61,193,106,238,91,223,76,208),(62,222,107,207,92,192,77,237),(63,191,108,236,93,221,78,206),(64,220,109,205,94,190,79,235),(65,189,110,234,95,219,80,204),(66,218,111,203,96,188,81,233),(67,187,112,232,97,217,82,202),(68,216,113,201,98,186,83,231),(69,185,114,230,99,215,84,200),(70,214,115,199,100,184,85,229),(71,183,116,228,101,213,86,198),(72,212,117,197,102,182,87,227),(73,181,118,226,103,211,88,196),(74,210,119,195,104,240,89,225),(75,239,120,224,105,209,90,194)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 10A | 10B | 10C | ··· | 10H | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | ··· | 20J | 30A | 30B | 30C | 30D | 30E | ··· | 30P | 60A | ··· | 60H | 60I | ··· | 60T |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 15 | 15 | 15 | 15 | 30 | ··· | 30 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | + | - | - | + | + | - | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D5 | D6 | Dic3 | Dic3 | D10 | Dic5 | Dic5 | D15 | C8○D4 | D30 | Dic15 | Dic15 | D4.Dic3 | D4.Dic5 | D4.Dic15 |
| kernel | D4.Dic15 | C2×C15⋊3C8 | C60.7C4 | C15×C4○D4 | D4×C15 | Q8×C15 | C5×C4○D4 | C3×C4○D4 | C2×C20 | C5×D4 | C5×Q8 | C2×C12 | C3×D4 | C3×Q8 | C4○D4 | C15 | C2×C4 | D4 | Q8 | C5 | C3 | C1 |
| # reps | 1 | 3 | 3 | 1 | 6 | 2 | 1 | 2 | 3 | 3 | 1 | 6 | 6 | 2 | 4 | 4 | 12 | 12 | 4 | 2 | 4 | 8 |
Matrix representation of D4.Dic15 ►in GL4(𝔽241) generated by
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 177 | 0 |
| 0 | 0 | 66 | 64 |
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 175 | 113 |
| 0 | 0 | 179 | 66 |
| 147 | 64 | 0 | 0 |
| 67 | 80 | 0 | 0 |
| 0 | 0 | 177 | 0 |
| 0 | 0 | 0 | 177 |
| 193 | 208 | 0 | 0 |
| 194 | 48 | 0 | 0 |
| 0 | 0 | 211 | 0 |
| 0 | 0 | 0 | 211 |
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,177,66,0,0,0,64],[240,0,0,0,0,240,0,0,0,0,175,179,0,0,113,66],[147,67,0,0,64,80,0,0,0,0,177,0,0,0,0,177],[193,194,0,0,208,48,0,0,0,0,211,0,0,0,0,211] >;
D4.Dic15 in GAP, Magma, Sage, TeX
D_4.{\rm Dic}_{15} % in TeX
G:=Group("D4.Dic15"); // GroupNames label
G:=SmallGroup(480,913);
// by ID
G=gap.SmallGroup(480,913);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,219,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^30=a^2,d^2=a^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^29>;
// generators/relations