metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.7D4, C30.43D8, D4⋊1Dic15, C30.27SD16, (D4×C15)⋊7C4, (C6×D4).1D5, C60.81(C2×C4), (C3×D4)⋊1Dic5, (C5×D4)⋊4Dic3, (D4×C10).1S3, C60⋊5C4⋊16C2, (D4×C30).1C2, (C2×D4).1D15, (C2×C20).72D6, (C2×C4).38D30, C6.21(D4⋊D5), C2.3(D4⋊D15), (C2×C12).73D10, (C2×C30).140D4, C3⋊3(D4⋊Dic5), C5⋊4(D4⋊Dic3), C6.9(D4.D5), C12.7(C2×Dic5), C4.1(C2×Dic15), C15⋊17(D4⋊C4), C10.21(D4⋊S3), C20.19(C3⋊D4), C12.21(C5⋊D4), C4.12(C15⋊7D4), (C2×C60).58C22, C2.3(D4.D15), C10.9(D4.S3), C20.28(C2×Dic3), C6.14(C23.D5), C30.102(C22⋊C4), C2.3(C30.38D4), C22.17(C15⋊7D4), C10.25(C6.D4), (C2×C15⋊3C8)⋊2C2, (C2×C6).72(C5⋊D4), (C2×C10).72(C3⋊D4), SmallGroup(480,192)
Series: Derived ►Chief ►Lower central ►Upper central
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, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, D4, C23, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C4⋊C4, C2×C8, C2×D4, Dic5, C20, C2×C10, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C3×D4, C3×D4, C22×C6, C30, C30, D4⋊C4, C5⋊2C8, C2×Dic5, C2×C20, C5×D4, C5×D4, C22×C10, C2×C3⋊C8, C4⋊Dic3, C6×D4, Dic15, C60, C2×C30, C2×C30, C2×C5⋊2C8, C4⋊Dic5, D4×C10, D4⋊Dic3, C15⋊3C8, C2×Dic15, C2×C60, D4×C15, D4×C15, C22×C30, D4⋊Dic5, C2×C15⋊3C8, C60⋊5C4, D4×C30, D4⋊Dic15
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, Dic3, D6, C22⋊C4, D8, SD16, Dic5, D10, C2×Dic3, C3⋊D4, D15, D4⋊C4, C2×Dic5, C5⋊D4, D4⋊S3, D4.S3, C6.D4, Dic15, D30, D4⋊D5, D4.D5, C23.D5, D4⋊Dic3, C2×Dic15, C15⋊7D4, D4⋊Dic5, D4⋊D15, D4.D15, C30.38D4, D4⋊Dic15
►On 240 points
Generators in S240
(1 53 128 117)(2 54 129 118)(3 55 130 119)(4 56 131 120)(5 57 132 91)(6 58 133 92)(7 59 134 93)(8 60 135 94)(9 31 136 95)(10 32 137 96)(11 33 138 97)(12 34 139 98)(13 35 140 99)(14 36 141 100)(15 37 142 101)(16 38 143 102)(17 39 144 103)(18 40 145 104)(19 41 146 105)(20 42 147 106)(21 43 148 107)(22 44 149 108)(23 45 150 109)(24 46 121 110)(25 47 122 111)(26 48 123 112)(27 49 124 113)(28 50 125 114)(29 51 126 115)(30 52 127 116)(61 170 233 184)(62 171 234 185)(63 172 235 186)(64 173 236 187)(65 174 237 188)(66 175 238 189)(67 176 239 190)(68 177 240 191)(69 178 211 192)(70 179 212 193)(71 180 213 194)(72 151 214 195)(73 152 215 196)(74 153 216 197)(75 154 217 198)(76 155 218 199)(77 156 219 200)(78 157 220 201)(79 158 221 202)(80 159 222 203)(81 160 223 204)(82 161 224 205)(83 162 225 206)(84 163 226 207)(85 164 227 208)(86 165 228 209)(87 166 229 210)(88 167 230 181)(89 168 231 182)(90 169 232 183)
(1 102)(2 103)(3 104)(4 105)(5 106)(6 107)(7 108)(8 109)(9 110)(10 111)(11 112)(12 113)(13 114)(14 115)(15 116)(16 117)(17 118)(18 119)(19 120)(20 91)(21 92)(22 93)(23 94)(24 95)(25 96)(26 97)(27 98)(28 99)(29 100)(30 101)(31 121)(32 122)(33 123)(34 124)(35 125)(36 126)(37 127)(38 128)(39 129)(40 130)(41 131)(42 132)(43 133)(44 134)(45 135)(46 136)(47 137)(48 138)(49 139)(50 140)(51 141)(52 142)(53 143)(54 144)(55 145)(56 146)(57 147)(58 148)(59 149)(60 150)(61 218)(62 219)(63 220)(64 221)(65 222)(66 223)(67 224)(68 225)(69 226)(70 227)(71 228)(72 229)(73 230)(74 231)(75 232)(76 233)(77 234)(78 235)(79 236)(80 237)(81 238)(82 239)(83 240)(84 211)(85 212)(86 213)(87 214)(88 215)(89 216)(90 217)(151 166)(152 167)(153 168)(154 169)(155 170)(156 171)(157 172)(158 173)(159 174)(160 175)(161 176)(162 177)(163 178)(164 179)(165 180)(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 75 16 90)(2 74 17 89)(3 73 18 88)(4 72 19 87)(5 71 20 86)(6 70 21 85)(7 69 22 84)(8 68 23 83)(9 67 24 82)(10 66 25 81)(11 65 26 80)(12 64 27 79)(13 63 28 78)(14 62 29 77)(15 61 30 76)(31 190 46 205)(32 189 47 204)(33 188 48 203)(34 187 49 202)(35 186 50 201)(36 185 51 200)(37 184 52 199)(38 183 53 198)(39 182 54 197)(40 181 55 196)(41 210 56 195)(42 209 57 194)(43 208 58 193)(44 207 59 192)(45 206 60 191)(91 180 106 165)(92 179 107 164)(93 178 108 163)(94 177 109 162)(95 176 110 161)(96 175 111 160)(97 174 112 159)(98 173 113 158)(99 172 114 157)(100 171 115 156)(101 170 116 155)(102 169 117 154)(103 168 118 153)(104 167 119 152)(105 166 120 151)(121 224 136 239)(122 223 137 238)(123 222 138 237)(124 221 139 236)(125 220 140 235)(126 219 141 234)(127 218 142 233)(128 217 143 232)(129 216 144 231)(130 215 145 230)(131 214 146 229)(132 213 147 228)(133 212 148 227)(134 211 149 226)(135 240 150 225)
G:=sub<Sym(240)| (1,53,128,117)(2,54,129,118)(3,55,130,119)(4,56,131,120)(5,57,132,91)(6,58,133,92)(7,59,134,93)(8,60,135,94)(9,31,136,95)(10,32,137,96)(11,33,138,97)(12,34,139,98)(13,35,140,99)(14,36,141,100)(15,37,142,101)(16,38,143,102)(17,39,144,103)(18,40,145,104)(19,41,146,105)(20,42,147,106)(21,43,148,107)(22,44,149,108)(23,45,150,109)(24,46,121,110)(25,47,122,111)(26,48,123,112)(27,49,124,113)(28,50,125,114)(29,51,126,115)(30,52,127,116)(61,170,233,184)(62,171,234,185)(63,172,235,186)(64,173,236,187)(65,174,237,188)(66,175,238,189)(67,176,239,190)(68,177,240,191)(69,178,211,192)(70,179,212,193)(71,180,213,194)(72,151,214,195)(73,152,215,196)(74,153,216,197)(75,154,217,198)(76,155,218,199)(77,156,219,200)(78,157,220,201)(79,158,221,202)(80,159,222,203)(81,160,223,204)(82,161,224,205)(83,162,225,206)(84,163,226,207)(85,164,227,208)(86,165,228,209)(87,166,229,210)(88,167,230,181)(89,168,231,182)(90,169,232,183), (1,102)(2,103)(3,104)(4,105)(5,106)(6,107)(7,108)(8,109)(9,110)(10,111)(11,112)(12,113)(13,114)(14,115)(15,116)(16,117)(17,118)(18,119)(19,120)(20,91)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,100)(30,101)(31,121)(32,122)(33,123)(34,124)(35,125)(36,126)(37,127)(38,128)(39,129)(40,130)(41,131)(42,132)(43,133)(44,134)(45,135)(46,136)(47,137)(48,138)(49,139)(50,140)(51,141)(52,142)(53,143)(54,144)(55,145)(56,146)(57,147)(58,148)(59,149)(60,150)(61,218)(62,219)(63,220)(64,221)(65,222)(66,223)(67,224)(68,225)(69,226)(70,227)(71,228)(72,229)(73,230)(74,231)(75,232)(76,233)(77,234)(78,235)(79,236)(80,237)(81,238)(82,239)(83,240)(84,211)(85,212)(86,213)(87,214)(88,215)(89,216)(90,217)(151,166)(152,167)(153,168)(154,169)(155,170)(156,171)(157,172)(158,173)(159,174)(160,175)(161,176)(162,177)(163,178)(164,179)(165,180)(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), (1,75,16,90)(2,74,17,89)(3,73,18,88)(4,72,19,87)(5,71,20,86)(6,70,21,85)(7,69,22,84)(8,68,23,83)(9,67,24,82)(10,66,25,81)(11,65,26,80)(12,64,27,79)(13,63,28,78)(14,62,29,77)(15,61,30,76)(31,190,46,205)(32,189,47,204)(33,188,48,203)(34,187,49,202)(35,186,50,201)(36,185,51,200)(37,184,52,199)(38,183,53,198)(39,182,54,197)(40,181,55,196)(41,210,56,195)(42,209,57,194)(43,208,58,193)(44,207,59,192)(45,206,60,191)(91,180,106,165)(92,179,107,164)(93,178,108,163)(94,177,109,162)(95,176,110,161)(96,175,111,160)(97,174,112,159)(98,173,113,158)(99,172,114,157)(100,171,115,156)(101,170,116,155)(102,169,117,154)(103,168,118,153)(104,167,119,152)(105,166,120,151)(121,224,136,239)(122,223,137,238)(123,222,138,237)(124,221,139,236)(125,220,140,235)(126,219,141,234)(127,218,142,233)(128,217,143,232)(129,216,144,231)(130,215,145,230)(131,214,146,229)(132,213,147,228)(133,212,148,227)(134,211,149,226)(135,240,150,225)>;
G:=Group( (1,53,128,117)(2,54,129,118)(3,55,130,119)(4,56,131,120)(5,57,132,91)(6,58,133,92)(7,59,134,93)(8,60,135,94)(9,31,136,95)(10,32,137,96)(11,33,138,97)(12,34,139,98)(13,35,140,99)(14,36,141,100)(15,37,142,101)(16,38,143,102)(17,39,144,103)(18,40,145,104)(19,41,146,105)(20,42,147,106)(21,43,148,107)(22,44,149,108)(23,45,150,109)(24,46,121,110)(25,47,122,111)(26,48,123,112)(27,49,124,113)(28,50,125,114)(29,51,126,115)(30,52,127,116)(61,170,233,184)(62,171,234,185)(63,172,235,186)(64,173,236,187)(65,174,237,188)(66,175,238,189)(67,176,239,190)(68,177,240,191)(69,178,211,192)(70,179,212,193)(71,180,213,194)(72,151,214,195)(73,152,215,196)(74,153,216,197)(75,154,217,198)(76,155,218,199)(77,156,219,200)(78,157,220,201)(79,158,221,202)(80,159,222,203)(81,160,223,204)(82,161,224,205)(83,162,225,206)(84,163,226,207)(85,164,227,208)(86,165,228,209)(87,166,229,210)(88,167,230,181)(89,168,231,182)(90,169,232,183), (1,102)(2,103)(3,104)(4,105)(5,106)(6,107)(7,108)(8,109)(9,110)(10,111)(11,112)(12,113)(13,114)(14,115)(15,116)(16,117)(17,118)(18,119)(19,120)(20,91)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,100)(30,101)(31,121)(32,122)(33,123)(34,124)(35,125)(36,126)(37,127)(38,128)(39,129)(40,130)(41,131)(42,132)(43,133)(44,134)(45,135)(46,136)(47,137)(48,138)(49,139)(50,140)(51,141)(52,142)(53,143)(54,144)(55,145)(56,146)(57,147)(58,148)(59,149)(60,150)(61,218)(62,219)(63,220)(64,221)(65,222)(66,223)(67,224)(68,225)(69,226)(70,227)(71,228)(72,229)(73,230)(74,231)(75,232)(76,233)(77,234)(78,235)(79,236)(80,237)(81,238)(82,239)(83,240)(84,211)(85,212)(86,213)(87,214)(88,215)(89,216)(90,217)(151,166)(152,167)(153,168)(154,169)(155,170)(156,171)(157,172)(158,173)(159,174)(160,175)(161,176)(162,177)(163,178)(164,179)(165,180)(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), (1,75,16,90)(2,74,17,89)(3,73,18,88)(4,72,19,87)(5,71,20,86)(6,70,21,85)(7,69,22,84)(8,68,23,83)(9,67,24,82)(10,66,25,81)(11,65,26,80)(12,64,27,79)(13,63,28,78)(14,62,29,77)(15,61,30,76)(31,190,46,205)(32,189,47,204)(33,188,48,203)(34,187,49,202)(35,186,50,201)(36,185,51,200)(37,184,52,199)(38,183,53,198)(39,182,54,197)(40,181,55,196)(41,210,56,195)(42,209,57,194)(43,208,58,193)(44,207,59,192)(45,206,60,191)(91,180,106,165)(92,179,107,164)(93,178,108,163)(94,177,109,162)(95,176,110,161)(96,175,111,160)(97,174,112,159)(98,173,113,158)(99,172,114,157)(100,171,115,156)(101,170,116,155)(102,169,117,154)(103,168,118,153)(104,167,119,152)(105,166,120,151)(121,224,136,239)(122,223,137,238)(123,222,138,237)(124,221,139,236)(125,220,140,235)(126,219,141,234)(127,218,142,233)(128,217,143,232)(129,216,144,231)(130,215,145,230)(131,214,146,229)(132,213,147,228)(133,212,148,227)(134,211,149,226)(135,240,150,225) );
G=PermutationGroup([[(1,53,128,117),(2,54,129,118),(3,55,130,119),(4,56,131,120),(5,57,132,91),(6,58,133,92),(7,59,134,93),(8,60,135,94),(9,31,136,95),(10,32,137,96),(11,33,138,97),(12,34,139,98),(13,35,140,99),(14,36,141,100),(15,37,142,101),(16,38,143,102),(17,39,144,103),(18,40,145,104),(19,41,146,105),(20,42,147,106),(21,43,148,107),(22,44,149,108),(23,45,150,109),(24,46,121,110),(25,47,122,111),(26,48,123,112),(27,49,124,113),(28,50,125,114),(29,51,126,115),(30,52,127,116),(61,170,233,184),(62,171,234,185),(63,172,235,186),(64,173,236,187),(65,174,237,188),(66,175,238,189),(67,176,239,190),(68,177,240,191),(69,178,211,192),(70,179,212,193),(71,180,213,194),(72,151,214,195),(73,152,215,196),(74,153,216,197),(75,154,217,198),(76,155,218,199),(77,156,219,200),(78,157,220,201),(79,158,221,202),(80,159,222,203),(81,160,223,204),(82,161,224,205),(83,162,225,206),(84,163,226,207),(85,164,227,208),(86,165,228,209),(87,166,229,210),(88,167,230,181),(89,168,231,182),(90,169,232,183)], [(1,102),(2,103),(3,104),(4,105),(5,106),(6,107),(7,108),(8,109),(9,110),(10,111),(11,112),(12,113),(13,114),(14,115),(15,116),(16,117),(17,118),(18,119),(19,120),(20,91),(21,92),(22,93),(23,94),(24,95),(25,96),(26,97),(27,98),(28,99),(29,100),(30,101),(31,121),(32,122),(33,123),(34,124),(35,125),(36,126),(37,127),(38,128),(39,129),(40,130),(41,131),(42,132),(43,133),(44,134),(45,135),(46,136),(47,137),(48,138),(49,139),(50,140),(51,141),(52,142),(53,143),(54,144),(55,145),(56,146),(57,147),(58,148),(59,149),(60,150),(61,218),(62,219),(63,220),(64,221),(65,222),(66,223),(67,224),(68,225),(69,226),(70,227),(71,228),(72,229),(73,230),(74,231),(75,232),(76,233),(77,234),(78,235),(79,236),(80,237),(81,238),(82,239),(83,240),(84,211),(85,212),(86,213),(87,214),(88,215),(89,216),(90,217),(151,166),(152,167),(153,168),(154,169),(155,170),(156,171),(157,172),(158,173),(159,174),(160,175),(161,176),(162,177),(163,178),(164,179),(165,180),(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)], [(1,75,16,90),(2,74,17,89),(3,73,18,88),(4,72,19,87),(5,71,20,86),(6,70,21,85),(7,69,22,84),(8,68,23,83),(9,67,24,82),(10,66,25,81),(11,65,26,80),(12,64,27,79),(13,63,28,78),(14,62,29,77),(15,61,30,76),(31,190,46,205),(32,189,47,204),(33,188,48,203),(34,187,49,202),(35,186,50,201),(36,185,51,200),(37,184,52,199),(38,183,53,198),(39,182,54,197),(40,181,55,196),(41,210,56,195),(42,209,57,194),(43,208,58,193),(44,207,59,192),(45,206,60,191),(91,180,106,165),(92,179,107,164),(93,178,108,163),(94,177,109,162),(95,176,110,161),(96,175,111,160),(97,174,112,159),(98,173,113,158),(99,172,114,157),(100,171,115,156),(101,170,116,155),(102,169,117,154),(103,168,118,153),(104,167,119,152),(105,166,120,151),(121,224,136,239),(122,223,137,238),(123,222,138,237),(124,221,139,236),(125,220,140,235),(126,219,141,234),(127,218,142,233),(128,217,143,232),(129,216,144,231),(130,215,145,230),(131,214,146,229),(132,213,147,228),(133,212,148,227),(134,211,149,226),(135,240,150,225)]])
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