metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic15⋊9D4, D6⋊4(C4×D5), C15⋊18(C4×D4), D6⋊C4⋊11D5, D10⋊6(C4×S3), C15⋊D4⋊5C4, C6.45(D4×D5), C30.57(C2×D4), C10.46(S3×D4), (C2×C20).199D6, D10⋊C4⋊11S3, Dic15⋊18(C2×C4), (C4×Dic15)⋊12C2, (C2×C12).197D10, C5⋊4(Dic3⋊4D4), C3⋊3(Dic5⋊4D4), C2.4(C20⋊D6), C30.61(C22×C4), Dic15⋊5C4⋊25C2, (C22×D5).56D6, C30.125(C4○D4), C6.73(D4⋊2D5), (C2×C30).132C23, (C2×C60).168C22, (C2×Dic5).116D6, (C22×S3).46D10, C10.72(D4⋊2S3), (C2×Dic3).111D10, C2.4(C30.C23), (C6×Dic5).82C22, (C10×Dic3).83C22, (C2×Dic15).210C22, C2.31(C4×S3×D5), C6.29(C2×C4×D5), (C6×D5)⋊6(C2×C4), C10.62(S3×C2×C4), (C2×S3×Dic5)⋊9C2, (C5×D6⋊C4)⋊11C2, (C2×D5×Dic3)⋊11C2, (S3×C10)⋊13(C2×C4), C22.64(C2×S3×D5), (C2×C4).181(S3×D5), (C2×C15⋊D4).4C2, (D5×C2×C6).26C22, (S3×C2×C10).28C22, (C3×D10⋊C4)⋊11C2, (C2×C6).144(C22×D5), (C2×C10).144(C22×S3), SmallGroup(480,518)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic15⋊9D4
G = < a,b,c,d | a30=c4=d2=1, b2=a15, bab-1=cac-1=a-1, dad=a19, bc=cb, bd=db, dcd=c-1 >
Subgroups: 908 in 188 conjugacy classes, 60 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, D6, D6, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, D10, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C5×S3, C3×D5, C30, C4×D4, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, C6×D5, S3×C10, S3×C10, C2×C30, C4×Dic5, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C22×Dic5, C2×C5⋊D4, Dic3⋊4D4, D5×Dic3, S3×Dic5, C15⋊D4, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, S3×C2×C10, Dic5⋊4D4, Dic15⋊5C4, C3×D10⋊C4, C5×D6⋊C4, C4×Dic15, C2×D5×Dic3, C2×S3×Dic5, C2×C15⋊D4, Dic15⋊9D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, D6, C22×C4, C2×D4, C4○D4, D10, C4×S3, C22×S3, C4×D4, C4×D5, C22×D5, S3×C2×C4, S3×D4, D4⋊2S3, S3×D5, C2×C4×D5, D4×D5, D4⋊2D5, Dic3⋊4D4, C2×S3×D5, Dic5⋊4D4, C4×S3×D5, C20⋊D6, C30.C23, Dic15⋊9D4
►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 93 16 108)(2 92 17 107)(3 91 18 106)(4 120 19 105)(5 119 20 104)(6 118 21 103)(7 117 22 102)(8 116 23 101)(9 115 24 100)(10 114 25 99)(11 113 26 98)(12 112 27 97)(13 111 28 96)(14 110 29 95)(15 109 30 94)(31 189 46 204)(32 188 47 203)(33 187 48 202)(34 186 49 201)(35 185 50 200)(36 184 51 199)(37 183 52 198)(38 182 53 197)(39 181 54 196)(40 210 55 195)(41 209 56 194)(42 208 57 193)(43 207 58 192)(44 206 59 191)(45 205 60 190)(61 226 76 211)(62 225 77 240)(63 224 78 239)(64 223 79 238)(65 222 80 237)(66 221 81 236)(67 220 82 235)(68 219 83 234)(69 218 84 233)(70 217 85 232)(71 216 86 231)(72 215 87 230)(73 214 88 229)(74 213 89 228)(75 212 90 227)(121 160 136 175)(122 159 137 174)(123 158 138 173)(124 157 139 172)(125 156 140 171)(126 155 141 170)(127 154 142 169)(128 153 143 168)(129 152 144 167)(130 151 145 166)(131 180 146 165)(132 179 147 164)(133 178 148 163)(134 177 149 162)(135 176 150 161)
(1 80 164 60)(2 79 165 59)(3 78 166 58)(4 77 167 57)(5 76 168 56)(6 75 169 55)(7 74 170 54)(8 73 171 53)(9 72 172 52)(10 71 173 51)(11 70 174 50)(12 69 175 49)(13 68 176 48)(14 67 177 47)(15 66 178 46)(16 65 179 45)(17 64 180 44)(18 63 151 43)(19 62 152 42)(20 61 153 41)(21 90 154 40)(22 89 155 39)(23 88 156 38)(24 87 157 37)(25 86 158 36)(26 85 159 35)(27 84 160 34)(28 83 161 33)(29 82 162 32)(30 81 163 31)(91 239 130 192)(92 238 131 191)(93 237 132 190)(94 236 133 189)(95 235 134 188)(96 234 135 187)(97 233 136 186)(98 232 137 185)(99 231 138 184)(100 230 139 183)(101 229 140 182)(102 228 141 181)(103 227 142 210)(104 226 143 209)(105 225 144 208)(106 224 145 207)(107 223 146 206)(108 222 147 205)(109 221 148 204)(110 220 149 203)(111 219 150 202)(112 218 121 201)(113 217 122 200)(114 216 123 199)(115 215 124 198)(116 214 125 197)(117 213 126 196)(118 212 127 195)(119 211 128 194)(120 240 129 193)
(1 164)(2 153)(3 172)(4 161)(5 180)(6 169)(7 158)(8 177)(9 166)(10 155)(11 174)(12 163)(13 152)(14 171)(15 160)(16 179)(17 168)(18 157)(19 176)(20 165)(21 154)(22 173)(23 162)(24 151)(25 170)(26 159)(27 178)(28 167)(29 156)(30 175)(31 49)(32 38)(33 57)(34 46)(36 54)(37 43)(39 51)(41 59)(42 48)(44 56)(47 53)(52 58)(61 79)(62 68)(63 87)(64 76)(66 84)(67 73)(69 81)(71 89)(72 78)(74 86)(77 83)(82 88)(91 124)(92 143)(93 132)(94 121)(95 140)(96 129)(97 148)(98 137)(99 126)(100 145)(101 134)(102 123)(103 142)(104 131)(105 150)(106 139)(107 128)(108 147)(109 136)(110 125)(111 144)(112 133)(113 122)(114 141)(115 130)(116 149)(117 138)(118 127)(119 146)(120 135)(181 199)(182 188)(183 207)(184 196)(186 204)(187 193)(189 201)(191 209)(192 198)(194 206)(197 203)(202 208)(211 223)(213 231)(214 220)(215 239)(216 228)(218 236)(219 225)(221 233)(224 230)(226 238)(229 235)(234 240)
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), (1,93,16,108)(2,92,17,107)(3,91,18,106)(4,120,19,105)(5,119,20,104)(6,118,21,103)(7,117,22,102)(8,116,23,101)(9,115,24,100)(10,114,25,99)(11,113,26,98)(12,112,27,97)(13,111,28,96)(14,110,29,95)(15,109,30,94)(31,189,46,204)(32,188,47,203)(33,187,48,202)(34,186,49,201)(35,185,50,200)(36,184,51,199)(37,183,52,198)(38,182,53,197)(39,181,54,196)(40,210,55,195)(41,209,56,194)(42,208,57,193)(43,207,58,192)(44,206,59,191)(45,205,60,190)(61,226,76,211)(62,225,77,240)(63,224,78,239)(64,223,79,238)(65,222,80,237)(66,221,81,236)(67,220,82,235)(68,219,83,234)(69,218,84,233)(70,217,85,232)(71,216,86,231)(72,215,87,230)(73,214,88,229)(74,213,89,228)(75,212,90,227)(121,160,136,175)(122,159,137,174)(123,158,138,173)(124,157,139,172)(125,156,140,171)(126,155,141,170)(127,154,142,169)(128,153,143,168)(129,152,144,167)(130,151,145,166)(131,180,146,165)(132,179,147,164)(133,178,148,163)(134,177,149,162)(135,176,150,161), (1,80,164,60)(2,79,165,59)(3,78,166,58)(4,77,167,57)(5,76,168,56)(6,75,169,55)(7,74,170,54)(8,73,171,53)(9,72,172,52)(10,71,173,51)(11,70,174,50)(12,69,175,49)(13,68,176,48)(14,67,177,47)(15,66,178,46)(16,65,179,45)(17,64,180,44)(18,63,151,43)(19,62,152,42)(20,61,153,41)(21,90,154,40)(22,89,155,39)(23,88,156,38)(24,87,157,37)(25,86,158,36)(26,85,159,35)(27,84,160,34)(28,83,161,33)(29,82,162,32)(30,81,163,31)(91,239,130,192)(92,238,131,191)(93,237,132,190)(94,236,133,189)(95,235,134,188)(96,234,135,187)(97,233,136,186)(98,232,137,185)(99,231,138,184)(100,230,139,183)(101,229,140,182)(102,228,141,181)(103,227,142,210)(104,226,143,209)(105,225,144,208)(106,224,145,207)(107,223,146,206)(108,222,147,205)(109,221,148,204)(110,220,149,203)(111,219,150,202)(112,218,121,201)(113,217,122,200)(114,216,123,199)(115,215,124,198)(116,214,125,197)(117,213,126,196)(118,212,127,195)(119,211,128,194)(120,240,129,193), (1,164)(2,153)(3,172)(4,161)(5,180)(6,169)(7,158)(8,177)(9,166)(10,155)(11,174)(12,163)(13,152)(14,171)(15,160)(16,179)(17,168)(18,157)(19,176)(20,165)(21,154)(22,173)(23,162)(24,151)(25,170)(26,159)(27,178)(28,167)(29,156)(30,175)(31,49)(32,38)(33,57)(34,46)(36,54)(37,43)(39,51)(41,59)(42,48)(44,56)(47,53)(52,58)(61,79)(62,68)(63,87)(64,76)(66,84)(67,73)(69,81)(71,89)(72,78)(74,86)(77,83)(82,88)(91,124)(92,143)(93,132)(94,121)(95,140)(96,129)(97,148)(98,137)(99,126)(100,145)(101,134)(102,123)(103,142)(104,131)(105,150)(106,139)(107,128)(108,147)(109,136)(110,125)(111,144)(112,133)(113,122)(114,141)(115,130)(116,149)(117,138)(118,127)(119,146)(120,135)(181,199)(182,188)(183,207)(184,196)(186,204)(187,193)(189,201)(191,209)(192,198)(194,206)(197,203)(202,208)(211,223)(213,231)(214,220)(215,239)(216,228)(218,236)(219,225)(221,233)(224,230)(226,238)(229,235)(234,240)>;
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), (1,93,16,108)(2,92,17,107)(3,91,18,106)(4,120,19,105)(5,119,20,104)(6,118,21,103)(7,117,22,102)(8,116,23,101)(9,115,24,100)(10,114,25,99)(11,113,26,98)(12,112,27,97)(13,111,28,96)(14,110,29,95)(15,109,30,94)(31,189,46,204)(32,188,47,203)(33,187,48,202)(34,186,49,201)(35,185,50,200)(36,184,51,199)(37,183,52,198)(38,182,53,197)(39,181,54,196)(40,210,55,195)(41,209,56,194)(42,208,57,193)(43,207,58,192)(44,206,59,191)(45,205,60,190)(61,226,76,211)(62,225,77,240)(63,224,78,239)(64,223,79,238)(65,222,80,237)(66,221,81,236)(67,220,82,235)(68,219,83,234)(69,218,84,233)(70,217,85,232)(71,216,86,231)(72,215,87,230)(73,214,88,229)(74,213,89,228)(75,212,90,227)(121,160,136,175)(122,159,137,174)(123,158,138,173)(124,157,139,172)(125,156,140,171)(126,155,141,170)(127,154,142,169)(128,153,143,168)(129,152,144,167)(130,151,145,166)(131,180,146,165)(132,179,147,164)(133,178,148,163)(134,177,149,162)(135,176,150,161), (1,80,164,60)(2,79,165,59)(3,78,166,58)(4,77,167,57)(5,76,168,56)(6,75,169,55)(7,74,170,54)(8,73,171,53)(9,72,172,52)(10,71,173,51)(11,70,174,50)(12,69,175,49)(13,68,176,48)(14,67,177,47)(15,66,178,46)(16,65,179,45)(17,64,180,44)(18,63,151,43)(19,62,152,42)(20,61,153,41)(21,90,154,40)(22,89,155,39)(23,88,156,38)(24,87,157,37)(25,86,158,36)(26,85,159,35)(27,84,160,34)(28,83,161,33)(29,82,162,32)(30,81,163,31)(91,239,130,192)(92,238,131,191)(93,237,132,190)(94,236,133,189)(95,235,134,188)(96,234,135,187)(97,233,136,186)(98,232,137,185)(99,231,138,184)(100,230,139,183)(101,229,140,182)(102,228,141,181)(103,227,142,210)(104,226,143,209)(105,225,144,208)(106,224,145,207)(107,223,146,206)(108,222,147,205)(109,221,148,204)(110,220,149,203)(111,219,150,202)(112,218,121,201)(113,217,122,200)(114,216,123,199)(115,215,124,198)(116,214,125,197)(117,213,126,196)(118,212,127,195)(119,211,128,194)(120,240,129,193), (1,164)(2,153)(3,172)(4,161)(5,180)(6,169)(7,158)(8,177)(9,166)(10,155)(11,174)(12,163)(13,152)(14,171)(15,160)(16,179)(17,168)(18,157)(19,176)(20,165)(21,154)(22,173)(23,162)(24,151)(25,170)(26,159)(27,178)(28,167)(29,156)(30,175)(31,49)(32,38)(33,57)(34,46)(36,54)(37,43)(39,51)(41,59)(42,48)(44,56)(47,53)(52,58)(61,79)(62,68)(63,87)(64,76)(66,84)(67,73)(69,81)(71,89)(72,78)(74,86)(77,83)(82,88)(91,124)(92,143)(93,132)(94,121)(95,140)(96,129)(97,148)(98,137)(99,126)(100,145)(101,134)(102,123)(103,142)(104,131)(105,150)(106,139)(107,128)(108,147)(109,136)(110,125)(111,144)(112,133)(113,122)(114,141)(115,130)(116,149)(117,138)(118,127)(119,146)(120,135)(181,199)(182,188)(183,207)(184,196)(186,204)(187,193)(189,201)(191,209)(192,198)(194,206)(197,203)(202,208)(211,223)(213,231)(214,220)(215,239)(216,228)(218,236)(219,225)(221,233)(224,230)(226,238)(229,235)(234,240) );
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)], [(1,93,16,108),(2,92,17,107),(3,91,18,106),(4,120,19,105),(5,119,20,104),(6,118,21,103),(7,117,22,102),(8,116,23,101),(9,115,24,100),(10,114,25,99),(11,113,26,98),(12,112,27,97),(13,111,28,96),(14,110,29,95),(15,109,30,94),(31,189,46,204),(32,188,47,203),(33,187,48,202),(34,186,49,201),(35,185,50,200),(36,184,51,199),(37,183,52,198),(38,182,53,197),(39,181,54,196),(40,210,55,195),(41,209,56,194),(42,208,57,193),(43,207,58,192),(44,206,59,191),(45,205,60,190),(61,226,76,211),(62,225,77,240),(63,224,78,239),(64,223,79,238),(65,222,80,237),(66,221,81,236),(67,220,82,235),(68,219,83,234),(69,218,84,233),(70,217,85,232),(71,216,86,231),(72,215,87,230),(73,214,88,229),(74,213,89,228),(75,212,90,227),(121,160,136,175),(122,159,137,174),(123,158,138,173),(124,157,139,172),(125,156,140,171),(126,155,141,170),(127,154,142,169),(128,153,143,168),(129,152,144,167),(130,151,145,166),(131,180,146,165),(132,179,147,164),(133,178,148,163),(134,177,149,162),(135,176,150,161)], [(1,80,164,60),(2,79,165,59),(3,78,166,58),(4,77,167,57),(5,76,168,56),(6,75,169,55),(7,74,170,54),(8,73,171,53),(9,72,172,52),(10,71,173,51),(11,70,174,50),(12,69,175,49),(13,68,176,48),(14,67,177,47),(15,66,178,46),(16,65,179,45),(17,64,180,44),(18,63,151,43),(19,62,152,42),(20,61,153,41),(21,90,154,40),(22,89,155,39),(23,88,156,38),(24,87,157,37),(25,86,158,36),(26,85,159,35),(27,84,160,34),(28,83,161,33),(29,82,162,32),(30,81,163,31),(91,239,130,192),(92,238,131,191),(93,237,132,190),(94,236,133,189),(95,235,134,188),(96,234,135,187),(97,233,136,186),(98,232,137,185),(99,231,138,184),(100,230,139,183),(101,229,140,182),(102,228,141,181),(103,227,142,210),(104,226,143,209),(105,225,144,208),(106,224,145,207),(107,223,146,206),(108,222,147,205),(109,221,148,204),(110,220,149,203),(111,219,150,202),(112,218,121,201),(113,217,122,200),(114,216,123,199),(115,215,124,198),(116,214,125,197),(117,213,126,196),(118,212,127,195),(119,211,128,194),(120,240,129,193)], [(1,164),(2,153),(3,172),(4,161),(5,180),(6,169),(7,158),(8,177),(9,166),(10,155),(11,174),(12,163),(13,152),(14,171),(15,160),(16,179),(17,168),(18,157),(19,176),(20,165),(21,154),(22,173),(23,162),(24,151),(25,170),(26,159),(27,178),(28,167),(29,156),(30,175),(31,49),(32,38),(33,57),(34,46),(36,54),(37,43),(39,51),(41,59),(42,48),(44,56),(47,53),(52,58),(61,79),(62,68),(63,87),(64,76),(66,84),(67,73),(69,81),(71,89),(72,78),(74,86),(77,83),(82,88),(91,124),(92,143),(93,132),(94,121),(95,140),(96,129),(97,148),(98,137),(99,126),(100,145),(101,134),(102,123),(103,142),(104,131),(105,150),(106,139),(107,128),(108,147),(109,136),(110,125),(111,144),(112,133),(113,122),(114,141),(115,130),(116,149),(117,138),(118,127),(119,146),(120,135),(181,199),(182,188),(183,207),(184,196),(186,204),(187,193),(189,201),(191,209),(192,198),(194,206),(197,203),(202,208),(211,223),(213,231),(214,220),(215,239),(216,228),(218,236),(219,225),(221,233),(224,230),(226,238),(229,235),(234,240)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 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 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 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 | 6 | 6 | 10 | 10 | 2 | 2 | 2 | 6 | 6 | 10 | 10 | 15 | 15 | 15 | 15 | 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 |
66 irreducible representations
| dim | 1 | 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 | C4 | S3 | D4 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D10 | C4×S3 | C4×D5 | S3×D4 | D4⋊2S3 | S3×D5 | D4×D5 | D4⋊2D5 | C2×S3×D5 | C4×S3×D5 | C20⋊D6 | C30.C23 |
| kernel | Dic15⋊9D4 | Dic15⋊5C4 | C3×D10⋊C4 | C5×D6⋊C4 | C4×Dic15 | C2×D5×Dic3 | C2×S3×Dic5 | C2×C15⋊D4 | C15⋊D4 | D10⋊C4 | Dic15 | D6⋊C4 | C2×Dic5 | C2×C20 | C22×D5 | C30 | C2×Dic3 | C2×C12 | C22×S3 | D10 | D6 | C10 | C10 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of Dic15⋊9D4 ►in GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 1 | 0 | 0 |
| 0 | 0 | 42 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 60 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 1 | 0 | 0 |
| 0 | 0 | 43 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 30 |
| 0 | 0 | 0 | 0 | 34 | 57 |
| 40 | 2 | 0 | 0 | 0 | 0 |
| 23 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 1 | 0 | 0 |
| 0 | 0 | 43 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 44 | 25 |
| 0 | 0 | 0 | 0 | 8 | 17 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 21 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 1 | 0 | 0 |
| 0 | 0 | 43 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,42,0,0,0,0,1,18,0,0,0,0,0,0,1,1,0,0,0,0,60,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,43,43,0,0,0,0,1,18,0,0,0,0,0,0,4,34,0,0,0,0,30,57],[40,23,0,0,0,0,2,21,0,0,0,0,0,0,43,43,0,0,0,0,1,18,0,0,0,0,0,0,44,8,0,0,0,0,25,17],[1,21,0,0,0,0,0,60,0,0,0,0,0,0,43,43,0,0,0,0,1,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
Dic15⋊9D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{15}\rtimes_9D_4 % in TeX
G:=Group("Dic15:9D4"); // GroupNames label
G:=SmallGroup(480,518);
// by ID
G=gap.SmallGroup(480,518);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,422,219,58,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=c^4=d^2=1,b^2=a^15,b*a*b^-1=c*a*c^-1=a^-1,d*a*d=a^19,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations