metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic15⋊4D4, (S3×C10)⋊6D4, C6.86(D4×D5), D6⋊3(C5⋊D4), C5⋊7(Dic3⋊D4), C23.D5⋊9S3, C15⋊20(C4⋊D4), C10.161(S3×D4), C30.238(C2×D4), D30⋊4C4⋊31C2, C23.22(S3×D5), C3⋊3(Dic5⋊D4), Dic15⋊5C4⋊37C2, (C2×Dic5).63D6, (C22×C6).37D10, (C22×C10).52D6, C10.85(C4○D12), C30.152(C4○D4), C6.58(D4⋊2D5), (C2×C30).200C23, (C2×Dic3).62D10, (C22×S3).54D10, C2.40(D10⋊D6), (C22×C30).62C22, C2.28(Dic3.D10), (C6×Dic5).116C22, (C22×D15).66C22, (C2×Dic15).139C22, (C10×Dic3).116C22, (C2×C3⋊D4)⋊3D5, (C10×C3⋊D4)⋊3C2, (C2×S3×Dic5)⋊16C2, C6.65(C2×C5⋊D4), C2.40(S3×C5⋊D4), (C2×C15⋊7D4)⋊15C2, (C2×C5⋊D12)⋊12C2, C22.232(C2×S3×D5), (S3×C2×C10).51C22, (C3×C23.D5)⋊11C2, (C2×C6).212(C22×D5), (C2×C10).212(C22×S3), SmallGroup(480,634)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic15⋊4D4
G = < a,b,c,d | a30=c4=d2=1, b2=a15, bab-1=a-1, cac-1=dad=a11, cbc-1=a15b, bd=db, dcd=c-1 >
Subgroups: 1068 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, D6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×S3, C22×C6, C5×S3, D15, C30, C30, C4⋊D4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C22×C10, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C2×C3⋊D4, C5×Dic3, C3×Dic5, Dic15, S3×C10, S3×C10, D30, C2×C30, C2×C30, C10.D4, D10⋊C4, C23.D5, C22×Dic5, C2×C5⋊D4, D4×C10, Dic3⋊D4, S3×Dic5, C5⋊D12, C6×Dic5, C10×Dic3, C5×C3⋊D4, C2×Dic15, C15⋊7D4, S3×C2×C10, C22×D15, C22×C30, Dic5⋊D4, D30⋊4C4, Dic15⋊5C4, C3×C23.D5, C2×S3×Dic5, C2×C5⋊D12, C10×C3⋊D4, C2×C15⋊7D4, Dic15⋊4D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C22×S3, C4⋊D4, C5⋊D4, C22×D5, C4○D12, S3×D4, S3×D5, D4×D5, D4⋊2D5, C2×C5⋊D4, Dic3⋊D4, C2×S3×D5, Dic5⋊D4, Dic3.D10, S3×C5⋊D4, D10⋊D6, Dic15⋊4D4
►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)
(1 120 16 105)(2 119 17 104)(3 118 18 103)(4 117 19 102)(5 116 20 101)(6 115 21 100)(7 114 22 99)(8 113 23 98)(9 112 24 97)(10 111 25 96)(11 110 26 95)(12 109 27 94)(13 108 28 93)(14 107 29 92)(15 106 30 91)(31 86 46 71)(32 85 47 70)(33 84 48 69)(34 83 49 68)(35 82 50 67)(36 81 51 66)(37 80 52 65)(38 79 53 64)(39 78 54 63)(40 77 55 62)(41 76 56 61)(42 75 57 90)(43 74 58 89)(44 73 59 88)(45 72 60 87)(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)(151 189 166 204)(152 188 167 203)(153 187 168 202)(154 186 169 201)(155 185 170 200)(156 184 171 199)(157 183 172 198)(158 182 173 197)(159 181 174 196)(160 210 175 195)(161 209 176 194)(162 208 177 193)(163 207 178 192)(164 206 179 191)(165 205 180 190)
(1 76 201 237)(2 87 202 218)(3 68 203 229)(4 79 204 240)(5 90 205 221)(6 71 206 232)(7 82 207 213)(8 63 208 224)(9 74 209 235)(10 85 210 216)(11 66 181 227)(12 77 182 238)(13 88 183 219)(14 69 184 230)(15 80 185 211)(16 61 186 222)(17 72 187 233)(18 83 188 214)(19 64 189 225)(20 75 190 236)(21 86 191 217)(22 67 192 228)(23 78 193 239)(24 89 194 220)(25 70 195 231)(26 81 196 212)(27 62 197 223)(28 73 198 234)(29 84 199 215)(30 65 200 226)(31 164 139 100)(32 175 140 111)(33 156 141 92)(34 167 142 103)(35 178 143 114)(36 159 144 95)(37 170 145 106)(38 151 146 117)(39 162 147 98)(40 173 148 109)(41 154 149 120)(42 165 150 101)(43 176 121 112)(44 157 122 93)(45 168 123 104)(46 179 124 115)(47 160 125 96)(48 171 126 107)(49 152 127 118)(50 163 128 99)(51 174 129 110)(52 155 130 91)(53 166 131 102)(54 177 132 113)(55 158 133 94)(56 169 134 105)(57 180 135 116)(58 161 136 97)(59 172 137 108)(60 153 138 119)
(2 12)(3 23)(5 15)(6 26)(8 18)(9 29)(11 21)(14 24)(17 27)(20 30)(31 129)(32 140)(33 121)(34 132)(35 143)(36 124)(37 135)(38 146)(39 127)(40 138)(41 149)(42 130)(43 141)(44 122)(45 133)(46 144)(47 125)(48 136)(49 147)(50 128)(51 139)(52 150)(53 131)(54 142)(55 123)(56 134)(57 145)(58 126)(59 137)(60 148)(61 222)(62 233)(63 214)(64 225)(65 236)(66 217)(67 228)(68 239)(69 220)(70 231)(71 212)(72 223)(73 234)(74 215)(75 226)(76 237)(77 218)(78 229)(79 240)(80 221)(81 232)(82 213)(83 224)(84 235)(85 216)(86 227)(87 238)(88 219)(89 230)(90 211)(91 101)(92 112)(94 104)(95 115)(97 107)(98 118)(100 110)(103 113)(106 116)(109 119)(152 162)(153 173)(155 165)(156 176)(158 168)(159 179)(161 171)(164 174)(167 177)(170 180)(181 191)(182 202)(184 194)(185 205)(187 197)(188 208)(190 200)(193 203)(196 206)(199 209)
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), (1,120,16,105)(2,119,17,104)(3,118,18,103)(4,117,19,102)(5,116,20,101)(6,115,21,100)(7,114,22,99)(8,113,23,98)(9,112,24,97)(10,111,25,96)(11,110,26,95)(12,109,27,94)(13,108,28,93)(14,107,29,92)(15,106,30,91)(31,86,46,71)(32,85,47,70)(33,84,48,69)(34,83,49,68)(35,82,50,67)(36,81,51,66)(37,80,52,65)(38,79,53,64)(39,78,54,63)(40,77,55,62)(41,76,56,61)(42,75,57,90)(43,74,58,89)(44,73,59,88)(45,72,60,87)(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)(151,189,166,204)(152,188,167,203)(153,187,168,202)(154,186,169,201)(155,185,170,200)(156,184,171,199)(157,183,172,198)(158,182,173,197)(159,181,174,196)(160,210,175,195)(161,209,176,194)(162,208,177,193)(163,207,178,192)(164,206,179,191)(165,205,180,190), (1,76,201,237)(2,87,202,218)(3,68,203,229)(4,79,204,240)(5,90,205,221)(6,71,206,232)(7,82,207,213)(8,63,208,224)(9,74,209,235)(10,85,210,216)(11,66,181,227)(12,77,182,238)(13,88,183,219)(14,69,184,230)(15,80,185,211)(16,61,186,222)(17,72,187,233)(18,83,188,214)(19,64,189,225)(20,75,190,236)(21,86,191,217)(22,67,192,228)(23,78,193,239)(24,89,194,220)(25,70,195,231)(26,81,196,212)(27,62,197,223)(28,73,198,234)(29,84,199,215)(30,65,200,226)(31,164,139,100)(32,175,140,111)(33,156,141,92)(34,167,142,103)(35,178,143,114)(36,159,144,95)(37,170,145,106)(38,151,146,117)(39,162,147,98)(40,173,148,109)(41,154,149,120)(42,165,150,101)(43,176,121,112)(44,157,122,93)(45,168,123,104)(46,179,124,115)(47,160,125,96)(48,171,126,107)(49,152,127,118)(50,163,128,99)(51,174,129,110)(52,155,130,91)(53,166,131,102)(54,177,132,113)(55,158,133,94)(56,169,134,105)(57,180,135,116)(58,161,136,97)(59,172,137,108)(60,153,138,119), (2,12)(3,23)(5,15)(6,26)(8,18)(9,29)(11,21)(14,24)(17,27)(20,30)(31,129)(32,140)(33,121)(34,132)(35,143)(36,124)(37,135)(38,146)(39,127)(40,138)(41,149)(42,130)(43,141)(44,122)(45,133)(46,144)(47,125)(48,136)(49,147)(50,128)(51,139)(52,150)(53,131)(54,142)(55,123)(56,134)(57,145)(58,126)(59,137)(60,148)(61,222)(62,233)(63,214)(64,225)(65,236)(66,217)(67,228)(68,239)(69,220)(70,231)(71,212)(72,223)(73,234)(74,215)(75,226)(76,237)(77,218)(78,229)(79,240)(80,221)(81,232)(82,213)(83,224)(84,235)(85,216)(86,227)(87,238)(88,219)(89,230)(90,211)(91,101)(92,112)(94,104)(95,115)(97,107)(98,118)(100,110)(103,113)(106,116)(109,119)(152,162)(153,173)(155,165)(156,176)(158,168)(159,179)(161,171)(164,174)(167,177)(170,180)(181,191)(182,202)(184,194)(185,205)(187,197)(188,208)(190,200)(193,203)(196,206)(199,209)>;
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), (1,120,16,105)(2,119,17,104)(3,118,18,103)(4,117,19,102)(5,116,20,101)(6,115,21,100)(7,114,22,99)(8,113,23,98)(9,112,24,97)(10,111,25,96)(11,110,26,95)(12,109,27,94)(13,108,28,93)(14,107,29,92)(15,106,30,91)(31,86,46,71)(32,85,47,70)(33,84,48,69)(34,83,49,68)(35,82,50,67)(36,81,51,66)(37,80,52,65)(38,79,53,64)(39,78,54,63)(40,77,55,62)(41,76,56,61)(42,75,57,90)(43,74,58,89)(44,73,59,88)(45,72,60,87)(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)(151,189,166,204)(152,188,167,203)(153,187,168,202)(154,186,169,201)(155,185,170,200)(156,184,171,199)(157,183,172,198)(158,182,173,197)(159,181,174,196)(160,210,175,195)(161,209,176,194)(162,208,177,193)(163,207,178,192)(164,206,179,191)(165,205,180,190), (1,76,201,237)(2,87,202,218)(3,68,203,229)(4,79,204,240)(5,90,205,221)(6,71,206,232)(7,82,207,213)(8,63,208,224)(9,74,209,235)(10,85,210,216)(11,66,181,227)(12,77,182,238)(13,88,183,219)(14,69,184,230)(15,80,185,211)(16,61,186,222)(17,72,187,233)(18,83,188,214)(19,64,189,225)(20,75,190,236)(21,86,191,217)(22,67,192,228)(23,78,193,239)(24,89,194,220)(25,70,195,231)(26,81,196,212)(27,62,197,223)(28,73,198,234)(29,84,199,215)(30,65,200,226)(31,164,139,100)(32,175,140,111)(33,156,141,92)(34,167,142,103)(35,178,143,114)(36,159,144,95)(37,170,145,106)(38,151,146,117)(39,162,147,98)(40,173,148,109)(41,154,149,120)(42,165,150,101)(43,176,121,112)(44,157,122,93)(45,168,123,104)(46,179,124,115)(47,160,125,96)(48,171,126,107)(49,152,127,118)(50,163,128,99)(51,174,129,110)(52,155,130,91)(53,166,131,102)(54,177,132,113)(55,158,133,94)(56,169,134,105)(57,180,135,116)(58,161,136,97)(59,172,137,108)(60,153,138,119), (2,12)(3,23)(5,15)(6,26)(8,18)(9,29)(11,21)(14,24)(17,27)(20,30)(31,129)(32,140)(33,121)(34,132)(35,143)(36,124)(37,135)(38,146)(39,127)(40,138)(41,149)(42,130)(43,141)(44,122)(45,133)(46,144)(47,125)(48,136)(49,147)(50,128)(51,139)(52,150)(53,131)(54,142)(55,123)(56,134)(57,145)(58,126)(59,137)(60,148)(61,222)(62,233)(63,214)(64,225)(65,236)(66,217)(67,228)(68,239)(69,220)(70,231)(71,212)(72,223)(73,234)(74,215)(75,226)(76,237)(77,218)(78,229)(79,240)(80,221)(81,232)(82,213)(83,224)(84,235)(85,216)(86,227)(87,238)(88,219)(89,230)(90,211)(91,101)(92,112)(94,104)(95,115)(97,107)(98,118)(100,110)(103,113)(106,116)(109,119)(152,162)(153,173)(155,165)(156,176)(158,168)(159,179)(161,171)(164,174)(167,177)(170,180)(181,191)(182,202)(184,194)(185,205)(187,197)(188,208)(190,200)(193,203)(196,206)(199,209) );
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)], [(1,120,16,105),(2,119,17,104),(3,118,18,103),(4,117,19,102),(5,116,20,101),(6,115,21,100),(7,114,22,99),(8,113,23,98),(9,112,24,97),(10,111,25,96),(11,110,26,95),(12,109,27,94),(13,108,28,93),(14,107,29,92),(15,106,30,91),(31,86,46,71),(32,85,47,70),(33,84,48,69),(34,83,49,68),(35,82,50,67),(36,81,51,66),(37,80,52,65),(38,79,53,64),(39,78,54,63),(40,77,55,62),(41,76,56,61),(42,75,57,90),(43,74,58,89),(44,73,59,88),(45,72,60,87),(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),(151,189,166,204),(152,188,167,203),(153,187,168,202),(154,186,169,201),(155,185,170,200),(156,184,171,199),(157,183,172,198),(158,182,173,197),(159,181,174,196),(160,210,175,195),(161,209,176,194),(162,208,177,193),(163,207,178,192),(164,206,179,191),(165,205,180,190)], [(1,76,201,237),(2,87,202,218),(3,68,203,229),(4,79,204,240),(5,90,205,221),(6,71,206,232),(7,82,207,213),(8,63,208,224),(9,74,209,235),(10,85,210,216),(11,66,181,227),(12,77,182,238),(13,88,183,219),(14,69,184,230),(15,80,185,211),(16,61,186,222),(17,72,187,233),(18,83,188,214),(19,64,189,225),(20,75,190,236),(21,86,191,217),(22,67,192,228),(23,78,193,239),(24,89,194,220),(25,70,195,231),(26,81,196,212),(27,62,197,223),(28,73,198,234),(29,84,199,215),(30,65,200,226),(31,164,139,100),(32,175,140,111),(33,156,141,92),(34,167,142,103),(35,178,143,114),(36,159,144,95),(37,170,145,106),(38,151,146,117),(39,162,147,98),(40,173,148,109),(41,154,149,120),(42,165,150,101),(43,176,121,112),(44,157,122,93),(45,168,123,104),(46,179,124,115),(47,160,125,96),(48,171,126,107),(49,152,127,118),(50,163,128,99),(51,174,129,110),(52,155,130,91),(53,166,131,102),(54,177,132,113),(55,158,133,94),(56,169,134,105),(57,180,135,116),(58,161,136,97),(59,172,137,108),(60,153,138,119)], [(2,12),(3,23),(5,15),(6,26),(8,18),(9,29),(11,21),(14,24),(17,27),(20,30),(31,129),(32,140),(33,121),(34,132),(35,143),(36,124),(37,135),(38,146),(39,127),(40,138),(41,149),(42,130),(43,141),(44,122),(45,133),(46,144),(47,125),(48,136),(49,147),(50,128),(51,139),(52,150),(53,131),(54,142),(55,123),(56,134),(57,145),(58,126),(59,137),(60,148),(61,222),(62,233),(63,214),(64,225),(65,236),(66,217),(67,228),(68,239),(69,220),(70,231),(71,212),(72,223),(73,234),(74,215),(75,226),(76,237),(77,218),(78,229),(79,240),(80,221),(81,232),(82,213),(83,224),(84,235),(85,216),(86,227),(87,238),(88,219),(89,230),(90,211),(91,101),(92,112),(94,104),(95,115),(97,107),(98,118),(100,110),(103,113),(106,116),(109,119),(152,162),(153,173),(155,165),(156,176),(158,168),(159,179),(161,171),(164,174),(167,177),(170,180),(181,191),(182,202),(184,194),(185,205),(187,197),(188,208),(190,200),(193,203),(196,206),(199,209)]])
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 | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 12A | 12B | 12C | 12D | 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 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 60 | 2 | 10 | 10 | 12 | 20 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 20 | 20 | 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 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | C4○D12 | S3×D4 | S3×D5 | D4×D5 | D4⋊2D5 | C2×S3×D5 | Dic3.D10 | S3×C5⋊D4 | D10⋊D6 |
| kernel | Dic15⋊4D4 | D30⋊4C4 | Dic15⋊5C4 | C3×C23.D5 | C2×S3×Dic5 | C2×C5⋊D12 | C10×C3⋊D4 | C2×C15⋊7D4 | C23.D5 | Dic15 | S3×C10 | C2×C3⋊D4 | C2×Dic5 | C22×C10 | C30 | C2×Dic3 | C22×S3 | C22×C6 | D6 | C10 | C10 | C23 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 8 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of Dic15⋊4D4 ►in GL4(𝔽61) generated by
| 1 | 60 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 1 | 44 |
| 0 | 11 | 0 | 0 |
| 11 | 0 | 0 | 0 |
| 0 | 0 | 36 | 30 |
| 0 | 0 | 32 | 25 |
| 43 | 9 | 0 | 0 |
| 52 | 18 | 0 | 0 |
| 0 | 0 | 47 | 16 |
| 0 | 0 | 45 | 14 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(61))| [1,1,0,0,60,0,0,0,0,0,0,1,0,0,60,44],[0,11,0,0,11,0,0,0,0,0,36,32,0,0,30,25],[43,52,0,0,9,18,0,0,0,0,47,45,0,0,16,14],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;
Dic15⋊4D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{15}\rtimes_4D_4 % in TeX
G:=Group("Dic15:4D4"); // GroupNames label
G:=SmallGroup(480,634);
// by ID
G=gap.SmallGroup(480,634);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,422,303,1356,18822]);
// 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=a^15*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations