metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic15.10D4, D6⋊C4⋊7D5, C6.65(D4×D5), (C2×C20).30D6, C10.67(S3×D4), D10⋊C4⋊7S3, D6⋊Dic5⋊19C2, (C2×Dic30)⋊3C2, C30.156(C2×D4), (C2×C12).30D10, C30.94(C4○D4), C6.59(C4○D20), C15⋊11(C4.4D4), (C2×C60).15C22, (C22×D5).22D6, (Dic3×Dic5)⋊24C2, C10.63(C4○D12), C6.32(D4⋊2D5), D10⋊Dic3⋊19C2, (C2×C30).152C23, (C2×Dic5).122D6, C3⋊2(Dic5.5D4), C5⋊2(C23.11D6), (C22×S3).20D10, C10.34(D4⋊2S3), C2.19(D20⋊5S3), C2.19(D12⋊5D5), C2.18(D10⋊D6), (C2×Dic3).114D10, (C6×Dic5).92C22, (C10×Dic3).92C22, (C2×Dic15).115C22, (C5×D6⋊C4)⋊7C2, (C2×C4).63(S3×D5), (C2×C15⋊D4).8C2, (C3×D10⋊C4)⋊7C2, (D5×C2×C6).36C22, C22.204(C2×S3×D5), (S3×C2×C10).36C22, (C2×C6).164(C22×D5), (C2×C10).164(C22×S3), SmallGroup(480,538)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic15.10D4
G = < a,b,c,d | a30=c4=d2=1, b2=a15, bab-1=a-1, cac-1=dad=a19, bc=cb, dbd=a15b, dcd=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, C6×Dic5, C10×Dic3, Dic30, C2×Dic15, C2×C60, D5×C2×C6, S3×C2×C10, Dic5.5D4, Dic3×Dic5, D10⋊Dic3, D6⋊Dic5, C3×D10⋊C4, C5×D6⋊C4, C2×C15⋊D4, C2×Dic30, Dic15.10D4
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, D20⋊5S3, D12⋊5D5, D10⋊D6, Dic15.10D4
►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)
(1 130 16 145)(2 129 17 144)(3 128 18 143)(4 127 19 142)(5 126 20 141)(6 125 21 140)(7 124 22 139)(8 123 23 138)(9 122 24 137)(10 121 25 136)(11 150 26 135)(12 149 27 134)(13 148 28 133)(14 147 29 132)(15 146 30 131)(31 215 46 230)(32 214 47 229)(33 213 48 228)(34 212 49 227)(35 211 50 226)(36 240 51 225)(37 239 52 224)(38 238 53 223)(39 237 54 222)(40 236 55 221)(41 235 56 220)(42 234 57 219)(43 233 58 218)(44 232 59 217)(45 231 60 216)(61 105 76 120)(62 104 77 119)(63 103 78 118)(64 102 79 117)(65 101 80 116)(66 100 81 115)(67 99 82 114)(68 98 83 113)(69 97 84 112)(70 96 85 111)(71 95 86 110)(72 94 87 109)(73 93 88 108)(74 92 89 107)(75 91 90 106)(151 182 166 197)(152 181 167 196)(153 210 168 195)(154 209 169 194)(155 208 170 193)(156 207 171 192)(157 206 172 191)(158 205 173 190)(159 204 174 189)(160 203 175 188)(161 202 176 187)(162 201 177 186)(163 200 178 185)(164 199 179 184)(165 198 180 183)
(1 68 42 153)(2 87 43 172)(3 76 44 161)(4 65 45 180)(5 84 46 169)(6 73 47 158)(7 62 48 177)(8 81 49 166)(9 70 50 155)(10 89 51 174)(11 78 52 163)(12 67 53 152)(13 86 54 171)(14 75 55 160)(15 64 56 179)(16 83 57 168)(17 72 58 157)(18 61 59 176)(19 80 60 165)(20 69 31 154)(21 88 32 173)(22 77 33 162)(23 66 34 151)(24 85 35 170)(25 74 36 159)(26 63 37 178)(27 82 38 167)(28 71 39 156)(29 90 40 175)(30 79 41 164)(91 221 203 147)(92 240 204 136)(93 229 205 125)(94 218 206 144)(95 237 207 133)(96 226 208 122)(97 215 209 141)(98 234 210 130)(99 223 181 149)(100 212 182 138)(101 231 183 127)(102 220 184 146)(103 239 185 135)(104 228 186 124)(105 217 187 143)(106 236 188 132)(107 225 189 121)(108 214 190 140)(109 233 191 129)(110 222 192 148)(111 211 193 137)(112 230 194 126)(113 219 195 145)(114 238 196 134)(115 227 197 123)(116 216 198 142)(117 235 199 131)(118 224 200 150)(119 213 201 139)(120 232 202 128)
(1 57)(2 46)(3 35)(4 54)(5 43)(6 32)(7 51)(8 40)(9 59)(10 48)(11 37)(12 56)(13 45)(14 34)(15 53)(16 42)(17 31)(18 50)(19 39)(20 58)(21 47)(22 36)(23 55)(24 44)(25 33)(26 52)(27 41)(28 60)(29 49)(30 38)(61 85)(62 74)(64 82)(65 71)(66 90)(67 79)(69 87)(70 76)(72 84)(75 81)(77 89)(80 86)(91 100)(92 119)(93 108)(94 97)(95 116)(96 105)(98 113)(99 102)(101 110)(103 118)(104 107)(106 115)(109 112)(111 120)(114 117)(121 213)(122 232)(123 221)(124 240)(125 229)(126 218)(127 237)(128 226)(129 215)(130 234)(131 223)(132 212)(133 231)(134 220)(135 239)(136 228)(137 217)(138 236)(139 225)(140 214)(141 233)(142 222)(143 211)(144 230)(145 219)(146 238)(147 227)(148 216)(149 235)(150 224)(151 175)(152 164)(154 172)(155 161)(156 180)(157 169)(159 177)(160 166)(162 174)(165 171)(167 179)(170 176)(181 184)(182 203)(183 192)(185 200)(186 189)(187 208)(188 197)(190 205)(191 194)(193 202)(195 210)(196 199)(198 207)(201 204)(206 209)
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), (1,130,16,145)(2,129,17,144)(3,128,18,143)(4,127,19,142)(5,126,20,141)(6,125,21,140)(7,124,22,139)(8,123,23,138)(9,122,24,137)(10,121,25,136)(11,150,26,135)(12,149,27,134)(13,148,28,133)(14,147,29,132)(15,146,30,131)(31,215,46,230)(32,214,47,229)(33,213,48,228)(34,212,49,227)(35,211,50,226)(36,240,51,225)(37,239,52,224)(38,238,53,223)(39,237,54,222)(40,236,55,221)(41,235,56,220)(42,234,57,219)(43,233,58,218)(44,232,59,217)(45,231,60,216)(61,105,76,120)(62,104,77,119)(63,103,78,118)(64,102,79,117)(65,101,80,116)(66,100,81,115)(67,99,82,114)(68,98,83,113)(69,97,84,112)(70,96,85,111)(71,95,86,110)(72,94,87,109)(73,93,88,108)(74,92,89,107)(75,91,90,106)(151,182,166,197)(152,181,167,196)(153,210,168,195)(154,209,169,194)(155,208,170,193)(156,207,171,192)(157,206,172,191)(158,205,173,190)(159,204,174,189)(160,203,175,188)(161,202,176,187)(162,201,177,186)(163,200,178,185)(164,199,179,184)(165,198,180,183), (1,68,42,153)(2,87,43,172)(3,76,44,161)(4,65,45,180)(5,84,46,169)(6,73,47,158)(7,62,48,177)(8,81,49,166)(9,70,50,155)(10,89,51,174)(11,78,52,163)(12,67,53,152)(13,86,54,171)(14,75,55,160)(15,64,56,179)(16,83,57,168)(17,72,58,157)(18,61,59,176)(19,80,60,165)(20,69,31,154)(21,88,32,173)(22,77,33,162)(23,66,34,151)(24,85,35,170)(25,74,36,159)(26,63,37,178)(27,82,38,167)(28,71,39,156)(29,90,40,175)(30,79,41,164)(91,221,203,147)(92,240,204,136)(93,229,205,125)(94,218,206,144)(95,237,207,133)(96,226,208,122)(97,215,209,141)(98,234,210,130)(99,223,181,149)(100,212,182,138)(101,231,183,127)(102,220,184,146)(103,239,185,135)(104,228,186,124)(105,217,187,143)(106,236,188,132)(107,225,189,121)(108,214,190,140)(109,233,191,129)(110,222,192,148)(111,211,193,137)(112,230,194,126)(113,219,195,145)(114,238,196,134)(115,227,197,123)(116,216,198,142)(117,235,199,131)(118,224,200,150)(119,213,201,139)(120,232,202,128), (1,57)(2,46)(3,35)(4,54)(5,43)(6,32)(7,51)(8,40)(9,59)(10,48)(11,37)(12,56)(13,45)(14,34)(15,53)(16,42)(17,31)(18,50)(19,39)(20,58)(21,47)(22,36)(23,55)(24,44)(25,33)(26,52)(27,41)(28,60)(29,49)(30,38)(61,85)(62,74)(64,82)(65,71)(66,90)(67,79)(69,87)(70,76)(72,84)(75,81)(77,89)(80,86)(91,100)(92,119)(93,108)(94,97)(95,116)(96,105)(98,113)(99,102)(101,110)(103,118)(104,107)(106,115)(109,112)(111,120)(114,117)(121,213)(122,232)(123,221)(124,240)(125,229)(126,218)(127,237)(128,226)(129,215)(130,234)(131,223)(132,212)(133,231)(134,220)(135,239)(136,228)(137,217)(138,236)(139,225)(140,214)(141,233)(142,222)(143,211)(144,230)(145,219)(146,238)(147,227)(148,216)(149,235)(150,224)(151,175)(152,164)(154,172)(155,161)(156,180)(157,169)(159,177)(160,166)(162,174)(165,171)(167,179)(170,176)(181,184)(182,203)(183,192)(185,200)(186,189)(187,208)(188,197)(190,205)(191,194)(193,202)(195,210)(196,199)(198,207)(201,204)(206,209)>;
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), (1,130,16,145)(2,129,17,144)(3,128,18,143)(4,127,19,142)(5,126,20,141)(6,125,21,140)(7,124,22,139)(8,123,23,138)(9,122,24,137)(10,121,25,136)(11,150,26,135)(12,149,27,134)(13,148,28,133)(14,147,29,132)(15,146,30,131)(31,215,46,230)(32,214,47,229)(33,213,48,228)(34,212,49,227)(35,211,50,226)(36,240,51,225)(37,239,52,224)(38,238,53,223)(39,237,54,222)(40,236,55,221)(41,235,56,220)(42,234,57,219)(43,233,58,218)(44,232,59,217)(45,231,60,216)(61,105,76,120)(62,104,77,119)(63,103,78,118)(64,102,79,117)(65,101,80,116)(66,100,81,115)(67,99,82,114)(68,98,83,113)(69,97,84,112)(70,96,85,111)(71,95,86,110)(72,94,87,109)(73,93,88,108)(74,92,89,107)(75,91,90,106)(151,182,166,197)(152,181,167,196)(153,210,168,195)(154,209,169,194)(155,208,170,193)(156,207,171,192)(157,206,172,191)(158,205,173,190)(159,204,174,189)(160,203,175,188)(161,202,176,187)(162,201,177,186)(163,200,178,185)(164,199,179,184)(165,198,180,183), (1,68,42,153)(2,87,43,172)(3,76,44,161)(4,65,45,180)(5,84,46,169)(6,73,47,158)(7,62,48,177)(8,81,49,166)(9,70,50,155)(10,89,51,174)(11,78,52,163)(12,67,53,152)(13,86,54,171)(14,75,55,160)(15,64,56,179)(16,83,57,168)(17,72,58,157)(18,61,59,176)(19,80,60,165)(20,69,31,154)(21,88,32,173)(22,77,33,162)(23,66,34,151)(24,85,35,170)(25,74,36,159)(26,63,37,178)(27,82,38,167)(28,71,39,156)(29,90,40,175)(30,79,41,164)(91,221,203,147)(92,240,204,136)(93,229,205,125)(94,218,206,144)(95,237,207,133)(96,226,208,122)(97,215,209,141)(98,234,210,130)(99,223,181,149)(100,212,182,138)(101,231,183,127)(102,220,184,146)(103,239,185,135)(104,228,186,124)(105,217,187,143)(106,236,188,132)(107,225,189,121)(108,214,190,140)(109,233,191,129)(110,222,192,148)(111,211,193,137)(112,230,194,126)(113,219,195,145)(114,238,196,134)(115,227,197,123)(116,216,198,142)(117,235,199,131)(118,224,200,150)(119,213,201,139)(120,232,202,128), (1,57)(2,46)(3,35)(4,54)(5,43)(6,32)(7,51)(8,40)(9,59)(10,48)(11,37)(12,56)(13,45)(14,34)(15,53)(16,42)(17,31)(18,50)(19,39)(20,58)(21,47)(22,36)(23,55)(24,44)(25,33)(26,52)(27,41)(28,60)(29,49)(30,38)(61,85)(62,74)(64,82)(65,71)(66,90)(67,79)(69,87)(70,76)(72,84)(75,81)(77,89)(80,86)(91,100)(92,119)(93,108)(94,97)(95,116)(96,105)(98,113)(99,102)(101,110)(103,118)(104,107)(106,115)(109,112)(111,120)(114,117)(121,213)(122,232)(123,221)(124,240)(125,229)(126,218)(127,237)(128,226)(129,215)(130,234)(131,223)(132,212)(133,231)(134,220)(135,239)(136,228)(137,217)(138,236)(139,225)(140,214)(141,233)(142,222)(143,211)(144,230)(145,219)(146,238)(147,227)(148,216)(149,235)(150,224)(151,175)(152,164)(154,172)(155,161)(156,180)(157,169)(159,177)(160,166)(162,174)(165,171)(167,179)(170,176)(181,184)(182,203)(183,192)(185,200)(186,189)(187,208)(188,197)(190,205)(191,194)(193,202)(195,210)(196,199)(198,207)(201,204)(206,209) );
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)], [(1,130,16,145),(2,129,17,144),(3,128,18,143),(4,127,19,142),(5,126,20,141),(6,125,21,140),(7,124,22,139),(8,123,23,138),(9,122,24,137),(10,121,25,136),(11,150,26,135),(12,149,27,134),(13,148,28,133),(14,147,29,132),(15,146,30,131),(31,215,46,230),(32,214,47,229),(33,213,48,228),(34,212,49,227),(35,211,50,226),(36,240,51,225),(37,239,52,224),(38,238,53,223),(39,237,54,222),(40,236,55,221),(41,235,56,220),(42,234,57,219),(43,233,58,218),(44,232,59,217),(45,231,60,216),(61,105,76,120),(62,104,77,119),(63,103,78,118),(64,102,79,117),(65,101,80,116),(66,100,81,115),(67,99,82,114),(68,98,83,113),(69,97,84,112),(70,96,85,111),(71,95,86,110),(72,94,87,109),(73,93,88,108),(74,92,89,107),(75,91,90,106),(151,182,166,197),(152,181,167,196),(153,210,168,195),(154,209,169,194),(155,208,170,193),(156,207,171,192),(157,206,172,191),(158,205,173,190),(159,204,174,189),(160,203,175,188),(161,202,176,187),(162,201,177,186),(163,200,178,185),(164,199,179,184),(165,198,180,183)], [(1,68,42,153),(2,87,43,172),(3,76,44,161),(4,65,45,180),(5,84,46,169),(6,73,47,158),(7,62,48,177),(8,81,49,166),(9,70,50,155),(10,89,51,174),(11,78,52,163),(12,67,53,152),(13,86,54,171),(14,75,55,160),(15,64,56,179),(16,83,57,168),(17,72,58,157),(18,61,59,176),(19,80,60,165),(20,69,31,154),(21,88,32,173),(22,77,33,162),(23,66,34,151),(24,85,35,170),(25,74,36,159),(26,63,37,178),(27,82,38,167),(28,71,39,156),(29,90,40,175),(30,79,41,164),(91,221,203,147),(92,240,204,136),(93,229,205,125),(94,218,206,144),(95,237,207,133),(96,226,208,122),(97,215,209,141),(98,234,210,130),(99,223,181,149),(100,212,182,138),(101,231,183,127),(102,220,184,146),(103,239,185,135),(104,228,186,124),(105,217,187,143),(106,236,188,132),(107,225,189,121),(108,214,190,140),(109,233,191,129),(110,222,192,148),(111,211,193,137),(112,230,194,126),(113,219,195,145),(114,238,196,134),(115,227,197,123),(116,216,198,142),(117,235,199,131),(118,224,200,150),(119,213,201,139),(120,232,202,128)], [(1,57),(2,46),(3,35),(4,54),(5,43),(6,32),(7,51),(8,40),(9,59),(10,48),(11,37),(12,56),(13,45),(14,34),(15,53),(16,42),(17,31),(18,50),(19,39),(20,58),(21,47),(22,36),(23,55),(24,44),(25,33),(26,52),(27,41),(28,60),(29,49),(30,38),(61,85),(62,74),(64,82),(65,71),(66,90),(67,79),(69,87),(70,76),(72,84),(75,81),(77,89),(80,86),(91,100),(92,119),(93,108),(94,97),(95,116),(96,105),(98,113),(99,102),(101,110),(103,118),(104,107),(106,115),(109,112),(111,120),(114,117),(121,213),(122,232),(123,221),(124,240),(125,229),(126,218),(127,237),(128,226),(129,215),(130,234),(131,223),(132,212),(133,231),(134,220),(135,239),(136,228),(137,217),(138,236),(139,225),(140,214),(141,233),(142,222),(143,211),(144,230),(145,219),(146,238),(147,227),(148,216),(149,235),(150,224),(151,175),(152,164),(154,172),(155,161),(156,180),(157,169),(159,177),(160,166),(162,174),(165,171),(167,179),(170,176),(181,184),(182,203),(183,192),(185,200),(186,189),(187,208),(188,197),(190,205),(191,194),(193,202),(195,210),(196,199),(198,207),(201,204),(206,209)]])
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 | 4 | 6 | 6 | 10 | 10 | 30 | 30 | 60 | 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 | D20⋊5S3 | D12⋊5D5 | D10⋊D6 |
| kernel | Dic15.10D4 | Dic3×Dic5 | D10⋊Dic3 | D6⋊Dic5 | C3×D10⋊C4 | C5×D6⋊C4 | C2×C15⋊D4 | C2×Dic30 | 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 Dic15.10D4 ►in GL4(𝔽61) generated by
| 0 | 60 | 0 | 0 |
| 1 | 18 | 0 | 0 |
| 0 | 0 | 14 | 0 |
| 0 | 0 | 2 | 48 |
| 30 | 44 | 0 | 0 |
| 53 | 31 | 0 | 0 |
| 0 | 0 | 32 | 56 |
| 0 | 0 | 22 | 29 |
| 36 | 4 | 0 | 0 |
| 27 | 25 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 50 |
| 60 | 0 | 0 | 0 |
| 18 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 25 | 60 |
G:=sub<GL(4,GF(61))| [0,1,0,0,60,18,0,0,0,0,14,2,0,0,0,48],[30,53,0,0,44,31,0,0,0,0,32,22,0,0,56,29],[36,27,0,0,4,25,0,0,0,0,50,0,0,0,0,50],[60,18,0,0,0,1,0,0,0,0,1,25,0,0,0,60] >;
Dic15.10D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{15}._{10}D_4 % in TeX
G:=Group("Dic15.10D4"); // GroupNames label
G:=SmallGroup(480,538);
// by ID
G=gap.SmallGroup(480,538);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,422,303,58,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=c^4=d^2=1,b^2=a^15,b*a*b^-1=a^-1,c*a*c^-1=d*a*d=a^19,b*c=c*b,d*b*d=a^15*b,d*c*d=a^15*c^-1>;
// generators/relations