metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊3Dic6, Dic15.6D4, (C6×D5)⋊3Q8, C4⋊Dic3⋊8D5, C6.39(D4×D5), C6.35(Q8×D5), C30.51(C2×D4), C10.40(S3×D4), C30.47(C2×Q8), C3⋊3(D10⋊Q8), (C2×C12).22D10, (C2×C20).229D6, C15⋊18(C22⋊Q8), C2.17(D5×Dic6), D10⋊C4.9S3, C6.73(C4○D20), Dic15⋊5C4⋊19C2, C30.4Q8⋊20C2, (C2×Dic5).38D6, C10.17(C2×Dic6), (C22×D5).52D6, C30.122(C4○D4), C2.18(C20⋊D6), (C2×C30).120C23, (C2×C60).322C22, (C2×Dic3).38D10, C5⋊3(Dic3.D4), C10.47(D4⋊2S3), D10⋊Dic3.13C2, (C6×Dic5).73C22, C2.19(Dic5.D6), (C2×Dic15).97C22, (C10×Dic3).74C22, (C2×C15⋊Q8)⋊10C2, (C2×C4).53(S3×D5), (C2×D5×Dic3).7C2, (C5×C4⋊Dic3)⋊19C2, (D5×C2×C6).22C22, C22.183(C2×S3×D5), (C2×C6).132(C22×D5), (C3×D10⋊C4).12C2, (C2×C10).132(C22×S3), SmallGroup(480,506)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic15.D4
G = < a,b,c,d | a30=c4=1, b2=d2=a15, bab-1=a-1, ac=ca, dad-1=a11, cbc-1=dbd-1=a15b, dcd-1=c-1 >
Subgroups: 748 in 148 conjugacy classes, 50 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, Q8, C23, D5, C10, Dic3, C12, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, D10, D10, C2×C10, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C3×D5, C30, C22⋊Q8, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C22⋊C4, C2×Dic6, C22×Dic3, C5×Dic3, C3×Dic5, Dic15, Dic15, C60, C6×D5, C6×D5, C2×C30, C10.D4, D10⋊C4, D10⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, Dic3.D4, D5×Dic3, C15⋊Q8, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, D10⋊Q8, D10⋊Dic3, Dic15⋊5C4, C3×D10⋊C4, C5×C4⋊Dic3, C30.4Q8, C2×D5×Dic3, C2×C15⋊Q8, Dic15.D4
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, C4○D4, D10, Dic6, C22×S3, C22⋊Q8, C22×D5, C2×Dic6, S3×D4, D4⋊2S3, S3×D5, C4○D20, D4×D5, Q8×D5, Dic3.D4, C2×S3×D5, D10⋊Q8, D5×Dic6, C20⋊D6, Dic5.D6, Dic15.D4
►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)
(1 106 16 91)(2 105 17 120)(3 104 18 119)(4 103 19 118)(5 102 20 117)(6 101 21 116)(7 100 22 115)(8 99 23 114)(9 98 24 113)(10 97 25 112)(11 96 26 111)(12 95 27 110)(13 94 28 109)(14 93 29 108)(15 92 30 107)(31 76 46 61)(32 75 47 90)(33 74 48 89)(34 73 49 88)(35 72 50 87)(36 71 51 86)(37 70 52 85)(38 69 53 84)(39 68 54 83)(40 67 55 82)(41 66 56 81)(42 65 57 80)(43 64 58 79)(44 63 59 78)(45 62 60 77)(121 223 136 238)(122 222 137 237)(123 221 138 236)(124 220 139 235)(125 219 140 234)(126 218 141 233)(127 217 142 232)(128 216 143 231)(129 215 144 230)(130 214 145 229)(131 213 146 228)(132 212 147 227)(133 211 148 226)(134 240 149 225)(135 239 150 224)(151 205 166 190)(152 204 167 189)(153 203 168 188)(154 202 169 187)(155 201 170 186)(156 200 171 185)(157 199 172 184)(158 198 173 183)(159 197 174 182)(160 196 175 181)(161 195 176 210)(162 194 177 209)(163 193 178 208)(164 192 179 207)(165 191 180 206)
(1 50 230 168)(2 51 231 169)(3 52 232 170)(4 53 233 171)(5 54 234 172)(6 55 235 173)(7 56 236 174)(8 57 237 175)(9 58 238 176)(10 59 239 177)(11 60 240 178)(12 31 211 179)(13 32 212 180)(14 33 213 151)(15 34 214 152)(16 35 215 153)(17 36 216 154)(18 37 217 155)(19 38 218 156)(20 39 219 157)(21 40 220 158)(22 41 221 159)(23 42 222 160)(24 43 223 161)(25 44 224 162)(26 45 225 163)(27 46 226 164)(28 47 227 165)(29 48 228 166)(30 49 229 167)(61 148 192 95)(62 149 193 96)(63 150 194 97)(64 121 195 98)(65 122 196 99)(66 123 197 100)(67 124 198 101)(68 125 199 102)(69 126 200 103)(70 127 201 104)(71 128 202 105)(72 129 203 106)(73 130 204 107)(74 131 205 108)(75 132 206 109)(76 133 207 110)(77 134 208 111)(78 135 209 112)(79 136 210 113)(80 137 181 114)(81 138 182 115)(82 139 183 116)(83 140 184 117)(84 141 185 118)(85 142 186 119)(86 143 187 120)(87 144 188 91)(88 145 189 92)(89 146 190 93)(90 147 191 94)
(1 72 16 87)(2 83 17 68)(3 64 18 79)(4 75 19 90)(5 86 20 71)(6 67 21 82)(7 78 22 63)(8 89 23 74)(9 70 24 85)(10 81 25 66)(11 62 26 77)(12 73 27 88)(13 84 28 69)(14 65 29 80)(15 76 30 61)(31 107 46 92)(32 118 47 103)(33 99 48 114)(34 110 49 95)(35 91 50 106)(36 102 51 117)(37 113 52 98)(38 94 53 109)(39 105 54 120)(40 116 55 101)(41 97 56 112)(42 108 57 93)(43 119 58 104)(44 100 59 115)(45 111 60 96)(121 155 136 170)(122 166 137 151)(123 177 138 162)(124 158 139 173)(125 169 140 154)(126 180 141 165)(127 161 142 176)(128 172 143 157)(129 153 144 168)(130 164 145 179)(131 175 146 160)(132 156 147 171)(133 167 148 152)(134 178 149 163)(135 159 150 174)(181 213 196 228)(182 224 197 239)(183 235 198 220)(184 216 199 231)(185 227 200 212)(186 238 201 223)(187 219 202 234)(188 230 203 215)(189 211 204 226)(190 222 205 237)(191 233 206 218)(192 214 207 229)(193 225 208 240)(194 236 209 221)(195 217 210 232)
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), (1,106,16,91)(2,105,17,120)(3,104,18,119)(4,103,19,118)(5,102,20,117)(6,101,21,116)(7,100,22,115)(8,99,23,114)(9,98,24,113)(10,97,25,112)(11,96,26,111)(12,95,27,110)(13,94,28,109)(14,93,29,108)(15,92,30,107)(31,76,46,61)(32,75,47,90)(33,74,48,89)(34,73,49,88)(35,72,50,87)(36,71,51,86)(37,70,52,85)(38,69,53,84)(39,68,54,83)(40,67,55,82)(41,66,56,81)(42,65,57,80)(43,64,58,79)(44,63,59,78)(45,62,60,77)(121,223,136,238)(122,222,137,237)(123,221,138,236)(124,220,139,235)(125,219,140,234)(126,218,141,233)(127,217,142,232)(128,216,143,231)(129,215,144,230)(130,214,145,229)(131,213,146,228)(132,212,147,227)(133,211,148,226)(134,240,149,225)(135,239,150,224)(151,205,166,190)(152,204,167,189)(153,203,168,188)(154,202,169,187)(155,201,170,186)(156,200,171,185)(157,199,172,184)(158,198,173,183)(159,197,174,182)(160,196,175,181)(161,195,176,210)(162,194,177,209)(163,193,178,208)(164,192,179,207)(165,191,180,206), (1,50,230,168)(2,51,231,169)(3,52,232,170)(4,53,233,171)(5,54,234,172)(6,55,235,173)(7,56,236,174)(8,57,237,175)(9,58,238,176)(10,59,239,177)(11,60,240,178)(12,31,211,179)(13,32,212,180)(14,33,213,151)(15,34,214,152)(16,35,215,153)(17,36,216,154)(18,37,217,155)(19,38,218,156)(20,39,219,157)(21,40,220,158)(22,41,221,159)(23,42,222,160)(24,43,223,161)(25,44,224,162)(26,45,225,163)(27,46,226,164)(28,47,227,165)(29,48,228,166)(30,49,229,167)(61,148,192,95)(62,149,193,96)(63,150,194,97)(64,121,195,98)(65,122,196,99)(66,123,197,100)(67,124,198,101)(68,125,199,102)(69,126,200,103)(70,127,201,104)(71,128,202,105)(72,129,203,106)(73,130,204,107)(74,131,205,108)(75,132,206,109)(76,133,207,110)(77,134,208,111)(78,135,209,112)(79,136,210,113)(80,137,181,114)(81,138,182,115)(82,139,183,116)(83,140,184,117)(84,141,185,118)(85,142,186,119)(86,143,187,120)(87,144,188,91)(88,145,189,92)(89,146,190,93)(90,147,191,94), (1,72,16,87)(2,83,17,68)(3,64,18,79)(4,75,19,90)(5,86,20,71)(6,67,21,82)(7,78,22,63)(8,89,23,74)(9,70,24,85)(10,81,25,66)(11,62,26,77)(12,73,27,88)(13,84,28,69)(14,65,29,80)(15,76,30,61)(31,107,46,92)(32,118,47,103)(33,99,48,114)(34,110,49,95)(35,91,50,106)(36,102,51,117)(37,113,52,98)(38,94,53,109)(39,105,54,120)(40,116,55,101)(41,97,56,112)(42,108,57,93)(43,119,58,104)(44,100,59,115)(45,111,60,96)(121,155,136,170)(122,166,137,151)(123,177,138,162)(124,158,139,173)(125,169,140,154)(126,180,141,165)(127,161,142,176)(128,172,143,157)(129,153,144,168)(130,164,145,179)(131,175,146,160)(132,156,147,171)(133,167,148,152)(134,178,149,163)(135,159,150,174)(181,213,196,228)(182,224,197,239)(183,235,198,220)(184,216,199,231)(185,227,200,212)(186,238,201,223)(187,219,202,234)(188,230,203,215)(189,211,204,226)(190,222,205,237)(191,233,206,218)(192,214,207,229)(193,225,208,240)(194,236,209,221)(195,217,210,232)>;
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), (1,106,16,91)(2,105,17,120)(3,104,18,119)(4,103,19,118)(5,102,20,117)(6,101,21,116)(7,100,22,115)(8,99,23,114)(9,98,24,113)(10,97,25,112)(11,96,26,111)(12,95,27,110)(13,94,28,109)(14,93,29,108)(15,92,30,107)(31,76,46,61)(32,75,47,90)(33,74,48,89)(34,73,49,88)(35,72,50,87)(36,71,51,86)(37,70,52,85)(38,69,53,84)(39,68,54,83)(40,67,55,82)(41,66,56,81)(42,65,57,80)(43,64,58,79)(44,63,59,78)(45,62,60,77)(121,223,136,238)(122,222,137,237)(123,221,138,236)(124,220,139,235)(125,219,140,234)(126,218,141,233)(127,217,142,232)(128,216,143,231)(129,215,144,230)(130,214,145,229)(131,213,146,228)(132,212,147,227)(133,211,148,226)(134,240,149,225)(135,239,150,224)(151,205,166,190)(152,204,167,189)(153,203,168,188)(154,202,169,187)(155,201,170,186)(156,200,171,185)(157,199,172,184)(158,198,173,183)(159,197,174,182)(160,196,175,181)(161,195,176,210)(162,194,177,209)(163,193,178,208)(164,192,179,207)(165,191,180,206), (1,50,230,168)(2,51,231,169)(3,52,232,170)(4,53,233,171)(5,54,234,172)(6,55,235,173)(7,56,236,174)(8,57,237,175)(9,58,238,176)(10,59,239,177)(11,60,240,178)(12,31,211,179)(13,32,212,180)(14,33,213,151)(15,34,214,152)(16,35,215,153)(17,36,216,154)(18,37,217,155)(19,38,218,156)(20,39,219,157)(21,40,220,158)(22,41,221,159)(23,42,222,160)(24,43,223,161)(25,44,224,162)(26,45,225,163)(27,46,226,164)(28,47,227,165)(29,48,228,166)(30,49,229,167)(61,148,192,95)(62,149,193,96)(63,150,194,97)(64,121,195,98)(65,122,196,99)(66,123,197,100)(67,124,198,101)(68,125,199,102)(69,126,200,103)(70,127,201,104)(71,128,202,105)(72,129,203,106)(73,130,204,107)(74,131,205,108)(75,132,206,109)(76,133,207,110)(77,134,208,111)(78,135,209,112)(79,136,210,113)(80,137,181,114)(81,138,182,115)(82,139,183,116)(83,140,184,117)(84,141,185,118)(85,142,186,119)(86,143,187,120)(87,144,188,91)(88,145,189,92)(89,146,190,93)(90,147,191,94), (1,72,16,87)(2,83,17,68)(3,64,18,79)(4,75,19,90)(5,86,20,71)(6,67,21,82)(7,78,22,63)(8,89,23,74)(9,70,24,85)(10,81,25,66)(11,62,26,77)(12,73,27,88)(13,84,28,69)(14,65,29,80)(15,76,30,61)(31,107,46,92)(32,118,47,103)(33,99,48,114)(34,110,49,95)(35,91,50,106)(36,102,51,117)(37,113,52,98)(38,94,53,109)(39,105,54,120)(40,116,55,101)(41,97,56,112)(42,108,57,93)(43,119,58,104)(44,100,59,115)(45,111,60,96)(121,155,136,170)(122,166,137,151)(123,177,138,162)(124,158,139,173)(125,169,140,154)(126,180,141,165)(127,161,142,176)(128,172,143,157)(129,153,144,168)(130,164,145,179)(131,175,146,160)(132,156,147,171)(133,167,148,152)(134,178,149,163)(135,159,150,174)(181,213,196,228)(182,224,197,239)(183,235,198,220)(184,216,199,231)(185,227,200,212)(186,238,201,223)(187,219,202,234)(188,230,203,215)(189,211,204,226)(190,222,205,237)(191,233,206,218)(192,214,207,229)(193,225,208,240)(194,236,209,221)(195,217,210,232) );
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)], [(1,106,16,91),(2,105,17,120),(3,104,18,119),(4,103,19,118),(5,102,20,117),(6,101,21,116),(7,100,22,115),(8,99,23,114),(9,98,24,113),(10,97,25,112),(11,96,26,111),(12,95,27,110),(13,94,28,109),(14,93,29,108),(15,92,30,107),(31,76,46,61),(32,75,47,90),(33,74,48,89),(34,73,49,88),(35,72,50,87),(36,71,51,86),(37,70,52,85),(38,69,53,84),(39,68,54,83),(40,67,55,82),(41,66,56,81),(42,65,57,80),(43,64,58,79),(44,63,59,78),(45,62,60,77),(121,223,136,238),(122,222,137,237),(123,221,138,236),(124,220,139,235),(125,219,140,234),(126,218,141,233),(127,217,142,232),(128,216,143,231),(129,215,144,230),(130,214,145,229),(131,213,146,228),(132,212,147,227),(133,211,148,226),(134,240,149,225),(135,239,150,224),(151,205,166,190),(152,204,167,189),(153,203,168,188),(154,202,169,187),(155,201,170,186),(156,200,171,185),(157,199,172,184),(158,198,173,183),(159,197,174,182),(160,196,175,181),(161,195,176,210),(162,194,177,209),(163,193,178,208),(164,192,179,207),(165,191,180,206)], [(1,50,230,168),(2,51,231,169),(3,52,232,170),(4,53,233,171),(5,54,234,172),(6,55,235,173),(7,56,236,174),(8,57,237,175),(9,58,238,176),(10,59,239,177),(11,60,240,178),(12,31,211,179),(13,32,212,180),(14,33,213,151),(15,34,214,152),(16,35,215,153),(17,36,216,154),(18,37,217,155),(19,38,218,156),(20,39,219,157),(21,40,220,158),(22,41,221,159),(23,42,222,160),(24,43,223,161),(25,44,224,162),(26,45,225,163),(27,46,226,164),(28,47,227,165),(29,48,228,166),(30,49,229,167),(61,148,192,95),(62,149,193,96),(63,150,194,97),(64,121,195,98),(65,122,196,99),(66,123,197,100),(67,124,198,101),(68,125,199,102),(69,126,200,103),(70,127,201,104),(71,128,202,105),(72,129,203,106),(73,130,204,107),(74,131,205,108),(75,132,206,109),(76,133,207,110),(77,134,208,111),(78,135,209,112),(79,136,210,113),(80,137,181,114),(81,138,182,115),(82,139,183,116),(83,140,184,117),(84,141,185,118),(85,142,186,119),(86,143,187,120),(87,144,188,91),(88,145,189,92),(89,146,190,93),(90,147,191,94)], [(1,72,16,87),(2,83,17,68),(3,64,18,79),(4,75,19,90),(5,86,20,71),(6,67,21,82),(7,78,22,63),(8,89,23,74),(9,70,24,85),(10,81,25,66),(11,62,26,77),(12,73,27,88),(13,84,28,69),(14,65,29,80),(15,76,30,61),(31,107,46,92),(32,118,47,103),(33,99,48,114),(34,110,49,95),(35,91,50,106),(36,102,51,117),(37,113,52,98),(38,94,53,109),(39,105,54,120),(40,116,55,101),(41,97,56,112),(42,108,57,93),(43,119,58,104),(44,100,59,115),(45,111,60,96),(121,155,136,170),(122,166,137,151),(123,177,138,162),(124,158,139,173),(125,169,140,154),(126,180,141,165),(127,161,142,176),(128,172,143,157),(129,153,144,168),(130,164,145,179),(131,175,146,160),(132,156,147,171),(133,167,148,152),(134,178,149,163),(135,159,150,174),(181,213,196,228),(182,224,197,239),(183,235,198,220),(184,216,199,231),(185,227,200,212),(186,238,201,223),(187,219,202,234),(188,230,203,215),(189,211,204,226),(190,222,205,237),(191,233,206,218),(192,214,207,229),(193,225,208,240),(194,236,209,221),(195,217,210,232)]])
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 | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| 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 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 4 | 6 | 6 | 12 | 20 | 30 | 30 | 60 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 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 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | + | - | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | Dic6 | C4○D20 | S3×D4 | D4⋊2S3 | S3×D5 | D4×D5 | Q8×D5 | C2×S3×D5 | D5×Dic6 | C20⋊D6 | Dic5.D6 |
| kernel | Dic15.D4 | D10⋊Dic3 | Dic15⋊5C4 | C3×D10⋊C4 | C5×C4⋊Dic3 | C30.4Q8 | C2×D5×Dic3 | C2×C15⋊Q8 | D10⋊C4 | Dic15 | C6×D5 | C4⋊Dic3 | C2×Dic5 | C2×C20 | C22×D5 | C30 | C2×Dic3 | C2×C12 | D10 | C6 | C10 | C10 | C2×C4 | C6 | 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 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of Dic15.D4 ►in GL6(𝔽61)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 60 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 15 |
| 0 | 0 | 0 | 0 | 12 | 60 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 18 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 57 | 3 |
| 0 | 0 | 0 | 0 | 35 | 4 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 52 | 5 | 0 | 0 |
| 0 | 0 | 8 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 53 | 42 |
| 0 | 0 | 0 | 0 | 58 | 8 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 56 | 0 | 0 |
| 0 | 0 | 16 | 52 | 0 | 0 |
| 0 | 0 | 0 | 0 | 38 | 22 |
| 0 | 0 | 0 | 0 | 48 | 23 |
G:=sub<GL(6,GF(61))| [0,60,0,0,0,0,1,43,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,12,0,0,0,0,15,60],[60,18,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,57,35,0,0,0,0,3,4],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,52,8,0,0,0,0,5,9,0,0,0,0,0,0,53,58,0,0,0,0,42,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,16,0,0,0,0,56,52,0,0,0,0,0,0,38,48,0,0,0,0,22,23] >;
Dic15.D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{15}.D_4 % in TeX
G:=Group("Dic15.D4"); // GroupNames label
G:=SmallGroup(480,506);
// by ID
G=gap.SmallGroup(480,506);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,590,219,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=c^4=1,b^2=d^2=a^15,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^11,c*b*c^-1=d*b*d^-1=a^15*b,d*c*d^-1=c^-1>;
// generators/relations