metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic15.31D4, D6⋊C4⋊13D5, C6.48(D4×D5), C30.64(C2×D4), C10.49(S3×D4), D6⋊Dic5⋊20C2, (C2×C20).204D6, D10⋊C4⋊13S3, (C4×Dic15)⋊13C2, C6.42(C4○D20), (C2×C12).202D10, C15⋊12(C4.4D4), (C2×Dic5).48D6, (C22×D5).24D6, C30.127(C4○D4), C10.45(C4○D12), C6.74(D4⋊2D5), D10⋊Dic3⋊20C2, C2.24(C20⋊D6), (C2×C60).170C22, (C2×C30).154C23, (C2×Dic3).48D10, C3⋊3(Dic5.5D4), C5⋊3(C23.11D6), (C22×S3).22D10, C10.74(D4⋊2S3), (C6×Dic5).93C22, C2.19(C30.C23), C2.31(D6.D10), (C10×Dic3).93C22, (C2×Dic15).215C22, (C2×C15⋊Q8)⋊12C2, (C5×D6⋊C4)⋊13C2, (C2×C4).183(S3×D5), (C2×C15⋊D4).9C2, (D5×C2×C6).38C22, C22.206(C2×S3×D5), (S3×C2×C10).38C22, (C3×D10⋊C4)⋊13C2, (C2×C6).166(C22×D5), (C2×C10).166(C22×S3), SmallGroup(480,540)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic15.31D4
G = < a,b,c,d | a30=c4=1, b2=d2=a15, bab-1=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a15b, dcd-1=a15c-1 >
Subgroups: 812 in 152 conjugacy classes, 46 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C42, C22⋊C4, C2×D4, C2×Q8, Dic5, C20, D10, C2×C10, C2×C10, Dic6, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C5×S3, C3×D5, C30, C4.4D4, Dic10, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C4×Dic3, D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C2×Dic6, C2×C3⋊D4, C5×Dic3, C3×Dic5, Dic15, Dic15, C60, C6×D5, S3×C10, C2×C30, C4×Dic5, D10⋊C4, D10⋊C4, C23.D5, C5×C22⋊C4, C2×Dic10, C2×C5⋊D4, C23.11D6, C15⋊D4, C15⋊Q8, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, S3×C2×C10, Dic5.5D4, D10⋊Dic3, D6⋊Dic5, C3×D10⋊C4, C5×D6⋊C4, C4×Dic15, C2×C15⋊D4, C2×C15⋊Q8, Dic15.31D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C22×S3, C4.4D4, C22×D5, C4○D12, S3×D4, D4⋊2S3, S3×D5, C4○D20, D4×D5, D4⋊2D5, C23.11D6, C2×S3×D5, Dic5.5D4, D6.D10, C20⋊D6, C30.C23, Dic15.31D4
►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)
(1 147 16 132)(2 146 17 131)(3 145 18 130)(4 144 19 129)(5 143 20 128)(6 142 21 127)(7 141 22 126)(8 140 23 125)(9 139 24 124)(10 138 25 123)(11 137 26 122)(12 136 27 121)(13 135 28 150)(14 134 29 149)(15 133 30 148)(31 208 46 193)(32 207 47 192)(33 206 48 191)(34 205 49 190)(35 204 50 189)(36 203 51 188)(37 202 52 187)(38 201 53 186)(39 200 54 185)(40 199 55 184)(41 198 56 183)(42 197 57 182)(43 196 58 181)(44 195 59 210)(45 194 60 209)(61 101 76 116)(62 100 77 115)(63 99 78 114)(64 98 79 113)(65 97 80 112)(66 96 81 111)(67 95 82 110)(68 94 83 109)(69 93 84 108)(70 92 85 107)(71 91 86 106)(72 120 87 105)(73 119 88 104)(74 118 89 103)(75 117 90 102)(151 217 166 232)(152 216 167 231)(153 215 168 230)(154 214 169 229)(155 213 170 228)(156 212 171 227)(157 211 172 226)(158 240 173 225)(159 239 174 224)(160 238 175 223)(161 237 176 222)(162 236 177 221)(163 235 178 220)(164 234 179 219)(165 233 180 218)
(1 70 54 228)(2 71 55 229)(3 72 56 230)(4 73 57 231)(5 74 58 232)(6 75 59 233)(7 76 60 234)(8 77 31 235)(9 78 32 236)(10 79 33 237)(11 80 34 238)(12 81 35 239)(13 82 36 240)(14 83 37 211)(15 84 38 212)(16 85 39 213)(17 86 40 214)(18 87 41 215)(19 88 42 216)(20 89 43 217)(21 90 44 218)(22 61 45 219)(23 62 46 220)(24 63 47 221)(25 64 48 222)(26 65 49 223)(27 66 50 224)(28 67 51 225)(29 68 52 226)(30 69 53 227)(91 184 154 146)(92 185 155 147)(93 186 156 148)(94 187 157 149)(95 188 158 150)(96 189 159 121)(97 190 160 122)(98 191 161 123)(99 192 162 124)(100 193 163 125)(101 194 164 126)(102 195 165 127)(103 196 166 128)(104 197 167 129)(105 198 168 130)(106 199 169 131)(107 200 170 132)(108 201 171 133)(109 202 172 134)(110 203 173 135)(111 204 174 136)(112 205 175 137)(113 206 176 138)(114 207 177 139)(115 208 178 140)(116 209 179 141)(117 210 180 142)(118 181 151 143)(119 182 152 144)(120 183 153 145)
(1 92 16 107)(2 103 17 118)(3 114 18 99)(4 95 19 110)(5 106 20 91)(6 117 21 102)(7 98 22 113)(8 109 23 94)(9 120 24 105)(10 101 25 116)(11 112 26 97)(12 93 27 108)(13 104 28 119)(14 115 29 100)(15 96 30 111)(31 172 46 157)(32 153 47 168)(33 164 48 179)(34 175 49 160)(35 156 50 171)(36 167 51 152)(37 178 52 163)(38 159 53 174)(39 170 54 155)(40 151 55 166)(41 162 56 177)(42 173 57 158)(43 154 58 169)(44 165 59 180)(45 176 60 161)(61 123 76 138)(62 134 77 149)(63 145 78 130)(64 126 79 141)(65 137 80 122)(66 148 81 133)(67 129 82 144)(68 140 83 125)(69 121 84 136)(70 132 85 147)(71 143 86 128)(72 124 87 139)(73 135 88 150)(74 146 89 131)(75 127 90 142)(181 214 196 229)(182 225 197 240)(183 236 198 221)(184 217 199 232)(185 228 200 213)(186 239 201 224)(187 220 202 235)(188 231 203 216)(189 212 204 227)(190 223 205 238)(191 234 206 219)(192 215 207 230)(193 226 208 211)(194 237 209 222)(195 218 210 233)
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), (1,147,16,132)(2,146,17,131)(3,145,18,130)(4,144,19,129)(5,143,20,128)(6,142,21,127)(7,141,22,126)(8,140,23,125)(9,139,24,124)(10,138,25,123)(11,137,26,122)(12,136,27,121)(13,135,28,150)(14,134,29,149)(15,133,30,148)(31,208,46,193)(32,207,47,192)(33,206,48,191)(34,205,49,190)(35,204,50,189)(36,203,51,188)(37,202,52,187)(38,201,53,186)(39,200,54,185)(40,199,55,184)(41,198,56,183)(42,197,57,182)(43,196,58,181)(44,195,59,210)(45,194,60,209)(61,101,76,116)(62,100,77,115)(63,99,78,114)(64,98,79,113)(65,97,80,112)(66,96,81,111)(67,95,82,110)(68,94,83,109)(69,93,84,108)(70,92,85,107)(71,91,86,106)(72,120,87,105)(73,119,88,104)(74,118,89,103)(75,117,90,102)(151,217,166,232)(152,216,167,231)(153,215,168,230)(154,214,169,229)(155,213,170,228)(156,212,171,227)(157,211,172,226)(158,240,173,225)(159,239,174,224)(160,238,175,223)(161,237,176,222)(162,236,177,221)(163,235,178,220)(164,234,179,219)(165,233,180,218), (1,70,54,228)(2,71,55,229)(3,72,56,230)(4,73,57,231)(5,74,58,232)(6,75,59,233)(7,76,60,234)(8,77,31,235)(9,78,32,236)(10,79,33,237)(11,80,34,238)(12,81,35,239)(13,82,36,240)(14,83,37,211)(15,84,38,212)(16,85,39,213)(17,86,40,214)(18,87,41,215)(19,88,42,216)(20,89,43,217)(21,90,44,218)(22,61,45,219)(23,62,46,220)(24,63,47,221)(25,64,48,222)(26,65,49,223)(27,66,50,224)(28,67,51,225)(29,68,52,226)(30,69,53,227)(91,184,154,146)(92,185,155,147)(93,186,156,148)(94,187,157,149)(95,188,158,150)(96,189,159,121)(97,190,160,122)(98,191,161,123)(99,192,162,124)(100,193,163,125)(101,194,164,126)(102,195,165,127)(103,196,166,128)(104,197,167,129)(105,198,168,130)(106,199,169,131)(107,200,170,132)(108,201,171,133)(109,202,172,134)(110,203,173,135)(111,204,174,136)(112,205,175,137)(113,206,176,138)(114,207,177,139)(115,208,178,140)(116,209,179,141)(117,210,180,142)(118,181,151,143)(119,182,152,144)(120,183,153,145), (1,92,16,107)(2,103,17,118)(3,114,18,99)(4,95,19,110)(5,106,20,91)(6,117,21,102)(7,98,22,113)(8,109,23,94)(9,120,24,105)(10,101,25,116)(11,112,26,97)(12,93,27,108)(13,104,28,119)(14,115,29,100)(15,96,30,111)(31,172,46,157)(32,153,47,168)(33,164,48,179)(34,175,49,160)(35,156,50,171)(36,167,51,152)(37,178,52,163)(38,159,53,174)(39,170,54,155)(40,151,55,166)(41,162,56,177)(42,173,57,158)(43,154,58,169)(44,165,59,180)(45,176,60,161)(61,123,76,138)(62,134,77,149)(63,145,78,130)(64,126,79,141)(65,137,80,122)(66,148,81,133)(67,129,82,144)(68,140,83,125)(69,121,84,136)(70,132,85,147)(71,143,86,128)(72,124,87,139)(73,135,88,150)(74,146,89,131)(75,127,90,142)(181,214,196,229)(182,225,197,240)(183,236,198,221)(184,217,199,232)(185,228,200,213)(186,239,201,224)(187,220,202,235)(188,231,203,216)(189,212,204,227)(190,223,205,238)(191,234,206,219)(192,215,207,230)(193,226,208,211)(194,237,209,222)(195,218,210,233)>;
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), (1,147,16,132)(2,146,17,131)(3,145,18,130)(4,144,19,129)(5,143,20,128)(6,142,21,127)(7,141,22,126)(8,140,23,125)(9,139,24,124)(10,138,25,123)(11,137,26,122)(12,136,27,121)(13,135,28,150)(14,134,29,149)(15,133,30,148)(31,208,46,193)(32,207,47,192)(33,206,48,191)(34,205,49,190)(35,204,50,189)(36,203,51,188)(37,202,52,187)(38,201,53,186)(39,200,54,185)(40,199,55,184)(41,198,56,183)(42,197,57,182)(43,196,58,181)(44,195,59,210)(45,194,60,209)(61,101,76,116)(62,100,77,115)(63,99,78,114)(64,98,79,113)(65,97,80,112)(66,96,81,111)(67,95,82,110)(68,94,83,109)(69,93,84,108)(70,92,85,107)(71,91,86,106)(72,120,87,105)(73,119,88,104)(74,118,89,103)(75,117,90,102)(151,217,166,232)(152,216,167,231)(153,215,168,230)(154,214,169,229)(155,213,170,228)(156,212,171,227)(157,211,172,226)(158,240,173,225)(159,239,174,224)(160,238,175,223)(161,237,176,222)(162,236,177,221)(163,235,178,220)(164,234,179,219)(165,233,180,218), (1,70,54,228)(2,71,55,229)(3,72,56,230)(4,73,57,231)(5,74,58,232)(6,75,59,233)(7,76,60,234)(8,77,31,235)(9,78,32,236)(10,79,33,237)(11,80,34,238)(12,81,35,239)(13,82,36,240)(14,83,37,211)(15,84,38,212)(16,85,39,213)(17,86,40,214)(18,87,41,215)(19,88,42,216)(20,89,43,217)(21,90,44,218)(22,61,45,219)(23,62,46,220)(24,63,47,221)(25,64,48,222)(26,65,49,223)(27,66,50,224)(28,67,51,225)(29,68,52,226)(30,69,53,227)(91,184,154,146)(92,185,155,147)(93,186,156,148)(94,187,157,149)(95,188,158,150)(96,189,159,121)(97,190,160,122)(98,191,161,123)(99,192,162,124)(100,193,163,125)(101,194,164,126)(102,195,165,127)(103,196,166,128)(104,197,167,129)(105,198,168,130)(106,199,169,131)(107,200,170,132)(108,201,171,133)(109,202,172,134)(110,203,173,135)(111,204,174,136)(112,205,175,137)(113,206,176,138)(114,207,177,139)(115,208,178,140)(116,209,179,141)(117,210,180,142)(118,181,151,143)(119,182,152,144)(120,183,153,145), (1,92,16,107)(2,103,17,118)(3,114,18,99)(4,95,19,110)(5,106,20,91)(6,117,21,102)(7,98,22,113)(8,109,23,94)(9,120,24,105)(10,101,25,116)(11,112,26,97)(12,93,27,108)(13,104,28,119)(14,115,29,100)(15,96,30,111)(31,172,46,157)(32,153,47,168)(33,164,48,179)(34,175,49,160)(35,156,50,171)(36,167,51,152)(37,178,52,163)(38,159,53,174)(39,170,54,155)(40,151,55,166)(41,162,56,177)(42,173,57,158)(43,154,58,169)(44,165,59,180)(45,176,60,161)(61,123,76,138)(62,134,77,149)(63,145,78,130)(64,126,79,141)(65,137,80,122)(66,148,81,133)(67,129,82,144)(68,140,83,125)(69,121,84,136)(70,132,85,147)(71,143,86,128)(72,124,87,139)(73,135,88,150)(74,146,89,131)(75,127,90,142)(181,214,196,229)(182,225,197,240)(183,236,198,221)(184,217,199,232)(185,228,200,213)(186,239,201,224)(187,220,202,235)(188,231,203,216)(189,212,204,227)(190,223,205,238)(191,234,206,219)(192,215,207,230)(193,226,208,211)(194,237,209,222)(195,218,210,233) );
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)], [(1,147,16,132),(2,146,17,131),(3,145,18,130),(4,144,19,129),(5,143,20,128),(6,142,21,127),(7,141,22,126),(8,140,23,125),(9,139,24,124),(10,138,25,123),(11,137,26,122),(12,136,27,121),(13,135,28,150),(14,134,29,149),(15,133,30,148),(31,208,46,193),(32,207,47,192),(33,206,48,191),(34,205,49,190),(35,204,50,189),(36,203,51,188),(37,202,52,187),(38,201,53,186),(39,200,54,185),(40,199,55,184),(41,198,56,183),(42,197,57,182),(43,196,58,181),(44,195,59,210),(45,194,60,209),(61,101,76,116),(62,100,77,115),(63,99,78,114),(64,98,79,113),(65,97,80,112),(66,96,81,111),(67,95,82,110),(68,94,83,109),(69,93,84,108),(70,92,85,107),(71,91,86,106),(72,120,87,105),(73,119,88,104),(74,118,89,103),(75,117,90,102),(151,217,166,232),(152,216,167,231),(153,215,168,230),(154,214,169,229),(155,213,170,228),(156,212,171,227),(157,211,172,226),(158,240,173,225),(159,239,174,224),(160,238,175,223),(161,237,176,222),(162,236,177,221),(163,235,178,220),(164,234,179,219),(165,233,180,218)], [(1,70,54,228),(2,71,55,229),(3,72,56,230),(4,73,57,231),(5,74,58,232),(6,75,59,233),(7,76,60,234),(8,77,31,235),(9,78,32,236),(10,79,33,237),(11,80,34,238),(12,81,35,239),(13,82,36,240),(14,83,37,211),(15,84,38,212),(16,85,39,213),(17,86,40,214),(18,87,41,215),(19,88,42,216),(20,89,43,217),(21,90,44,218),(22,61,45,219),(23,62,46,220),(24,63,47,221),(25,64,48,222),(26,65,49,223),(27,66,50,224),(28,67,51,225),(29,68,52,226),(30,69,53,227),(91,184,154,146),(92,185,155,147),(93,186,156,148),(94,187,157,149),(95,188,158,150),(96,189,159,121),(97,190,160,122),(98,191,161,123),(99,192,162,124),(100,193,163,125),(101,194,164,126),(102,195,165,127),(103,196,166,128),(104,197,167,129),(105,198,168,130),(106,199,169,131),(107,200,170,132),(108,201,171,133),(109,202,172,134),(110,203,173,135),(111,204,174,136),(112,205,175,137),(113,206,176,138),(114,207,177,139),(115,208,178,140),(116,209,179,141),(117,210,180,142),(118,181,151,143),(119,182,152,144),(120,183,153,145)], [(1,92,16,107),(2,103,17,118),(3,114,18,99),(4,95,19,110),(5,106,20,91),(6,117,21,102),(7,98,22,113),(8,109,23,94),(9,120,24,105),(10,101,25,116),(11,112,26,97),(12,93,27,108),(13,104,28,119),(14,115,29,100),(15,96,30,111),(31,172,46,157),(32,153,47,168),(33,164,48,179),(34,175,49,160),(35,156,50,171),(36,167,51,152),(37,178,52,163),(38,159,53,174),(39,170,54,155),(40,151,55,166),(41,162,56,177),(42,173,57,158),(43,154,58,169),(44,165,59,180),(45,176,60,161),(61,123,76,138),(62,134,77,149),(63,145,78,130),(64,126,79,141),(65,137,80,122),(66,148,81,133),(67,129,82,144),(68,140,83,125),(69,121,84,136),(70,132,85,147),(71,143,86,128),(72,124,87,139),(73,135,88,150),(74,146,89,131),(75,127,90,142),(181,214,196,229),(182,225,197,240),(183,236,198,221),(184,217,199,232),(185,228,200,213),(186,239,201,224),(187,220,202,235),(188,231,203,216),(189,212,204,227),(190,223,205,238),(191,234,206,219),(192,215,207,230),(193,226,208,211),(194,237,209,222),(195,218,210,233)]])
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 | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 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 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 12 | 20 | 2 | 2 | 2 | 12 | 20 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 4 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 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 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D10 | C4○D12 | C4○D20 | S3×D4 | D4⋊2S3 | S3×D5 | D4×D5 | D4⋊2D5 | C2×S3×D5 | D6.D10 | C20⋊D6 | C30.C23 |
| kernel | Dic15.31D4 | D10⋊Dic3 | D6⋊Dic5 | C3×D10⋊C4 | C5×D6⋊C4 | C4×Dic15 | C2×C15⋊D4 | C2×C15⋊Q8 | D10⋊C4 | Dic15 | D6⋊C4 | C2×Dic5 | C2×C20 | C22×D5 | C30 | C2×Dic3 | C2×C12 | C22×S3 | C10 | C6 | C10 | C10 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of Dic15.31D4 ►in GL6(𝔽61)
| 1 | 16 | 0 | 0 | 0 | 0 |
| 1 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 48 | 0 | 0 | 0 |
| 0 | 0 | 20 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 39 | 8 | 0 | 0 | 0 | 0 |
| 8 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 56 | 0 | 0 |
| 0 | 0 | 33 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 52 | 35 |
| 0 | 0 | 0 | 0 | 36 | 9 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 50 | 0 | 0 | 0 |
| 0 | 0 | 0 | 50 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 19 |
| 0 | 0 | 0 | 0 | 30 | 38 |
| 32 | 58 | 0 | 0 | 0 | 0 |
| 57 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 6 | 0 | 0 |
| 0 | 0 | 17 | 43 | 0 | 0 |
| 0 | 0 | 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 0 | 17 | 11 |
G:=sub<GL(6,GF(61))| [1,1,0,0,0,0,16,17,0,0,0,0,0,0,48,20,0,0,0,0,0,14,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[39,8,0,0,0,0,8,22,0,0,0,0,0,0,46,33,0,0,0,0,56,15,0,0,0,0,0,0,52,36,0,0,0,0,35,9],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,23,30,0,0,0,0,19,38],[32,57,0,0,0,0,58,29,0,0,0,0,0,0,18,17,0,0,0,0,6,43,0,0,0,0,0,0,50,17,0,0,0,0,0,11] >;
Dic15.31D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{15}._{31}D_4 % in TeX
G:=Group("Dic15.31D4"); // GroupNames label
G:=SmallGroup(480,540);
// by ID
G=gap.SmallGroup(480,540);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,422,219,142,1356,18822]);
// 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,b*c=c*b,d*b*d^-1=a^15*b,d*c*d^-1=a^15*c^-1>;
// generators/relations