metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic15.48D4, (C2×C30)⋊4Q8, C6.94(D4×D5), (C2×C6)⋊2Dic10, (C2×C10)⋊5Dic6, C10.96(S3×D4), C22⋊4(C15⋊Q8), C30.64(C2×Q8), C30.256(C2×D4), C15⋊24(C22⋊Q8), C23.57(S3×D5), C23.D5.5S3, C6.Dic10⋊41C2, C30.Q8⋊41C2, Dic15⋊5C4⋊41C2, (C2×Dic5).68D6, C6.31(C2×Dic10), C10.31(C2×Dic6), C6.D4.5D5, (C22×C6).51D10, (C22×C10).66D6, C30.162(C4○D4), C6.87(D4⋊2D5), (C2×C30).218C23, (C2×Dic3).67D10, C5⋊4(Dic3.D4), C10.87(D4⋊2S3), C2.46(D10⋊D6), C3⋊4(Dic5.14D4), (C22×C30).80C22, C2.31(C30.C23), (C22×Dic15).13C2, (C6×Dic5).125C22, (C2×Dic15).230C22, (C10×Dic3).125C22, (C2×C15⋊Q8)⋊18C2, C2.13(C2×C15⋊Q8), C22.247(C2×S3×D5), (C3×C23.D5).6C2, (C5×C6.D4).6C2, (C2×C6).230(C22×D5), (C2×C10).230(C22×S3), SmallGroup(480,652)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic15.48D4
G = < a,b,c,d | a30=c4=1, b2=d2=a15, bab-1=a-1, cac-1=dad-1=a19, cbc-1=dbd-1=a15b, dcd-1=a15c-1 >
Subgroups: 684 in 148 conjugacy classes, 54 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C2×C4, Q8, C23, C10, C10, Dic3, C12, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, C2×C10, C2×C10, C2×C10, Dic6, C2×Dic3, C2×Dic3, C2×C12, C22×C6, C30, C30, C22⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C22×C10, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C22⋊C4, C2×Dic6, C22×Dic3, C5×Dic3, C3×Dic5, Dic15, Dic15, C2×C30, C2×C30, C2×C30, C10.D4, C4⋊Dic5, C23.D5, C5×C22⋊C4, C2×Dic10, C22×Dic5, Dic3.D4, C15⋊Q8, C6×Dic5, C10×Dic3, C2×Dic15, C2×Dic15, C22×C30, Dic5.14D4, C30.Q8, Dic15⋊5C4, C6.Dic10, C3×C23.D5, C5×C6.D4, C2×C15⋊Q8, C22×Dic15, Dic15.48D4
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, C4○D4, D10, Dic6, C22×S3, C22⋊Q8, Dic10, C22×D5, C2×Dic6, S3×D4, D4⋊2S3, S3×D5, C2×Dic10, D4×D5, D4⋊2D5, Dic3.D4, C15⋊Q8, C2×S3×D5, Dic5.14D4, C30.C23, C2×C15⋊Q8, D10⋊D6, Dic15.48D4
►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 36 16 51)(2 35 17 50)(3 34 18 49)(4 33 19 48)(5 32 20 47)(6 31 21 46)(7 60 22 45)(8 59 23 44)(9 58 24 43)(10 57 25 42)(11 56 26 41)(12 55 27 40)(13 54 28 39)(14 53 29 38)(15 52 30 37)(61 154 76 169)(62 153 77 168)(63 152 78 167)(64 151 79 166)(65 180 80 165)(66 179 81 164)(67 178 82 163)(68 177 83 162)(69 176 84 161)(70 175 85 160)(71 174 86 159)(72 173 87 158)(73 172 88 157)(74 171 89 156)(75 170 90 155)(91 186 106 201)(92 185 107 200)(93 184 108 199)(94 183 109 198)(95 182 110 197)(96 181 111 196)(97 210 112 195)(98 209 113 194)(99 208 114 193)(100 207 115 192)(101 206 116 191)(102 205 117 190)(103 204 118 189)(104 203 119 188)(105 202 120 187)(121 218 136 233)(122 217 137 232)(123 216 138 231)(124 215 139 230)(125 214 140 229)(126 213 141 228)(127 212 142 227)(128 211 143 226)(129 240 144 225)(130 239 145 224)(131 238 146 223)(132 237 147 222)(133 236 148 221)(134 235 149 220)(135 234 150 219)
(1 182 230 175)(2 201 231 164)(3 190 232 153)(4 209 233 172)(5 198 234 161)(6 187 235 180)(7 206 236 169)(8 195 237 158)(9 184 238 177)(10 203 239 166)(11 192 240 155)(12 181 211 174)(13 200 212 163)(14 189 213 152)(15 208 214 171)(16 197 215 160)(17 186 216 179)(18 205 217 168)(19 194 218 157)(20 183 219 176)(21 202 220 165)(22 191 221 154)(23 210 222 173)(24 199 223 162)(25 188 224 151)(26 207 225 170)(27 196 226 159)(28 185 227 178)(29 204 228 167)(30 193 229 156)(31 120 149 65)(32 109 150 84)(33 98 121 73)(34 117 122 62)(35 106 123 81)(36 95 124 70)(37 114 125 89)(38 103 126 78)(39 92 127 67)(40 111 128 86)(41 100 129 75)(42 119 130 64)(43 108 131 83)(44 97 132 72)(45 116 133 61)(46 105 134 80)(47 94 135 69)(48 113 136 88)(49 102 137 77)(50 91 138 66)(51 110 139 85)(52 99 140 74)(53 118 141 63)(54 107 142 82)(55 96 143 71)(56 115 144 90)(57 104 145 79)(58 93 146 68)(59 112 147 87)(60 101 148 76)
(1 160 16 175)(2 179 17 164)(3 168 18 153)(4 157 19 172)(5 176 20 161)(6 165 21 180)(7 154 22 169)(8 173 23 158)(9 162 24 177)(10 151 25 166)(11 170 26 155)(12 159 27 174)(13 178 28 163)(14 167 29 152)(15 156 30 171)(31 80 46 65)(32 69 47 84)(33 88 48 73)(34 77 49 62)(35 66 50 81)(36 85 51 70)(37 74 52 89)(38 63 53 78)(39 82 54 67)(40 71 55 86)(41 90 56 75)(42 79 57 64)(43 68 58 83)(44 87 59 72)(45 76 60 61)(91 138 106 123)(92 127 107 142)(93 146 108 131)(94 135 109 150)(95 124 110 139)(96 143 111 128)(97 132 112 147)(98 121 113 136)(99 140 114 125)(100 129 115 144)(101 148 116 133)(102 137 117 122)(103 126 118 141)(104 145 119 130)(105 134 120 149)(181 211 196 226)(182 230 197 215)(183 219 198 234)(184 238 199 223)(185 227 200 212)(186 216 201 231)(187 235 202 220)(188 224 203 239)(189 213 204 228)(190 232 205 217)(191 221 206 236)(192 240 207 225)(193 229 208 214)(194 218 209 233)(195 237 210 222)
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), (1,36,16,51)(2,35,17,50)(3,34,18,49)(4,33,19,48)(5,32,20,47)(6,31,21,46)(7,60,22,45)(8,59,23,44)(9,58,24,43)(10,57,25,42)(11,56,26,41)(12,55,27,40)(13,54,28,39)(14,53,29,38)(15,52,30,37)(61,154,76,169)(62,153,77,168)(63,152,78,167)(64,151,79,166)(65,180,80,165)(66,179,81,164)(67,178,82,163)(68,177,83,162)(69,176,84,161)(70,175,85,160)(71,174,86,159)(72,173,87,158)(73,172,88,157)(74,171,89,156)(75,170,90,155)(91,186,106,201)(92,185,107,200)(93,184,108,199)(94,183,109,198)(95,182,110,197)(96,181,111,196)(97,210,112,195)(98,209,113,194)(99,208,114,193)(100,207,115,192)(101,206,116,191)(102,205,117,190)(103,204,118,189)(104,203,119,188)(105,202,120,187)(121,218,136,233)(122,217,137,232)(123,216,138,231)(124,215,139,230)(125,214,140,229)(126,213,141,228)(127,212,142,227)(128,211,143,226)(129,240,144,225)(130,239,145,224)(131,238,146,223)(132,237,147,222)(133,236,148,221)(134,235,149,220)(135,234,150,219), (1,182,230,175)(2,201,231,164)(3,190,232,153)(4,209,233,172)(5,198,234,161)(6,187,235,180)(7,206,236,169)(8,195,237,158)(9,184,238,177)(10,203,239,166)(11,192,240,155)(12,181,211,174)(13,200,212,163)(14,189,213,152)(15,208,214,171)(16,197,215,160)(17,186,216,179)(18,205,217,168)(19,194,218,157)(20,183,219,176)(21,202,220,165)(22,191,221,154)(23,210,222,173)(24,199,223,162)(25,188,224,151)(26,207,225,170)(27,196,226,159)(28,185,227,178)(29,204,228,167)(30,193,229,156)(31,120,149,65)(32,109,150,84)(33,98,121,73)(34,117,122,62)(35,106,123,81)(36,95,124,70)(37,114,125,89)(38,103,126,78)(39,92,127,67)(40,111,128,86)(41,100,129,75)(42,119,130,64)(43,108,131,83)(44,97,132,72)(45,116,133,61)(46,105,134,80)(47,94,135,69)(48,113,136,88)(49,102,137,77)(50,91,138,66)(51,110,139,85)(52,99,140,74)(53,118,141,63)(54,107,142,82)(55,96,143,71)(56,115,144,90)(57,104,145,79)(58,93,146,68)(59,112,147,87)(60,101,148,76), (1,160,16,175)(2,179,17,164)(3,168,18,153)(4,157,19,172)(5,176,20,161)(6,165,21,180)(7,154,22,169)(8,173,23,158)(9,162,24,177)(10,151,25,166)(11,170,26,155)(12,159,27,174)(13,178,28,163)(14,167,29,152)(15,156,30,171)(31,80,46,65)(32,69,47,84)(33,88,48,73)(34,77,49,62)(35,66,50,81)(36,85,51,70)(37,74,52,89)(38,63,53,78)(39,82,54,67)(40,71,55,86)(41,90,56,75)(42,79,57,64)(43,68,58,83)(44,87,59,72)(45,76,60,61)(91,138,106,123)(92,127,107,142)(93,146,108,131)(94,135,109,150)(95,124,110,139)(96,143,111,128)(97,132,112,147)(98,121,113,136)(99,140,114,125)(100,129,115,144)(101,148,116,133)(102,137,117,122)(103,126,118,141)(104,145,119,130)(105,134,120,149)(181,211,196,226)(182,230,197,215)(183,219,198,234)(184,238,199,223)(185,227,200,212)(186,216,201,231)(187,235,202,220)(188,224,203,239)(189,213,204,228)(190,232,205,217)(191,221,206,236)(192,240,207,225)(193,229,208,214)(194,218,209,233)(195,237,210,222)>;
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), (1,36,16,51)(2,35,17,50)(3,34,18,49)(4,33,19,48)(5,32,20,47)(6,31,21,46)(7,60,22,45)(8,59,23,44)(9,58,24,43)(10,57,25,42)(11,56,26,41)(12,55,27,40)(13,54,28,39)(14,53,29,38)(15,52,30,37)(61,154,76,169)(62,153,77,168)(63,152,78,167)(64,151,79,166)(65,180,80,165)(66,179,81,164)(67,178,82,163)(68,177,83,162)(69,176,84,161)(70,175,85,160)(71,174,86,159)(72,173,87,158)(73,172,88,157)(74,171,89,156)(75,170,90,155)(91,186,106,201)(92,185,107,200)(93,184,108,199)(94,183,109,198)(95,182,110,197)(96,181,111,196)(97,210,112,195)(98,209,113,194)(99,208,114,193)(100,207,115,192)(101,206,116,191)(102,205,117,190)(103,204,118,189)(104,203,119,188)(105,202,120,187)(121,218,136,233)(122,217,137,232)(123,216,138,231)(124,215,139,230)(125,214,140,229)(126,213,141,228)(127,212,142,227)(128,211,143,226)(129,240,144,225)(130,239,145,224)(131,238,146,223)(132,237,147,222)(133,236,148,221)(134,235,149,220)(135,234,150,219), (1,182,230,175)(2,201,231,164)(3,190,232,153)(4,209,233,172)(5,198,234,161)(6,187,235,180)(7,206,236,169)(8,195,237,158)(9,184,238,177)(10,203,239,166)(11,192,240,155)(12,181,211,174)(13,200,212,163)(14,189,213,152)(15,208,214,171)(16,197,215,160)(17,186,216,179)(18,205,217,168)(19,194,218,157)(20,183,219,176)(21,202,220,165)(22,191,221,154)(23,210,222,173)(24,199,223,162)(25,188,224,151)(26,207,225,170)(27,196,226,159)(28,185,227,178)(29,204,228,167)(30,193,229,156)(31,120,149,65)(32,109,150,84)(33,98,121,73)(34,117,122,62)(35,106,123,81)(36,95,124,70)(37,114,125,89)(38,103,126,78)(39,92,127,67)(40,111,128,86)(41,100,129,75)(42,119,130,64)(43,108,131,83)(44,97,132,72)(45,116,133,61)(46,105,134,80)(47,94,135,69)(48,113,136,88)(49,102,137,77)(50,91,138,66)(51,110,139,85)(52,99,140,74)(53,118,141,63)(54,107,142,82)(55,96,143,71)(56,115,144,90)(57,104,145,79)(58,93,146,68)(59,112,147,87)(60,101,148,76), (1,160,16,175)(2,179,17,164)(3,168,18,153)(4,157,19,172)(5,176,20,161)(6,165,21,180)(7,154,22,169)(8,173,23,158)(9,162,24,177)(10,151,25,166)(11,170,26,155)(12,159,27,174)(13,178,28,163)(14,167,29,152)(15,156,30,171)(31,80,46,65)(32,69,47,84)(33,88,48,73)(34,77,49,62)(35,66,50,81)(36,85,51,70)(37,74,52,89)(38,63,53,78)(39,82,54,67)(40,71,55,86)(41,90,56,75)(42,79,57,64)(43,68,58,83)(44,87,59,72)(45,76,60,61)(91,138,106,123)(92,127,107,142)(93,146,108,131)(94,135,109,150)(95,124,110,139)(96,143,111,128)(97,132,112,147)(98,121,113,136)(99,140,114,125)(100,129,115,144)(101,148,116,133)(102,137,117,122)(103,126,118,141)(104,145,119,130)(105,134,120,149)(181,211,196,226)(182,230,197,215)(183,219,198,234)(184,238,199,223)(185,227,200,212)(186,216,201,231)(187,235,202,220)(188,224,203,239)(189,213,204,228)(190,232,205,217)(191,221,206,236)(192,240,207,225)(193,229,208,214)(194,218,209,233)(195,237,210,222) );
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)], [(1,36,16,51),(2,35,17,50),(3,34,18,49),(4,33,19,48),(5,32,20,47),(6,31,21,46),(7,60,22,45),(8,59,23,44),(9,58,24,43),(10,57,25,42),(11,56,26,41),(12,55,27,40),(13,54,28,39),(14,53,29,38),(15,52,30,37),(61,154,76,169),(62,153,77,168),(63,152,78,167),(64,151,79,166),(65,180,80,165),(66,179,81,164),(67,178,82,163),(68,177,83,162),(69,176,84,161),(70,175,85,160),(71,174,86,159),(72,173,87,158),(73,172,88,157),(74,171,89,156),(75,170,90,155),(91,186,106,201),(92,185,107,200),(93,184,108,199),(94,183,109,198),(95,182,110,197),(96,181,111,196),(97,210,112,195),(98,209,113,194),(99,208,114,193),(100,207,115,192),(101,206,116,191),(102,205,117,190),(103,204,118,189),(104,203,119,188),(105,202,120,187),(121,218,136,233),(122,217,137,232),(123,216,138,231),(124,215,139,230),(125,214,140,229),(126,213,141,228),(127,212,142,227),(128,211,143,226),(129,240,144,225),(130,239,145,224),(131,238,146,223),(132,237,147,222),(133,236,148,221),(134,235,149,220),(135,234,150,219)], [(1,182,230,175),(2,201,231,164),(3,190,232,153),(4,209,233,172),(5,198,234,161),(6,187,235,180),(7,206,236,169),(8,195,237,158),(9,184,238,177),(10,203,239,166),(11,192,240,155),(12,181,211,174),(13,200,212,163),(14,189,213,152),(15,208,214,171),(16,197,215,160),(17,186,216,179),(18,205,217,168),(19,194,218,157),(20,183,219,176),(21,202,220,165),(22,191,221,154),(23,210,222,173),(24,199,223,162),(25,188,224,151),(26,207,225,170),(27,196,226,159),(28,185,227,178),(29,204,228,167),(30,193,229,156),(31,120,149,65),(32,109,150,84),(33,98,121,73),(34,117,122,62),(35,106,123,81),(36,95,124,70),(37,114,125,89),(38,103,126,78),(39,92,127,67),(40,111,128,86),(41,100,129,75),(42,119,130,64),(43,108,131,83),(44,97,132,72),(45,116,133,61),(46,105,134,80),(47,94,135,69),(48,113,136,88),(49,102,137,77),(50,91,138,66),(51,110,139,85),(52,99,140,74),(53,118,141,63),(54,107,142,82),(55,96,143,71),(56,115,144,90),(57,104,145,79),(58,93,146,68),(59,112,147,87),(60,101,148,76)], [(1,160,16,175),(2,179,17,164),(3,168,18,153),(4,157,19,172),(5,176,20,161),(6,165,21,180),(7,154,22,169),(8,173,23,158),(9,162,24,177),(10,151,25,166),(11,170,26,155),(12,159,27,174),(13,178,28,163),(14,167,29,152),(15,156,30,171),(31,80,46,65),(32,69,47,84),(33,88,48,73),(34,77,49,62),(35,66,50,81),(36,85,51,70),(37,74,52,89),(38,63,53,78),(39,82,54,67),(40,71,55,86),(41,90,56,75),(42,79,57,64),(43,68,58,83),(44,87,59,72),(45,76,60,61),(91,138,106,123),(92,127,107,142),(93,146,108,131),(94,135,109,150),(95,124,110,139),(96,143,111,128),(97,132,112,147),(98,121,113,136),(99,140,114,125),(100,129,115,144),(101,148,116,133),(102,137,117,122),(103,126,118,141),(104,145,119,130),(105,134,120,149),(181,211,196,226),(182,230,197,215),(183,219,198,234),(184,238,199,223),(185,227,200,212),(186,216,201,231),(187,235,202,220),(188,224,203,239),(189,213,204,228),(190,232,205,217),(191,221,206,236),(192,240,207,225),(193,229,208,214),(194,218,209,233),(195,237,210,222)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 12 | 12 | 20 | 20 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 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 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | - | + | - | + | + | - | - | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D5 | D6 | D6 | C4○D4 | D10 | D10 | Dic6 | Dic10 | S3×D4 | D4⋊2S3 | S3×D5 | D4×D5 | D4⋊2D5 | C15⋊Q8 | C2×S3×D5 | C30.C23 | D10⋊D6 |
| kernel | Dic15.48D4 | C30.Q8 | Dic15⋊5C4 | C6.Dic10 | C3×C23.D5 | C5×C6.D4 | C2×C15⋊Q8 | C22×Dic15 | C23.D5 | Dic15 | C2×C30 | C6.D4 | C2×Dic5 | C22×C10 | C30 | C2×Dic3 | C22×C6 | C2×C10 | C2×C6 | C10 | C10 | C23 | C6 | C6 | C22 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of Dic15.48D4 ►in GL6(𝔽61)
| 0 | 60 | 0 | 0 | 0 | 0 |
| 1 | 44 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 20 | 0 | 0 |
| 0 | 0 | 49 | 53 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 10 | 8 | 0 | 0 | 0 | 0 |
| 56 | 51 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 38 | 0 | 0 |
| 0 | 0 | 20 | 41 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 20 | 58 | 0 | 0 | 0 | 0 |
| 32 | 41 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 60 |
| 0 | 0 | 0 | 0 | 26 | 56 |
| 20 | 58 | 0 | 0 | 0 | 0 |
| 32 | 41 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 60 |
| 0 | 0 | 0 | 0 | 24 | 56 |
G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,44,0,0,0,0,0,0,7,49,0,0,0,0,20,53,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,56,0,0,0,0,8,51,0,0,0,0,0,0,20,20,0,0,0,0,38,41,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[20,32,0,0,0,0,58,41,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,5,26,0,0,0,0,60,56],[20,32,0,0,0,0,58,41,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,5,24,0,0,0,0,60,56] >;
Dic15.48D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{15}._{48}D_4 % in TeX
G:=Group("Dic15.48D4"); // GroupNames label
G:=SmallGroup(480,652);
// by ID
G=gap.SmallGroup(480,652);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,64,422,219,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=c^4=1,b^2=d^2=a^15,b*a*b^-1=a^-1,c*a*c^-1=d*a*d^-1=a^19,c*b*c^-1=d*b*d^-1=a^15*b,d*c*d^-1=a^15*c^-1>;
// generators/relations