metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic15⋊18D4, (C2×C30)⋊10D4, C6.91(D4×D5), C10.93(S3×D4), C15⋊24(C4⋊D4), D6⋊Dic5⋊37C2, C30.251(C2×D4), C23.54(S3×D5), C3⋊6(Dic5⋊D4), C5⋊6(C23.14D6), C22⋊2(C15⋊D4), Dic15⋊5C4⋊38C2, (C2×Dic5).66D6, (C22×D5).33D6, (C22×C6).47D10, (C22×C10).62D6, C30.159(C4○D4), D10⋊Dic3⋊37C2, C6.86(D4⋊2D5), (C2×C30).213C23, (C2×Dic3).65D10, (C22×S3).31D10, C10.86(D4⋊2S3), C2.43(D10⋊D6), (C22×Dic15)⋊16C2, (C22×C30).75C22, C2.30(C30.C23), (C6×Dic5).123C22, (C10×Dic3).123C22, (C2×Dic15).229C22, (C2×C5⋊D4)⋊8S3, (C2×C3⋊D4)⋊8D5, (C6×C5⋊D4)⋊8C2, (C10×C3⋊D4)⋊8C2, (C2×C6)⋊3(C5⋊D4), C6.96(C2×C5⋊D4), (C2×C15⋊D4)⋊16C2, (C2×C10)⋊7(C3⋊D4), C10.96(C2×C3⋊D4), C2.26(C2×C15⋊D4), (D5×C2×C6).56C22, C22.242(C2×S3×D5), (S3×C2×C10).56C22, (C2×C6).225(C22×D5), (C2×C10).225(C22×S3), SmallGroup(480,647)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic15⋊18D4
G = < a,b,c,d | a30=c4=d2=1, b2=a15, bab-1=a-1, cac-1=dad=a11, cbc-1=dbd=a15b, dcd=c-1 >
Subgroups: 924 in 188 conjugacy classes, 54 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C5×S3, C3×D5, C30, C30, C4⋊D4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C22×C10, Dic3⋊C4, D6⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C6×D4, C5×Dic3, C3×Dic5, Dic15, Dic15, C6×D5, S3×C10, C2×C30, C2×C30, C2×C30, C10.D4, D10⋊C4, C23.D5, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, D4×C10, C23.14D6, C15⋊D4, C6×Dic5, C3×C5⋊D4, C10×Dic3, C5×C3⋊D4, C2×Dic15, C2×Dic15, D5×C2×C6, S3×C2×C10, C22×C30, Dic5⋊D4, D10⋊Dic3, D6⋊Dic5, Dic15⋊5C4, C2×C15⋊D4, C6×C5⋊D4, C10×C3⋊D4, C22×Dic15, Dic15⋊18D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C3⋊D4, C22×S3, C4⋊D4, C5⋊D4, C22×D5, S3×D4, D4⋊2S3, C2×C3⋊D4, S3×D5, D4×D5, D4⋊2D5, C2×C5⋊D4, C23.14D6, C15⋊D4, C2×S3×D5, Dic5⋊D4, C30.C23, C2×C15⋊D4, D10⋊D6, Dic15⋊18D4
►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)
(1 126 16 141)(2 125 17 140)(3 124 18 139)(4 123 19 138)(5 122 20 137)(6 121 21 136)(7 150 22 135)(8 149 23 134)(9 148 24 133)(10 147 25 132)(11 146 26 131)(12 145 27 130)(13 144 28 129)(14 143 29 128)(15 142 30 127)(31 154 46 169)(32 153 47 168)(33 152 48 167)(34 151 49 166)(35 180 50 165)(36 179 51 164)(37 178 52 163)(38 177 53 162)(39 176 54 161)(40 175 55 160)(41 174 56 159)(42 173 57 158)(43 172 58 157)(44 171 59 156)(45 170 60 155)(61 217 76 232)(62 216 77 231)(63 215 78 230)(64 214 79 229)(65 213 80 228)(66 212 81 227)(67 211 82 226)(68 240 83 225)(69 239 84 224)(70 238 85 223)(71 237 86 222)(72 236 87 221)(73 235 88 220)(74 234 89 219)(75 233 90 218)(91 196 106 181)(92 195 107 210)(93 194 108 209)(94 193 109 208)(95 192 110 207)(96 191 111 206)(97 190 112 205)(98 189 113 204)(99 188 114 203)(100 187 115 202)(101 186 116 201)(102 185 117 200)(103 184 118 199)(104 183 119 198)(105 182 120 197)
(1 210 240 170)(2 191 211 151)(3 202 212 162)(4 183 213 173)(5 194 214 154)(6 205 215 165)(7 186 216 176)(8 197 217 157)(9 208 218 168)(10 189 219 179)(11 200 220 160)(12 181 221 171)(13 192 222 152)(14 203 223 163)(15 184 224 174)(16 195 225 155)(17 206 226 166)(18 187 227 177)(19 198 228 158)(20 209 229 169)(21 190 230 180)(22 201 231 161)(23 182 232 172)(24 193 233 153)(25 204 234 164)(26 185 235 175)(27 196 236 156)(28 207 237 167)(29 188 238 178)(30 199 239 159)(31 122 93 79)(32 133 94 90)(33 144 95 71)(34 125 96 82)(35 136 97 63)(36 147 98 74)(37 128 99 85)(38 139 100 66)(39 150 101 77)(40 131 102 88)(41 142 103 69)(42 123 104 80)(43 134 105 61)(44 145 106 72)(45 126 107 83)(46 137 108 64)(47 148 109 75)(48 129 110 86)(49 140 111 67)(50 121 112 78)(51 132 113 89)(52 143 114 70)(53 124 115 81)(54 135 116 62)(55 146 117 73)(56 127 118 84)(57 138 119 65)(58 149 120 76)(59 130 91 87)(60 141 92 68)
(1 170)(2 151)(3 162)(4 173)(5 154)(6 165)(7 176)(8 157)(9 168)(10 179)(11 160)(12 171)(13 152)(14 163)(15 174)(16 155)(17 166)(18 177)(19 158)(20 169)(21 180)(22 161)(23 172)(24 153)(25 164)(26 175)(27 156)(28 167)(29 178)(30 159)(31 122)(32 133)(33 144)(34 125)(35 136)(36 147)(37 128)(38 139)(39 150)(40 131)(41 142)(42 123)(43 134)(44 145)(45 126)(46 137)(47 148)(48 129)(49 140)(50 121)(51 132)(52 143)(53 124)(54 135)(55 146)(56 127)(57 138)(58 149)(59 130)(60 141)(61 105)(62 116)(63 97)(64 108)(65 119)(66 100)(67 111)(68 92)(69 103)(70 114)(71 95)(72 106)(73 117)(74 98)(75 109)(76 120)(77 101)(78 112)(79 93)(80 104)(81 115)(82 96)(83 107)(84 118)(85 99)(86 110)(87 91)(88 102)(89 113)(90 94)(181 221)(182 232)(183 213)(184 224)(185 235)(186 216)(187 227)(188 238)(189 219)(190 230)(191 211)(192 222)(193 233)(194 214)(195 225)(196 236)(197 217)(198 228)(199 239)(200 220)(201 231)(202 212)(203 223)(204 234)(205 215)(206 226)(207 237)(208 218)(209 229)(210 240)
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), (1,126,16,141)(2,125,17,140)(3,124,18,139)(4,123,19,138)(5,122,20,137)(6,121,21,136)(7,150,22,135)(8,149,23,134)(9,148,24,133)(10,147,25,132)(11,146,26,131)(12,145,27,130)(13,144,28,129)(14,143,29,128)(15,142,30,127)(31,154,46,169)(32,153,47,168)(33,152,48,167)(34,151,49,166)(35,180,50,165)(36,179,51,164)(37,178,52,163)(38,177,53,162)(39,176,54,161)(40,175,55,160)(41,174,56,159)(42,173,57,158)(43,172,58,157)(44,171,59,156)(45,170,60,155)(61,217,76,232)(62,216,77,231)(63,215,78,230)(64,214,79,229)(65,213,80,228)(66,212,81,227)(67,211,82,226)(68,240,83,225)(69,239,84,224)(70,238,85,223)(71,237,86,222)(72,236,87,221)(73,235,88,220)(74,234,89,219)(75,233,90,218)(91,196,106,181)(92,195,107,210)(93,194,108,209)(94,193,109,208)(95,192,110,207)(96,191,111,206)(97,190,112,205)(98,189,113,204)(99,188,114,203)(100,187,115,202)(101,186,116,201)(102,185,117,200)(103,184,118,199)(104,183,119,198)(105,182,120,197), (1,210,240,170)(2,191,211,151)(3,202,212,162)(4,183,213,173)(5,194,214,154)(6,205,215,165)(7,186,216,176)(8,197,217,157)(9,208,218,168)(10,189,219,179)(11,200,220,160)(12,181,221,171)(13,192,222,152)(14,203,223,163)(15,184,224,174)(16,195,225,155)(17,206,226,166)(18,187,227,177)(19,198,228,158)(20,209,229,169)(21,190,230,180)(22,201,231,161)(23,182,232,172)(24,193,233,153)(25,204,234,164)(26,185,235,175)(27,196,236,156)(28,207,237,167)(29,188,238,178)(30,199,239,159)(31,122,93,79)(32,133,94,90)(33,144,95,71)(34,125,96,82)(35,136,97,63)(36,147,98,74)(37,128,99,85)(38,139,100,66)(39,150,101,77)(40,131,102,88)(41,142,103,69)(42,123,104,80)(43,134,105,61)(44,145,106,72)(45,126,107,83)(46,137,108,64)(47,148,109,75)(48,129,110,86)(49,140,111,67)(50,121,112,78)(51,132,113,89)(52,143,114,70)(53,124,115,81)(54,135,116,62)(55,146,117,73)(56,127,118,84)(57,138,119,65)(58,149,120,76)(59,130,91,87)(60,141,92,68), (1,170)(2,151)(3,162)(4,173)(5,154)(6,165)(7,176)(8,157)(9,168)(10,179)(11,160)(12,171)(13,152)(14,163)(15,174)(16,155)(17,166)(18,177)(19,158)(20,169)(21,180)(22,161)(23,172)(24,153)(25,164)(26,175)(27,156)(28,167)(29,178)(30,159)(31,122)(32,133)(33,144)(34,125)(35,136)(36,147)(37,128)(38,139)(39,150)(40,131)(41,142)(42,123)(43,134)(44,145)(45,126)(46,137)(47,148)(48,129)(49,140)(50,121)(51,132)(52,143)(53,124)(54,135)(55,146)(56,127)(57,138)(58,149)(59,130)(60,141)(61,105)(62,116)(63,97)(64,108)(65,119)(66,100)(67,111)(68,92)(69,103)(70,114)(71,95)(72,106)(73,117)(74,98)(75,109)(76,120)(77,101)(78,112)(79,93)(80,104)(81,115)(82,96)(83,107)(84,118)(85,99)(86,110)(87,91)(88,102)(89,113)(90,94)(181,221)(182,232)(183,213)(184,224)(185,235)(186,216)(187,227)(188,238)(189,219)(190,230)(191,211)(192,222)(193,233)(194,214)(195,225)(196,236)(197,217)(198,228)(199,239)(200,220)(201,231)(202,212)(203,223)(204,234)(205,215)(206,226)(207,237)(208,218)(209,229)(210,240)>;
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), (1,126,16,141)(2,125,17,140)(3,124,18,139)(4,123,19,138)(5,122,20,137)(6,121,21,136)(7,150,22,135)(8,149,23,134)(9,148,24,133)(10,147,25,132)(11,146,26,131)(12,145,27,130)(13,144,28,129)(14,143,29,128)(15,142,30,127)(31,154,46,169)(32,153,47,168)(33,152,48,167)(34,151,49,166)(35,180,50,165)(36,179,51,164)(37,178,52,163)(38,177,53,162)(39,176,54,161)(40,175,55,160)(41,174,56,159)(42,173,57,158)(43,172,58,157)(44,171,59,156)(45,170,60,155)(61,217,76,232)(62,216,77,231)(63,215,78,230)(64,214,79,229)(65,213,80,228)(66,212,81,227)(67,211,82,226)(68,240,83,225)(69,239,84,224)(70,238,85,223)(71,237,86,222)(72,236,87,221)(73,235,88,220)(74,234,89,219)(75,233,90,218)(91,196,106,181)(92,195,107,210)(93,194,108,209)(94,193,109,208)(95,192,110,207)(96,191,111,206)(97,190,112,205)(98,189,113,204)(99,188,114,203)(100,187,115,202)(101,186,116,201)(102,185,117,200)(103,184,118,199)(104,183,119,198)(105,182,120,197), (1,210,240,170)(2,191,211,151)(3,202,212,162)(4,183,213,173)(5,194,214,154)(6,205,215,165)(7,186,216,176)(8,197,217,157)(9,208,218,168)(10,189,219,179)(11,200,220,160)(12,181,221,171)(13,192,222,152)(14,203,223,163)(15,184,224,174)(16,195,225,155)(17,206,226,166)(18,187,227,177)(19,198,228,158)(20,209,229,169)(21,190,230,180)(22,201,231,161)(23,182,232,172)(24,193,233,153)(25,204,234,164)(26,185,235,175)(27,196,236,156)(28,207,237,167)(29,188,238,178)(30,199,239,159)(31,122,93,79)(32,133,94,90)(33,144,95,71)(34,125,96,82)(35,136,97,63)(36,147,98,74)(37,128,99,85)(38,139,100,66)(39,150,101,77)(40,131,102,88)(41,142,103,69)(42,123,104,80)(43,134,105,61)(44,145,106,72)(45,126,107,83)(46,137,108,64)(47,148,109,75)(48,129,110,86)(49,140,111,67)(50,121,112,78)(51,132,113,89)(52,143,114,70)(53,124,115,81)(54,135,116,62)(55,146,117,73)(56,127,118,84)(57,138,119,65)(58,149,120,76)(59,130,91,87)(60,141,92,68), (1,170)(2,151)(3,162)(4,173)(5,154)(6,165)(7,176)(8,157)(9,168)(10,179)(11,160)(12,171)(13,152)(14,163)(15,174)(16,155)(17,166)(18,177)(19,158)(20,169)(21,180)(22,161)(23,172)(24,153)(25,164)(26,175)(27,156)(28,167)(29,178)(30,159)(31,122)(32,133)(33,144)(34,125)(35,136)(36,147)(37,128)(38,139)(39,150)(40,131)(41,142)(42,123)(43,134)(44,145)(45,126)(46,137)(47,148)(48,129)(49,140)(50,121)(51,132)(52,143)(53,124)(54,135)(55,146)(56,127)(57,138)(58,149)(59,130)(60,141)(61,105)(62,116)(63,97)(64,108)(65,119)(66,100)(67,111)(68,92)(69,103)(70,114)(71,95)(72,106)(73,117)(74,98)(75,109)(76,120)(77,101)(78,112)(79,93)(80,104)(81,115)(82,96)(83,107)(84,118)(85,99)(86,110)(87,91)(88,102)(89,113)(90,94)(181,221)(182,232)(183,213)(184,224)(185,235)(186,216)(187,227)(188,238)(189,219)(190,230)(191,211)(192,222)(193,233)(194,214)(195,225)(196,236)(197,217)(198,228)(199,239)(200,220)(201,231)(202,212)(203,223)(204,234)(205,215)(206,226)(207,237)(208,218)(209,229)(210,240) );
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)], [(1,126,16,141),(2,125,17,140),(3,124,18,139),(4,123,19,138),(5,122,20,137),(6,121,21,136),(7,150,22,135),(8,149,23,134),(9,148,24,133),(10,147,25,132),(11,146,26,131),(12,145,27,130),(13,144,28,129),(14,143,29,128),(15,142,30,127),(31,154,46,169),(32,153,47,168),(33,152,48,167),(34,151,49,166),(35,180,50,165),(36,179,51,164),(37,178,52,163),(38,177,53,162),(39,176,54,161),(40,175,55,160),(41,174,56,159),(42,173,57,158),(43,172,58,157),(44,171,59,156),(45,170,60,155),(61,217,76,232),(62,216,77,231),(63,215,78,230),(64,214,79,229),(65,213,80,228),(66,212,81,227),(67,211,82,226),(68,240,83,225),(69,239,84,224),(70,238,85,223),(71,237,86,222),(72,236,87,221),(73,235,88,220),(74,234,89,219),(75,233,90,218),(91,196,106,181),(92,195,107,210),(93,194,108,209),(94,193,109,208),(95,192,110,207),(96,191,111,206),(97,190,112,205),(98,189,113,204),(99,188,114,203),(100,187,115,202),(101,186,116,201),(102,185,117,200),(103,184,118,199),(104,183,119,198),(105,182,120,197)], [(1,210,240,170),(2,191,211,151),(3,202,212,162),(4,183,213,173),(5,194,214,154),(6,205,215,165),(7,186,216,176),(8,197,217,157),(9,208,218,168),(10,189,219,179),(11,200,220,160),(12,181,221,171),(13,192,222,152),(14,203,223,163),(15,184,224,174),(16,195,225,155),(17,206,226,166),(18,187,227,177),(19,198,228,158),(20,209,229,169),(21,190,230,180),(22,201,231,161),(23,182,232,172),(24,193,233,153),(25,204,234,164),(26,185,235,175),(27,196,236,156),(28,207,237,167),(29,188,238,178),(30,199,239,159),(31,122,93,79),(32,133,94,90),(33,144,95,71),(34,125,96,82),(35,136,97,63),(36,147,98,74),(37,128,99,85),(38,139,100,66),(39,150,101,77),(40,131,102,88),(41,142,103,69),(42,123,104,80),(43,134,105,61),(44,145,106,72),(45,126,107,83),(46,137,108,64),(47,148,109,75),(48,129,110,86),(49,140,111,67),(50,121,112,78),(51,132,113,89),(52,143,114,70),(53,124,115,81),(54,135,116,62),(55,146,117,73),(56,127,118,84),(57,138,119,65),(58,149,120,76),(59,130,91,87),(60,141,92,68)], [(1,170),(2,151),(3,162),(4,173),(5,154),(6,165),(7,176),(8,157),(9,168),(10,179),(11,160),(12,171),(13,152),(14,163),(15,174),(16,155),(17,166),(18,177),(19,158),(20,169),(21,180),(22,161),(23,172),(24,153),(25,164),(26,175),(27,156),(28,167),(29,178),(30,159),(31,122),(32,133),(33,144),(34,125),(35,136),(36,147),(37,128),(38,139),(39,150),(40,131),(41,142),(42,123),(43,134),(44,145),(45,126),(46,137),(47,148),(48,129),(49,140),(50,121),(51,132),(52,143),(53,124),(54,135),(55,146),(56,127),(57,138),(58,149),(59,130),(60,141),(61,105),(62,116),(63,97),(64,108),(65,119),(66,100),(67,111),(68,92),(69,103),(70,114),(71,95),(72,106),(73,117),(74,98),(75,109),(76,120),(77,101),(78,112),(79,93),(80,104),(81,115),(82,96),(83,107),(84,118),(85,99),(86,110),(87,91),(88,102),(89,113),(90,94),(181,221),(182,232),(183,213),(184,224),(185,235),(186,216),(187,227),(188,238),(189,219),(190,230),(191,211),(192,222),(193,233),(194,214),(195,225),(196,236),(197,217),(198,228),(199,239),(200,220),(201,231),(202,212),(203,223),(204,234),(205,215),(206,226),(207,237),(208,218),(209,229),(210,240)]])
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 | 10K | 10L | 10M | 10N | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 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 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 20 | 2 | 12 | 20 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 20 | 20 | 4 | 4 | 12 | 12 | 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 | 2 | 4 | 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 | D10 | C3⋊D4 | C5⋊D4 | S3×D4 | D4⋊2S3 | S3×D5 | D4×D5 | D4⋊2D5 | C15⋊D4 | C2×S3×D5 | C30.C23 | D10⋊D6 |
| kernel | Dic15⋊18D4 | D10⋊Dic3 | D6⋊Dic5 | Dic15⋊5C4 | C2×C15⋊D4 | C6×C5⋊D4 | C10×C3⋊D4 | C22×Dic15 | C2×C5⋊D4 | Dic15 | C2×C30 | C2×C3⋊D4 | C2×Dic5 | C22×D5 | C22×C10 | C30 | C2×Dic3 | C22×S3 | C22×C6 | C2×C10 | C2×C6 | C10 | C10 | C23 | C6 | C6 | C22 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of Dic15⋊18D4 ►in GL8(𝔽61)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 60 | 43 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 47 |
| 8 | 38 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 53 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 56 | 37 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 5 |
| 60 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 48 | 41 | 0 | 0 | 0 | 0 |
| 0 | 0 | 45 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 59 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 | 24 |
| 0 | 0 | 0 | 0 | 0 | 0 | 60 | 56 |
| 60 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 48 | 41 | 0 | 0 | 0 | 0 |
| 0 | 0 | 45 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 59 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 | 24 |
| 0 | 0 | 0 | 0 | 0 | 0 | 60 | 56 |
G:=sub<GL(8,GF(61))| [0,60,0,0,0,0,0,0,1,43,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,13,2,0,0,0,0,0,0,0,47],[8,16,0,0,0,0,0,0,38,53,0,0,0,0,0,0,0,0,46,20,0,0,0,0,0,0,7,15,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,56,1,0,0,0,0,0,0,37,5],[60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,48,45,0,0,0,0,0,0,41,13,0,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,59,1,0,0,0,0,0,0,0,0,5,60,0,0,0,0,0,0,24,56],[60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,48,45,0,0,0,0,0,0,41,13,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,59,1,0,0,0,0,0,0,0,0,5,60,0,0,0,0,0,0,24,56] >;
Dic15⋊18D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{15}\rtimes_{18}D_4 % in TeX
G:=Group("Dic15:18D4"); // GroupNames label
G:=SmallGroup(480,647);
// by ID
G=gap.SmallGroup(480,647);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,422,219,1356,18822]);
// 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^11,c*b*c^-1=d*b*d=a^15*b,d*c*d=c^-1>;
// generators/relations