metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5⋊4D12, C15⋊7(C4×D4), C5⋊3(C4×D12), D6⋊1(C4×D5), D6⋊C4⋊20D5, D30⋊9(C2×C4), C5⋊D12⋊3C4, C2.1(D5×D12), C6.12(D4×D5), Dic5⋊5(C4×S3), (C3×Dic5)⋊6D4, C30.35(C2×D4), (C4×Dic5)⋊15S3, (C2×C20).196D6, C10.12(C2×D12), D30⋊3C4⋊25C2, (C12×Dic5)⋊25C2, (C2×C12).265D10, C3⋊1(Dic5⋊4D4), (C2×C30).95C23, C30.56(C22×C4), C30.Q8⋊15C2, C30.116(C4○D4), C10.69(C4○D12), C6.44(D4⋊2D5), (C2×C60).388C22, (C2×Dic5).174D6, (C2×Dic3).96D10, (C22×S3).34D10, C2.4(Dic3.D10), (C2×Dic15).76C22, (C6×Dic5).199C22, (C10×Dic3).55C22, (C22×D15).30C22, C6.24(C2×C4×D5), C2.26(C4×S3×D5), C10.57(S3×C2×C4), (C2×S3×Dic5)⋊1C2, (S3×C10)⋊8(C2×C4), (C5×D6⋊C4)⋊25C2, C22.50(C2×S3×D5), (C2×D30.C2)⋊3C2, (C2×C4).128(S3×D5), (C2×C5⋊D12).5C2, (C3×Dic5)⋊15(C2×C4), (S3×C2×C10).12C22, (C2×C6).107(C22×D5), (C2×C10).107(C22×S3), SmallGroup(480,481)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5⋊4D12
G = < a,b,c,d | a10=c12=d2=1, b2=a5, bab-1=cac-1=dad=a-1, bc=cb, bd=db, dcd=c-1 >
Subgroups: 1004 in 188 conjugacy classes, 62 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C5×S3, D15, C30, C4×D4, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C4⋊Dic3, D6⋊C4, D6⋊C4, C4×C12, S3×C2×C4, C2×D12, C5×Dic3, C3×Dic5, Dic15, C60, S3×C10, S3×C10, D30, D30, C2×C30, C4×Dic5, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C22×Dic5, C2×C5⋊D4, C4×D12, S3×Dic5, D30.C2, C5⋊D12, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, S3×C2×C10, C22×D15, Dic5⋊4D4, C30.Q8, C12×Dic5, C5×D6⋊C4, D30⋊3C4, C2×S3×Dic5, C2×D30.C2, C2×C5⋊D12, Dic5⋊4D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, D6, C22×C4, C2×D4, C4○D4, D10, C4×S3, D12, C22×S3, C4×D4, C4×D5, C22×D5, S3×C2×C4, C2×D12, C4○D12, S3×D5, C2×C4×D5, D4×D5, D4⋊2D5, C4×D12, C2×S3×D5, Dic5⋊4D4, C4×S3×D5, D5×D12, Dic3.D10, Dic5⋊4D12
►On 240 points
Generators in S240
(1 184 174 152 58 72 94 240 108 13)(2 14 97 229 95 61 59 153 175 185)(3 186 176 154 60 62 96 230 98 15)(4 16 99 231 85 63 49 155 177 187)(5 188 178 156 50 64 86 232 100 17)(6 18 101 233 87 65 51 145 179 189)(7 190 180 146 52 66 88 234 102 19)(8 20 103 235 89 67 53 147 169 191)(9 192 170 148 54 68 90 236 104 21)(10 22 105 237 91 69 55 149 171 181)(11 182 172 150 56 70 92 238 106 23)(12 24 107 239 93 71 57 151 173 183)(25 144 80 159 218 39 201 213 129 115)(26 116 130 214 202 40 219 160 81 133)(27 134 82 161 220 41 203 215 131 117)(28 118 132 216 204 42 221 162 83 135)(29 136 84 163 222 43 193 205 121 119)(30 120 122 206 194 44 223 164 73 137)(31 138 74 165 224 45 195 207 123 109)(32 110 124 208 196 46 225 166 75 139)(33 140 76 167 226 47 197 209 125 111)(34 112 126 210 198 48 227 168 77 141)(35 142 78 157 228 37 199 211 127 113)(36 114 128 212 200 38 217 158 79 143)
(1 202 72 133)(2 203 61 134)(3 204 62 135)(4 193 63 136)(5 194 64 137)(6 195 65 138)(7 196 66 139)(8 197 67 140)(9 198 68 141)(10 199 69 142)(11 200 70 143)(12 201 71 144)(13 40 58 26)(14 41 59 27)(15 42 60 28)(16 43 49 29)(17 44 50 30)(18 45 51 31)(19 46 52 32)(20 47 53 33)(21 48 54 34)(22 37 55 35)(23 38 56 36)(24 39 57 25)(73 188 206 86)(74 189 207 87)(75 190 208 88)(76 191 209 89)(77 192 210 90)(78 181 211 91)(79 182 212 92)(80 183 213 93)(81 184 214 94)(82 185 215 95)(83 186 216 96)(84 187 205 85)(97 220 153 117)(98 221 154 118)(99 222 155 119)(100 223 156 120)(101 224 145 109)(102 225 146 110)(103 226 147 111)(104 227 148 112)(105 228 149 113)(106 217 150 114)(107 218 151 115)(108 219 152 116)(121 231 163 177)(122 232 164 178)(123 233 165 179)(124 234 166 180)(125 235 167 169)(126 236 168 170)(127 237 157 171)(128 238 158 172)(129 239 159 173)(130 240 160 174)(131 229 161 175)(132 230 162 176)
(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 68)(2 67)(3 66)(4 65)(5 64)(6 63)(7 62)(8 61)(9 72)(10 71)(11 70)(12 69)(13 90)(14 89)(15 88)(16 87)(17 86)(18 85)(19 96)(20 95)(21 94)(22 93)(23 92)(24 91)(25 211)(26 210)(27 209)(28 208)(29 207)(30 206)(31 205)(32 216)(33 215)(34 214)(35 213)(36 212)(37 80)(38 79)(39 78)(40 77)(41 76)(42 75)(43 74)(44 73)(45 84)(46 83)(47 82)(48 81)(49 189)(50 188)(51 187)(52 186)(53 185)(54 184)(55 183)(56 182)(57 181)(58 192)(59 191)(60 190)(97 235)(98 234)(99 233)(100 232)(101 231)(102 230)(103 229)(104 240)(105 239)(106 238)(107 237)(108 236)(109 121)(110 132)(111 131)(112 130)(113 129)(114 128)(115 127)(116 126)(117 125)(118 124)(119 123)(120 122)(133 198)(134 197)(135 196)(136 195)(137 194)(138 193)(139 204)(140 203)(141 202)(142 201)(143 200)(144 199)(145 177)(146 176)(147 175)(148 174)(149 173)(150 172)(151 171)(152 170)(153 169)(154 180)(155 179)(156 178)(157 218)(158 217)(159 228)(160 227)(161 226)(162 225)(163 224)(164 223)(165 222)(166 221)(167 220)(168 219)
G:=sub<Sym(240)| (1,184,174,152,58,72,94,240,108,13)(2,14,97,229,95,61,59,153,175,185)(3,186,176,154,60,62,96,230,98,15)(4,16,99,231,85,63,49,155,177,187)(5,188,178,156,50,64,86,232,100,17)(6,18,101,233,87,65,51,145,179,189)(7,190,180,146,52,66,88,234,102,19)(8,20,103,235,89,67,53,147,169,191)(9,192,170,148,54,68,90,236,104,21)(10,22,105,237,91,69,55,149,171,181)(11,182,172,150,56,70,92,238,106,23)(12,24,107,239,93,71,57,151,173,183)(25,144,80,159,218,39,201,213,129,115)(26,116,130,214,202,40,219,160,81,133)(27,134,82,161,220,41,203,215,131,117)(28,118,132,216,204,42,221,162,83,135)(29,136,84,163,222,43,193,205,121,119)(30,120,122,206,194,44,223,164,73,137)(31,138,74,165,224,45,195,207,123,109)(32,110,124,208,196,46,225,166,75,139)(33,140,76,167,226,47,197,209,125,111)(34,112,126,210,198,48,227,168,77,141)(35,142,78,157,228,37,199,211,127,113)(36,114,128,212,200,38,217,158,79,143), (1,202,72,133)(2,203,61,134)(3,204,62,135)(4,193,63,136)(5,194,64,137)(6,195,65,138)(7,196,66,139)(8,197,67,140)(9,198,68,141)(10,199,69,142)(11,200,70,143)(12,201,71,144)(13,40,58,26)(14,41,59,27)(15,42,60,28)(16,43,49,29)(17,44,50,30)(18,45,51,31)(19,46,52,32)(20,47,53,33)(21,48,54,34)(22,37,55,35)(23,38,56,36)(24,39,57,25)(73,188,206,86)(74,189,207,87)(75,190,208,88)(76,191,209,89)(77,192,210,90)(78,181,211,91)(79,182,212,92)(80,183,213,93)(81,184,214,94)(82,185,215,95)(83,186,216,96)(84,187,205,85)(97,220,153,117)(98,221,154,118)(99,222,155,119)(100,223,156,120)(101,224,145,109)(102,225,146,110)(103,226,147,111)(104,227,148,112)(105,228,149,113)(106,217,150,114)(107,218,151,115)(108,219,152,116)(121,231,163,177)(122,232,164,178)(123,233,165,179)(124,234,166,180)(125,235,167,169)(126,236,168,170)(127,237,157,171)(128,238,158,172)(129,239,159,173)(130,240,160,174)(131,229,161,175)(132,230,162,176), (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), (1,68)(2,67)(3,66)(4,65)(5,64)(6,63)(7,62)(8,61)(9,72)(10,71)(11,70)(12,69)(13,90)(14,89)(15,88)(16,87)(17,86)(18,85)(19,96)(20,95)(21,94)(22,93)(23,92)(24,91)(25,211)(26,210)(27,209)(28,208)(29,207)(30,206)(31,205)(32,216)(33,215)(34,214)(35,213)(36,212)(37,80)(38,79)(39,78)(40,77)(41,76)(42,75)(43,74)(44,73)(45,84)(46,83)(47,82)(48,81)(49,189)(50,188)(51,187)(52,186)(53,185)(54,184)(55,183)(56,182)(57,181)(58,192)(59,191)(60,190)(97,235)(98,234)(99,233)(100,232)(101,231)(102,230)(103,229)(104,240)(105,239)(106,238)(107,237)(108,236)(109,121)(110,132)(111,131)(112,130)(113,129)(114,128)(115,127)(116,126)(117,125)(118,124)(119,123)(120,122)(133,198)(134,197)(135,196)(136,195)(137,194)(138,193)(139,204)(140,203)(141,202)(142,201)(143,200)(144,199)(145,177)(146,176)(147,175)(148,174)(149,173)(150,172)(151,171)(152,170)(153,169)(154,180)(155,179)(156,178)(157,218)(158,217)(159,228)(160,227)(161,226)(162,225)(163,224)(164,223)(165,222)(166,221)(167,220)(168,219)>;
G:=Group( (1,184,174,152,58,72,94,240,108,13)(2,14,97,229,95,61,59,153,175,185)(3,186,176,154,60,62,96,230,98,15)(4,16,99,231,85,63,49,155,177,187)(5,188,178,156,50,64,86,232,100,17)(6,18,101,233,87,65,51,145,179,189)(7,190,180,146,52,66,88,234,102,19)(8,20,103,235,89,67,53,147,169,191)(9,192,170,148,54,68,90,236,104,21)(10,22,105,237,91,69,55,149,171,181)(11,182,172,150,56,70,92,238,106,23)(12,24,107,239,93,71,57,151,173,183)(25,144,80,159,218,39,201,213,129,115)(26,116,130,214,202,40,219,160,81,133)(27,134,82,161,220,41,203,215,131,117)(28,118,132,216,204,42,221,162,83,135)(29,136,84,163,222,43,193,205,121,119)(30,120,122,206,194,44,223,164,73,137)(31,138,74,165,224,45,195,207,123,109)(32,110,124,208,196,46,225,166,75,139)(33,140,76,167,226,47,197,209,125,111)(34,112,126,210,198,48,227,168,77,141)(35,142,78,157,228,37,199,211,127,113)(36,114,128,212,200,38,217,158,79,143), (1,202,72,133)(2,203,61,134)(3,204,62,135)(4,193,63,136)(5,194,64,137)(6,195,65,138)(7,196,66,139)(8,197,67,140)(9,198,68,141)(10,199,69,142)(11,200,70,143)(12,201,71,144)(13,40,58,26)(14,41,59,27)(15,42,60,28)(16,43,49,29)(17,44,50,30)(18,45,51,31)(19,46,52,32)(20,47,53,33)(21,48,54,34)(22,37,55,35)(23,38,56,36)(24,39,57,25)(73,188,206,86)(74,189,207,87)(75,190,208,88)(76,191,209,89)(77,192,210,90)(78,181,211,91)(79,182,212,92)(80,183,213,93)(81,184,214,94)(82,185,215,95)(83,186,216,96)(84,187,205,85)(97,220,153,117)(98,221,154,118)(99,222,155,119)(100,223,156,120)(101,224,145,109)(102,225,146,110)(103,226,147,111)(104,227,148,112)(105,228,149,113)(106,217,150,114)(107,218,151,115)(108,219,152,116)(121,231,163,177)(122,232,164,178)(123,233,165,179)(124,234,166,180)(125,235,167,169)(126,236,168,170)(127,237,157,171)(128,238,158,172)(129,239,159,173)(130,240,160,174)(131,229,161,175)(132,230,162,176), (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), (1,68)(2,67)(3,66)(4,65)(5,64)(6,63)(7,62)(8,61)(9,72)(10,71)(11,70)(12,69)(13,90)(14,89)(15,88)(16,87)(17,86)(18,85)(19,96)(20,95)(21,94)(22,93)(23,92)(24,91)(25,211)(26,210)(27,209)(28,208)(29,207)(30,206)(31,205)(32,216)(33,215)(34,214)(35,213)(36,212)(37,80)(38,79)(39,78)(40,77)(41,76)(42,75)(43,74)(44,73)(45,84)(46,83)(47,82)(48,81)(49,189)(50,188)(51,187)(52,186)(53,185)(54,184)(55,183)(56,182)(57,181)(58,192)(59,191)(60,190)(97,235)(98,234)(99,233)(100,232)(101,231)(102,230)(103,229)(104,240)(105,239)(106,238)(107,237)(108,236)(109,121)(110,132)(111,131)(112,130)(113,129)(114,128)(115,127)(116,126)(117,125)(118,124)(119,123)(120,122)(133,198)(134,197)(135,196)(136,195)(137,194)(138,193)(139,204)(140,203)(141,202)(142,201)(143,200)(144,199)(145,177)(146,176)(147,175)(148,174)(149,173)(150,172)(151,171)(152,170)(153,169)(154,180)(155,179)(156,178)(157,218)(158,217)(159,228)(160,227)(161,226)(162,225)(163,224)(164,223)(165,222)(166,221)(167,220)(168,219) );
G=PermutationGroup([[(1,184,174,152,58,72,94,240,108,13),(2,14,97,229,95,61,59,153,175,185),(3,186,176,154,60,62,96,230,98,15),(4,16,99,231,85,63,49,155,177,187),(5,188,178,156,50,64,86,232,100,17),(6,18,101,233,87,65,51,145,179,189),(7,190,180,146,52,66,88,234,102,19),(8,20,103,235,89,67,53,147,169,191),(9,192,170,148,54,68,90,236,104,21),(10,22,105,237,91,69,55,149,171,181),(11,182,172,150,56,70,92,238,106,23),(12,24,107,239,93,71,57,151,173,183),(25,144,80,159,218,39,201,213,129,115),(26,116,130,214,202,40,219,160,81,133),(27,134,82,161,220,41,203,215,131,117),(28,118,132,216,204,42,221,162,83,135),(29,136,84,163,222,43,193,205,121,119),(30,120,122,206,194,44,223,164,73,137),(31,138,74,165,224,45,195,207,123,109),(32,110,124,208,196,46,225,166,75,139),(33,140,76,167,226,47,197,209,125,111),(34,112,126,210,198,48,227,168,77,141),(35,142,78,157,228,37,199,211,127,113),(36,114,128,212,200,38,217,158,79,143)], [(1,202,72,133),(2,203,61,134),(3,204,62,135),(4,193,63,136),(5,194,64,137),(6,195,65,138),(7,196,66,139),(8,197,67,140),(9,198,68,141),(10,199,69,142),(11,200,70,143),(12,201,71,144),(13,40,58,26),(14,41,59,27),(15,42,60,28),(16,43,49,29),(17,44,50,30),(18,45,51,31),(19,46,52,32),(20,47,53,33),(21,48,54,34),(22,37,55,35),(23,38,56,36),(24,39,57,25),(73,188,206,86),(74,189,207,87),(75,190,208,88),(76,191,209,89),(77,192,210,90),(78,181,211,91),(79,182,212,92),(80,183,213,93),(81,184,214,94),(82,185,215,95),(83,186,216,96),(84,187,205,85),(97,220,153,117),(98,221,154,118),(99,222,155,119),(100,223,156,120),(101,224,145,109),(102,225,146,110),(103,226,147,111),(104,227,148,112),(105,228,149,113),(106,217,150,114),(107,218,151,115),(108,219,152,116),(121,231,163,177),(122,232,164,178),(123,233,165,179),(124,234,166,180),(125,235,167,169),(126,236,168,170),(127,237,157,171),(128,238,158,172),(129,239,159,173),(130,240,160,174),(131,229,161,175),(132,230,162,176)], [(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)], [(1,68),(2,67),(3,66),(4,65),(5,64),(6,63),(7,62),(8,61),(9,72),(10,71),(11,70),(12,69),(13,90),(14,89),(15,88),(16,87),(17,86),(18,85),(19,96),(20,95),(21,94),(22,93),(23,92),(24,91),(25,211),(26,210),(27,209),(28,208),(29,207),(30,206),(31,205),(32,216),(33,215),(34,214),(35,213),(36,212),(37,80),(38,79),(39,78),(40,77),(41,76),(42,75),(43,74),(44,73),(45,84),(46,83),(47,82),(48,81),(49,189),(50,188),(51,187),(52,186),(53,185),(54,184),(55,183),(56,182),(57,181),(58,192),(59,191),(60,190),(97,235),(98,234),(99,233),(100,232),(101,231),(102,230),(103,229),(104,240),(105,239),(106,238),(107,237),(108,236),(109,121),(110,132),(111,131),(112,130),(113,129),(114,128),(115,127),(116,126),(117,125),(118,124),(119,123),(120,122),(133,198),(134,197),(135,196),(136,195),(137,194),(138,193),(139,204),(140,203),(141,202),(142,201),(143,200),(144,199),(145,177),(146,176),(147,175),(148,174),(149,173),(150,172),(151,171),(152,170),(153,169),(154,180),(155,179),(156,178),(157,218),(158,217),(159,228),(160,227),(161,226),(162,225),(163,224),(164,223),(165,222),(166,221),(167,220),(168,219)]])
72 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 | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 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 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 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 | 30 | 30 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 6 | 6 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D10 | C4×S3 | D12 | C4×D5 | C4○D12 | S3×D5 | D4×D5 | D4⋊2D5 | C2×S3×D5 | C4×S3×D5 | D5×D12 | Dic3.D10 |
| kernel | Dic5⋊4D12 | C30.Q8 | C12×Dic5 | C5×D6⋊C4 | D30⋊3C4 | C2×S3×Dic5 | C2×D30.C2 | C2×C5⋊D12 | C5⋊D12 | C4×Dic5 | C3×Dic5 | D6⋊C4 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C22×S3 | Dic5 | Dic5 | D6 | C10 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of Dic5⋊4D12 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 60 |
| 0 | 0 | 45 | 17 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 57 | 50 |
| 0 | 0 | 57 | 4 |
| 23 | 38 | 0 | 0 |
| 23 | 46 | 0 | 0 |
| 0 | 0 | 17 | 1 |
| 0 | 0 | 17 | 44 |
| 60 | 60 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 44 | 60 |
| 0 | 0 | 44 | 17 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,1,45,0,0,60,17],[1,0,0,0,0,1,0,0,0,0,57,57,0,0,50,4],[23,23,0,0,38,46,0,0,0,0,17,17,0,0,1,44],[60,0,0,0,60,1,0,0,0,0,44,44,0,0,60,17] >;
Dic5⋊4D12 in GAP, Magma, Sage, TeX
{\rm Dic}_5\rtimes_4D_{12} % in TeX
G:=Group("Dic5:4D12"); // GroupNames label
G:=SmallGroup(480,481);
// by ID
G=gap.SmallGroup(480,481);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,135,142,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^12=d^2=1,b^2=a^5,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations