metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic15⋊3D4, (C6×D5)⋊7D4, C10.86(S3×D4), C6.158(D4×D5), C3⋊7(D10⋊D4), D10⋊4(C3⋊D4), C15⋊18(C4⋊D4), C30.230(C2×D4), C6.D4⋊9D5, D30⋊4C4⋊28C2, C23.19(S3×D5), C6.83(C4○D20), C5⋊3(C23.14D6), Dic15⋊5C4⋊33C2, (C2×Dic5).61D6, (C22×D5).63D6, (C22×C6).31D10, (C22×C10).47D6, C30.145(C4○D4), (C2×C30).192C23, (C2×Dic3).60D10, C2.38(D10⋊D6), C10.55(D4⋊2S3), (C22×C30).54C22, C2.28(Dic5.D6), (C6×Dic5).111C22, (C22×D15).62C22, (C2×Dic15).132C22, (C10×Dic3).111C22, (C2×C5⋊D4)⋊3S3, (C6×C5⋊D4)⋊3C2, (C2×D5×Dic3)⋊16C2, C2.40(D5×C3⋊D4), (C2×C3⋊D20)⋊12C2, (C2×C15⋊7D4)⋊13C2, C10.62(C2×C3⋊D4), (D5×C2×C6).50C22, C22.229(C2×S3×D5), (C5×C6.D4)⋊10C2, (C2×C6).204(C22×D5), (C2×C10).204(C22×S3), SmallGroup(480,626)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic15⋊3D4
G = < a,b,c,d | a30=c4=d2=1, b2=a15, bab-1=a-1, cac-1=dad=a19, cbc-1=a15b, bd=db, dcd=c-1 >
Subgroups: 1116 in 188 conjugacy classes, 50 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C3×D5, D15, C30, C30, C4⋊D4, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×D5, C22×C10, Dic3⋊C4, D6⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4, C6×D4, C5×Dic3, C3×Dic5, Dic15, C6×D5, C6×D5, D30, C2×C30, C2×C30, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C2×C5⋊D4, C23.14D6, D5×Dic3, C3⋊D20, C6×Dic5, C3×C5⋊D4, C10×Dic3, C2×Dic15, C15⋊7D4, D5×C2×C6, C22×D15, C22×C30, D10⋊D4, D30⋊4C4, Dic15⋊5C4, C5×C6.D4, C2×D5×Dic3, C2×C3⋊D20, C6×C5⋊D4, C2×C15⋊7D4, Dic15⋊3D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C3⋊D4, C22×S3, C4⋊D4, C22×D5, S3×D4, D4⋊2S3, C2×C3⋊D4, S3×D5, C4○D20, D4×D5, C23.14D6, C2×S3×D5, D10⋊D4, Dic5.D6, D5×C3⋊D4, D10⋊D6, Dic15⋊3D4
►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)
(1 170 16 155)(2 169 17 154)(3 168 18 153)(4 167 19 152)(5 166 20 151)(6 165 21 180)(7 164 22 179)(8 163 23 178)(9 162 24 177)(10 161 25 176)(11 160 26 175)(12 159 27 174)(13 158 28 173)(14 157 29 172)(15 156 30 171)(31 95 46 110)(32 94 47 109)(33 93 48 108)(34 92 49 107)(35 91 50 106)(36 120 51 105)(37 119 52 104)(38 118 53 103)(39 117 54 102)(40 116 55 101)(41 115 56 100)(42 114 57 99)(43 113 58 98)(44 112 59 97)(45 111 60 96)(61 185 76 200)(62 184 77 199)(63 183 78 198)(64 182 79 197)(65 181 80 196)(66 210 81 195)(67 209 82 194)(68 208 83 193)(69 207 84 192)(70 206 85 191)(71 205 86 190)(72 204 87 189)(73 203 88 188)(74 202 89 187)(75 201 90 186)(121 235 136 220)(122 234 137 219)(123 233 138 218)(124 232 139 217)(125 231 140 216)(126 230 141 215)(127 229 142 214)(128 228 143 213)(129 227 144 212)(130 226 145 211)(131 225 146 240)(132 224 147 239)(133 223 148 238)(134 222 149 237)(135 221 150 236)
(1 67 45 122)(2 86 46 141)(3 75 47 130)(4 64 48 149)(5 83 49 138)(6 72 50 127)(7 61 51 146)(8 80 52 135)(9 69 53 124)(10 88 54 143)(11 77 55 132)(12 66 56 121)(13 85 57 140)(14 74 58 129)(15 63 59 148)(16 82 60 137)(17 71 31 126)(18 90 32 145)(19 79 33 134)(20 68 34 123)(21 87 35 142)(22 76 36 131)(23 65 37 150)(24 84 38 139)(25 73 39 128)(26 62 40 147)(27 81 41 136)(28 70 42 125)(29 89 43 144)(30 78 44 133)(91 229 180 204)(92 218 151 193)(93 237 152 182)(94 226 153 201)(95 215 154 190)(96 234 155 209)(97 223 156 198)(98 212 157 187)(99 231 158 206)(100 220 159 195)(101 239 160 184)(102 228 161 203)(103 217 162 192)(104 236 163 181)(105 225 164 200)(106 214 165 189)(107 233 166 208)(108 222 167 197)(109 211 168 186)(110 230 169 205)(111 219 170 194)(112 238 171 183)(113 227 172 202)(114 216 173 191)(115 235 174 210)(116 224 175 199)(117 213 176 188)(118 232 177 207)(119 221 178 196)(120 240 179 185)
(1 60)(2 49)(3 38)(4 57)(5 46)(6 35)(7 54)(8 43)(9 32)(10 51)(11 40)(12 59)(13 48)(14 37)(15 56)(16 45)(17 34)(18 53)(19 42)(20 31)(21 50)(22 39)(23 58)(24 47)(25 36)(26 55)(27 44)(28 33)(29 52)(30 41)(61 88)(62 77)(63 66)(64 85)(65 74)(67 82)(68 71)(69 90)(70 79)(72 87)(73 76)(75 84)(78 81)(80 89)(83 86)(91 165)(92 154)(93 173)(94 162)(95 151)(96 170)(97 159)(98 178)(99 167)(100 156)(101 175)(102 164)(103 153)(104 172)(105 161)(106 180)(107 169)(108 158)(109 177)(110 166)(111 155)(112 174)(113 163)(114 152)(115 171)(116 160)(117 179)(118 168)(119 157)(120 176)(121 148)(122 137)(123 126)(124 145)(125 134)(127 142)(128 131)(129 150)(130 139)(132 147)(133 136)(135 144)(138 141)(140 149)(143 146)(181 202)(182 191)(183 210)(184 199)(185 188)(186 207)(187 196)(189 204)(190 193)(192 201)(194 209)(195 198)(197 206)(200 203)(205 208)(211 232)(212 221)(213 240)(214 229)(215 218)(216 237)(217 226)(219 234)(220 223)(222 231)(224 239)(225 228)(227 236)(230 233)(235 238)
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), (1,170,16,155)(2,169,17,154)(3,168,18,153)(4,167,19,152)(5,166,20,151)(6,165,21,180)(7,164,22,179)(8,163,23,178)(9,162,24,177)(10,161,25,176)(11,160,26,175)(12,159,27,174)(13,158,28,173)(14,157,29,172)(15,156,30,171)(31,95,46,110)(32,94,47,109)(33,93,48,108)(34,92,49,107)(35,91,50,106)(36,120,51,105)(37,119,52,104)(38,118,53,103)(39,117,54,102)(40,116,55,101)(41,115,56,100)(42,114,57,99)(43,113,58,98)(44,112,59,97)(45,111,60,96)(61,185,76,200)(62,184,77,199)(63,183,78,198)(64,182,79,197)(65,181,80,196)(66,210,81,195)(67,209,82,194)(68,208,83,193)(69,207,84,192)(70,206,85,191)(71,205,86,190)(72,204,87,189)(73,203,88,188)(74,202,89,187)(75,201,90,186)(121,235,136,220)(122,234,137,219)(123,233,138,218)(124,232,139,217)(125,231,140,216)(126,230,141,215)(127,229,142,214)(128,228,143,213)(129,227,144,212)(130,226,145,211)(131,225,146,240)(132,224,147,239)(133,223,148,238)(134,222,149,237)(135,221,150,236), (1,67,45,122)(2,86,46,141)(3,75,47,130)(4,64,48,149)(5,83,49,138)(6,72,50,127)(7,61,51,146)(8,80,52,135)(9,69,53,124)(10,88,54,143)(11,77,55,132)(12,66,56,121)(13,85,57,140)(14,74,58,129)(15,63,59,148)(16,82,60,137)(17,71,31,126)(18,90,32,145)(19,79,33,134)(20,68,34,123)(21,87,35,142)(22,76,36,131)(23,65,37,150)(24,84,38,139)(25,73,39,128)(26,62,40,147)(27,81,41,136)(28,70,42,125)(29,89,43,144)(30,78,44,133)(91,229,180,204)(92,218,151,193)(93,237,152,182)(94,226,153,201)(95,215,154,190)(96,234,155,209)(97,223,156,198)(98,212,157,187)(99,231,158,206)(100,220,159,195)(101,239,160,184)(102,228,161,203)(103,217,162,192)(104,236,163,181)(105,225,164,200)(106,214,165,189)(107,233,166,208)(108,222,167,197)(109,211,168,186)(110,230,169,205)(111,219,170,194)(112,238,171,183)(113,227,172,202)(114,216,173,191)(115,235,174,210)(116,224,175,199)(117,213,176,188)(118,232,177,207)(119,221,178,196)(120,240,179,185), (1,60)(2,49)(3,38)(4,57)(5,46)(6,35)(7,54)(8,43)(9,32)(10,51)(11,40)(12,59)(13,48)(14,37)(15,56)(16,45)(17,34)(18,53)(19,42)(20,31)(21,50)(22,39)(23,58)(24,47)(25,36)(26,55)(27,44)(28,33)(29,52)(30,41)(61,88)(62,77)(63,66)(64,85)(65,74)(67,82)(68,71)(69,90)(70,79)(72,87)(73,76)(75,84)(78,81)(80,89)(83,86)(91,165)(92,154)(93,173)(94,162)(95,151)(96,170)(97,159)(98,178)(99,167)(100,156)(101,175)(102,164)(103,153)(104,172)(105,161)(106,180)(107,169)(108,158)(109,177)(110,166)(111,155)(112,174)(113,163)(114,152)(115,171)(116,160)(117,179)(118,168)(119,157)(120,176)(121,148)(122,137)(123,126)(124,145)(125,134)(127,142)(128,131)(129,150)(130,139)(132,147)(133,136)(135,144)(138,141)(140,149)(143,146)(181,202)(182,191)(183,210)(184,199)(185,188)(186,207)(187,196)(189,204)(190,193)(192,201)(194,209)(195,198)(197,206)(200,203)(205,208)(211,232)(212,221)(213,240)(214,229)(215,218)(216,237)(217,226)(219,234)(220,223)(222,231)(224,239)(225,228)(227,236)(230,233)(235,238)>;
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), (1,170,16,155)(2,169,17,154)(3,168,18,153)(4,167,19,152)(5,166,20,151)(6,165,21,180)(7,164,22,179)(8,163,23,178)(9,162,24,177)(10,161,25,176)(11,160,26,175)(12,159,27,174)(13,158,28,173)(14,157,29,172)(15,156,30,171)(31,95,46,110)(32,94,47,109)(33,93,48,108)(34,92,49,107)(35,91,50,106)(36,120,51,105)(37,119,52,104)(38,118,53,103)(39,117,54,102)(40,116,55,101)(41,115,56,100)(42,114,57,99)(43,113,58,98)(44,112,59,97)(45,111,60,96)(61,185,76,200)(62,184,77,199)(63,183,78,198)(64,182,79,197)(65,181,80,196)(66,210,81,195)(67,209,82,194)(68,208,83,193)(69,207,84,192)(70,206,85,191)(71,205,86,190)(72,204,87,189)(73,203,88,188)(74,202,89,187)(75,201,90,186)(121,235,136,220)(122,234,137,219)(123,233,138,218)(124,232,139,217)(125,231,140,216)(126,230,141,215)(127,229,142,214)(128,228,143,213)(129,227,144,212)(130,226,145,211)(131,225,146,240)(132,224,147,239)(133,223,148,238)(134,222,149,237)(135,221,150,236), (1,67,45,122)(2,86,46,141)(3,75,47,130)(4,64,48,149)(5,83,49,138)(6,72,50,127)(7,61,51,146)(8,80,52,135)(9,69,53,124)(10,88,54,143)(11,77,55,132)(12,66,56,121)(13,85,57,140)(14,74,58,129)(15,63,59,148)(16,82,60,137)(17,71,31,126)(18,90,32,145)(19,79,33,134)(20,68,34,123)(21,87,35,142)(22,76,36,131)(23,65,37,150)(24,84,38,139)(25,73,39,128)(26,62,40,147)(27,81,41,136)(28,70,42,125)(29,89,43,144)(30,78,44,133)(91,229,180,204)(92,218,151,193)(93,237,152,182)(94,226,153,201)(95,215,154,190)(96,234,155,209)(97,223,156,198)(98,212,157,187)(99,231,158,206)(100,220,159,195)(101,239,160,184)(102,228,161,203)(103,217,162,192)(104,236,163,181)(105,225,164,200)(106,214,165,189)(107,233,166,208)(108,222,167,197)(109,211,168,186)(110,230,169,205)(111,219,170,194)(112,238,171,183)(113,227,172,202)(114,216,173,191)(115,235,174,210)(116,224,175,199)(117,213,176,188)(118,232,177,207)(119,221,178,196)(120,240,179,185), (1,60)(2,49)(3,38)(4,57)(5,46)(6,35)(7,54)(8,43)(9,32)(10,51)(11,40)(12,59)(13,48)(14,37)(15,56)(16,45)(17,34)(18,53)(19,42)(20,31)(21,50)(22,39)(23,58)(24,47)(25,36)(26,55)(27,44)(28,33)(29,52)(30,41)(61,88)(62,77)(63,66)(64,85)(65,74)(67,82)(68,71)(69,90)(70,79)(72,87)(73,76)(75,84)(78,81)(80,89)(83,86)(91,165)(92,154)(93,173)(94,162)(95,151)(96,170)(97,159)(98,178)(99,167)(100,156)(101,175)(102,164)(103,153)(104,172)(105,161)(106,180)(107,169)(108,158)(109,177)(110,166)(111,155)(112,174)(113,163)(114,152)(115,171)(116,160)(117,179)(118,168)(119,157)(120,176)(121,148)(122,137)(123,126)(124,145)(125,134)(127,142)(128,131)(129,150)(130,139)(132,147)(133,136)(135,144)(138,141)(140,149)(143,146)(181,202)(182,191)(183,210)(184,199)(185,188)(186,207)(187,196)(189,204)(190,193)(192,201)(194,209)(195,198)(197,206)(200,203)(205,208)(211,232)(212,221)(213,240)(214,229)(215,218)(216,237)(217,226)(219,234)(220,223)(222,231)(224,239)(225,228)(227,236)(230,233)(235,238) );
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)], [(1,170,16,155),(2,169,17,154),(3,168,18,153),(4,167,19,152),(5,166,20,151),(6,165,21,180),(7,164,22,179),(8,163,23,178),(9,162,24,177),(10,161,25,176),(11,160,26,175),(12,159,27,174),(13,158,28,173),(14,157,29,172),(15,156,30,171),(31,95,46,110),(32,94,47,109),(33,93,48,108),(34,92,49,107),(35,91,50,106),(36,120,51,105),(37,119,52,104),(38,118,53,103),(39,117,54,102),(40,116,55,101),(41,115,56,100),(42,114,57,99),(43,113,58,98),(44,112,59,97),(45,111,60,96),(61,185,76,200),(62,184,77,199),(63,183,78,198),(64,182,79,197),(65,181,80,196),(66,210,81,195),(67,209,82,194),(68,208,83,193),(69,207,84,192),(70,206,85,191),(71,205,86,190),(72,204,87,189),(73,203,88,188),(74,202,89,187),(75,201,90,186),(121,235,136,220),(122,234,137,219),(123,233,138,218),(124,232,139,217),(125,231,140,216),(126,230,141,215),(127,229,142,214),(128,228,143,213),(129,227,144,212),(130,226,145,211),(131,225,146,240),(132,224,147,239),(133,223,148,238),(134,222,149,237),(135,221,150,236)], [(1,67,45,122),(2,86,46,141),(3,75,47,130),(4,64,48,149),(5,83,49,138),(6,72,50,127),(7,61,51,146),(8,80,52,135),(9,69,53,124),(10,88,54,143),(11,77,55,132),(12,66,56,121),(13,85,57,140),(14,74,58,129),(15,63,59,148),(16,82,60,137),(17,71,31,126),(18,90,32,145),(19,79,33,134),(20,68,34,123),(21,87,35,142),(22,76,36,131),(23,65,37,150),(24,84,38,139),(25,73,39,128),(26,62,40,147),(27,81,41,136),(28,70,42,125),(29,89,43,144),(30,78,44,133),(91,229,180,204),(92,218,151,193),(93,237,152,182),(94,226,153,201),(95,215,154,190),(96,234,155,209),(97,223,156,198),(98,212,157,187),(99,231,158,206),(100,220,159,195),(101,239,160,184),(102,228,161,203),(103,217,162,192),(104,236,163,181),(105,225,164,200),(106,214,165,189),(107,233,166,208),(108,222,167,197),(109,211,168,186),(110,230,169,205),(111,219,170,194),(112,238,171,183),(113,227,172,202),(114,216,173,191),(115,235,174,210),(116,224,175,199),(117,213,176,188),(118,232,177,207),(119,221,178,196),(120,240,179,185)], [(1,60),(2,49),(3,38),(4,57),(5,46),(6,35),(7,54),(8,43),(9,32),(10,51),(11,40),(12,59),(13,48),(14,37),(15,56),(16,45),(17,34),(18,53),(19,42),(20,31),(21,50),(22,39),(23,58),(24,47),(25,36),(26,55),(27,44),(28,33),(29,52),(30,41),(61,88),(62,77),(63,66),(64,85),(65,74),(67,82),(68,71),(69,90),(70,79),(72,87),(73,76),(75,84),(78,81),(80,89),(83,86),(91,165),(92,154),(93,173),(94,162),(95,151),(96,170),(97,159),(98,178),(99,167),(100,156),(101,175),(102,164),(103,153),(104,172),(105,161),(106,180),(107,169),(108,158),(109,177),(110,166),(111,155),(112,174),(113,163),(114,152),(115,171),(116,160),(117,179),(118,168),(119,157),(120,176),(121,148),(122,137),(123,126),(124,145),(125,134),(127,142),(128,131),(129,150),(130,139),(132,147),(133,136),(135,144),(138,141),(140,149),(143,146),(181,202),(182,191),(183,210),(184,199),(185,188),(186,207),(187,196),(189,204),(190,193),(192,201),(194,209),(195,198),(197,206),(200,203),(205,208),(211,232),(212,221),(213,240),(214,229),(215,218),(216,237),(217,226),(219,234),(220,223),(222,231),(224,239),(225,228),(227,236),(230,233),(235,238)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 4 | 10 | 10 | 60 | 2 | 6 | 6 | 12 | 20 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 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 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | C3⋊D4 | C4○D20 | S3×D4 | D4⋊2S3 | S3×D5 | D4×D5 | C2×S3×D5 | Dic5.D6 | D5×C3⋊D4 | D10⋊D6 |
| kernel | Dic15⋊3D4 | D30⋊4C4 | Dic15⋊5C4 | C5×C6.D4 | C2×D5×Dic3 | C2×C3⋊D20 | C6×C5⋊D4 | C2×C15⋊7D4 | C2×C5⋊D4 | Dic15 | C6×D5 | C6.D4 | C2×Dic5 | C22×D5 | C22×C10 | C30 | C2×Dic3 | C22×C6 | D10 | C6 | C10 | C10 | C23 | 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 | 4 | 2 | 4 | 4 | 4 |
Matrix representation of Dic15⋊3D4 ►in GL4(𝔽61) generated by
| 60 | 56 | 0 | 0 |
| 25 | 2 | 0 | 0 |
| 0 | 0 | 17 | 60 |
| 0 | 0 | 45 | 1 |
| 26 | 28 | 0 | 0 |
| 39 | 35 | 0 | 0 |
| 0 | 0 | 4 | 15 |
| 0 | 0 | 7 | 57 |
| 34 | 32 | 0 | 0 |
| 23 | 27 | 0 | 0 |
| 0 | 0 | 39 | 8 |
| 0 | 0 | 8 | 22 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 17 | 18 |
| 0 | 0 | 45 | 44 |
G:=sub<GL(4,GF(61))| [60,25,0,0,56,2,0,0,0,0,17,45,0,0,60,1],[26,39,0,0,28,35,0,0,0,0,4,7,0,0,15,57],[34,23,0,0,32,27,0,0,0,0,39,8,0,0,8,22],[60,0,0,0,0,60,0,0,0,0,17,45,0,0,18,44] >;
Dic15⋊3D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{15}\rtimes_3D_4 % in TeX
G:=Group("Dic15:3D4"); // GroupNames label
G:=SmallGroup(480,626);
// by ID
G=gap.SmallGroup(480,626);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,590,219,1356,18822]);
// 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,c*b*c^-1=a^15*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations