direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5×D12, C15⋊9(C4×D4), C5⋊6(C4×D12), C20⋊5(C4×S3), C3⋊1(D4×Dic5), C60⋊18(C2×C4), (C5×D12)⋊9C4, C4⋊1(S3×Dic5), C2.3(D5×D12), C6.16(D4×D5), D6⋊Dic5⋊8C2, (C3×Dic5)⋊7D4, D6⋊1(C2×Dic5), (C4×Dic5)⋊5S3, C60⋊5C4⋊32C2, C12⋊4(C2×Dic5), C30.40(C2×D4), (C12×Dic5)⋊5C2, (C10×D12).5C2, (C2×D12).10D5, C10.16(C2×D12), (C2×C20).124D6, C30.60(C4○D4), (C2×C12).303D10, C10.56(C4○D12), C2.4(D12⋊5D5), C6.24(D4⋊2D5), (C2×C30).105C23, (C2×C60).147C22, C30.125(C22×C4), (C2×Dic5).176D6, (C22×S3).37D10, C6.11(C22×Dic5), (C6×Dic5).201C22, (C2×Dic15).85C22, (C2×S3×Dic5)⋊3C2, C10.118(S3×C2×C4), (S3×C10)⋊10(C2×C4), C2.12(C2×S3×Dic5), C22.53(C2×S3×D5), (C2×C4).160(S3×D5), (S3×C2×C10).17C22, (C2×C6).117(C22×D5), (C2×C10).117(C22×S3), SmallGroup(480,491)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5×D12
G = < a,b,c,d | a10=c12=d2=1, b2=a5, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 812 in 188 conjugacy classes, 72 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, D4, C23, C10, C10, Dic3, C12, C12, D6, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, Dic5, C20, C2×C10, C2×C10, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C5×S3, C30, C4×D4, C2×Dic5, C2×Dic5, C2×C20, C5×D4, C22×C10, C4⋊Dic3, D6⋊C4, C4×C12, S3×C2×C4, C2×D12, C3×Dic5, C3×Dic5, Dic15, C60, S3×C10, S3×C10, C2×C30, C4×Dic5, C4⋊Dic5, C23.D5, C22×Dic5, D4×C10, C4×D12, S3×Dic5, C6×Dic5, C5×D12, C2×Dic15, C2×C60, S3×C2×C10, D4×Dic5, D6⋊Dic5, C12×Dic5, C60⋊5C4, C2×S3×Dic5, C10×D12, Dic5×D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, D6, C22×C4, C2×D4, C4○D4, Dic5, D10, C4×S3, D12, C22×S3, C4×D4, C2×Dic5, C22×D5, S3×C2×C4, C2×D12, C4○D12, S3×D5, D4×D5, D4⋊2D5, C22×Dic5, C4×D12, S3×Dic5, C2×S3×D5, D4×Dic5, D12⋊5D5, D5×D12, C2×S3×Dic5, Dic5×D12
►On 240 points
Generators in S240
(1 150 128 195 185 144 49 101 46 113)(2 151 129 196 186 133 50 102 47 114)(3 152 130 197 187 134 51 103 48 115)(4 153 131 198 188 135 52 104 37 116)(5 154 132 199 189 136 53 105 38 117)(6 155 121 200 190 137 54 106 39 118)(7 156 122 201 191 138 55 107 40 119)(8 145 123 202 192 139 56 108 41 120)(9 146 124 203 181 140 57 97 42 109)(10 147 125 204 182 141 58 98 43 110)(11 148 126 193 183 142 59 99 44 111)(12 149 127 194 184 143 60 100 45 112)(13 85 33 84 225 229 162 205 177 62)(14 86 34 73 226 230 163 206 178 63)(15 87 35 74 227 231 164 207 179 64)(16 88 36 75 228 232 165 208 180 65)(17 89 25 76 217 233 166 209 169 66)(18 90 26 77 218 234 167 210 170 67)(19 91 27 78 219 235 168 211 171 68)(20 92 28 79 220 236 157 212 172 69)(21 93 29 80 221 237 158 213 173 70)(22 94 30 81 222 238 159 214 174 71)(23 95 31 82 223 239 160 215 175 72)(24 96 32 83 224 240 161 216 176 61)
(1 72 144 223)(2 61 133 224)(3 62 134 225)(4 63 135 226)(5 64 136 227)(6 65 137 228)(7 66 138 217)(8 67 139 218)(9 68 140 219)(10 69 141 220)(11 70 142 221)(12 71 143 222)(13 187 229 115)(14 188 230 116)(15 189 231 117)(16 190 232 118)(17 191 233 119)(18 192 234 120)(19 181 235 109)(20 182 236 110)(21 183 237 111)(22 184 238 112)(23 185 239 113)(24 186 240 114)(25 122 209 107)(26 123 210 108)(27 124 211 97)(28 125 212 98)(29 126 213 99)(30 127 214 100)(31 128 215 101)(32 129 216 102)(33 130 205 103)(34 131 206 104)(35 132 207 105)(36 121 208 106)(37 86 198 163)(38 87 199 164)(39 88 200 165)(40 89 201 166)(41 90 202 167)(42 91 203 168)(43 92 204 157)(44 93 193 158)(45 94 194 159)(46 95 195 160)(47 96 196 161)(48 85 197 162)(49 82 150 175)(50 83 151 176)(51 84 152 177)(52 73 153 178)(53 74 154 179)(54 75 155 180)(55 76 156 169)(56 77 145 170)(57 78 146 171)(58 79 147 172)(59 80 148 173)(60 81 149 174)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 23)(14 22)(15 21)(16 20)(17 19)(25 27)(28 36)(29 35)(30 34)(31 33)(37 45)(38 44)(39 43)(40 42)(46 48)(49 51)(52 60)(53 59)(54 58)(55 57)(62 72)(63 71)(64 70)(65 69)(66 68)(73 81)(74 80)(75 79)(76 78)(82 84)(85 95)(86 94)(87 93)(88 92)(89 91)(97 107)(98 106)(99 105)(100 104)(101 103)(109 119)(110 118)(111 117)(112 116)(113 115)(121 125)(122 124)(126 132)(127 131)(128 130)(134 144)(135 143)(136 142)(137 141)(138 140)(146 156)(147 155)(148 154)(149 153)(150 152)(157 165)(158 164)(159 163)(160 162)(166 168)(169 171)(172 180)(173 179)(174 178)(175 177)(181 191)(182 190)(183 189)(184 188)(185 187)(193 199)(194 198)(195 197)(200 204)(201 203)(205 215)(206 214)(207 213)(208 212)(209 211)(217 219)(220 228)(221 227)(222 226)(223 225)(229 239)(230 238)(231 237)(232 236)(233 235)
G:=sub<Sym(240)| (1,150,128,195,185,144,49,101,46,113)(2,151,129,196,186,133,50,102,47,114)(3,152,130,197,187,134,51,103,48,115)(4,153,131,198,188,135,52,104,37,116)(5,154,132,199,189,136,53,105,38,117)(6,155,121,200,190,137,54,106,39,118)(7,156,122,201,191,138,55,107,40,119)(8,145,123,202,192,139,56,108,41,120)(9,146,124,203,181,140,57,97,42,109)(10,147,125,204,182,141,58,98,43,110)(11,148,126,193,183,142,59,99,44,111)(12,149,127,194,184,143,60,100,45,112)(13,85,33,84,225,229,162,205,177,62)(14,86,34,73,226,230,163,206,178,63)(15,87,35,74,227,231,164,207,179,64)(16,88,36,75,228,232,165,208,180,65)(17,89,25,76,217,233,166,209,169,66)(18,90,26,77,218,234,167,210,170,67)(19,91,27,78,219,235,168,211,171,68)(20,92,28,79,220,236,157,212,172,69)(21,93,29,80,221,237,158,213,173,70)(22,94,30,81,222,238,159,214,174,71)(23,95,31,82,223,239,160,215,175,72)(24,96,32,83,224,240,161,216,176,61), (1,72,144,223)(2,61,133,224)(3,62,134,225)(4,63,135,226)(5,64,136,227)(6,65,137,228)(7,66,138,217)(8,67,139,218)(9,68,140,219)(10,69,141,220)(11,70,142,221)(12,71,143,222)(13,187,229,115)(14,188,230,116)(15,189,231,117)(16,190,232,118)(17,191,233,119)(18,192,234,120)(19,181,235,109)(20,182,236,110)(21,183,237,111)(22,184,238,112)(23,185,239,113)(24,186,240,114)(25,122,209,107)(26,123,210,108)(27,124,211,97)(28,125,212,98)(29,126,213,99)(30,127,214,100)(31,128,215,101)(32,129,216,102)(33,130,205,103)(34,131,206,104)(35,132,207,105)(36,121,208,106)(37,86,198,163)(38,87,199,164)(39,88,200,165)(40,89,201,166)(41,90,202,167)(42,91,203,168)(43,92,204,157)(44,93,193,158)(45,94,194,159)(46,95,195,160)(47,96,196,161)(48,85,197,162)(49,82,150,175)(50,83,151,176)(51,84,152,177)(52,73,153,178)(53,74,154,179)(54,75,155,180)(55,76,156,169)(56,77,145,170)(57,78,146,171)(58,79,147,172)(59,80,148,173)(60,81,149,174), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,23)(14,22)(15,21)(16,20)(17,19)(25,27)(28,36)(29,35)(30,34)(31,33)(37,45)(38,44)(39,43)(40,42)(46,48)(49,51)(52,60)(53,59)(54,58)(55,57)(62,72)(63,71)(64,70)(65,69)(66,68)(73,81)(74,80)(75,79)(76,78)(82,84)(85,95)(86,94)(87,93)(88,92)(89,91)(97,107)(98,106)(99,105)(100,104)(101,103)(109,119)(110,118)(111,117)(112,116)(113,115)(121,125)(122,124)(126,132)(127,131)(128,130)(134,144)(135,143)(136,142)(137,141)(138,140)(146,156)(147,155)(148,154)(149,153)(150,152)(157,165)(158,164)(159,163)(160,162)(166,168)(169,171)(172,180)(173,179)(174,178)(175,177)(181,191)(182,190)(183,189)(184,188)(185,187)(193,199)(194,198)(195,197)(200,204)(201,203)(205,215)(206,214)(207,213)(208,212)(209,211)(217,219)(220,228)(221,227)(222,226)(223,225)(229,239)(230,238)(231,237)(232,236)(233,235)>;
G:=Group( (1,150,128,195,185,144,49,101,46,113)(2,151,129,196,186,133,50,102,47,114)(3,152,130,197,187,134,51,103,48,115)(4,153,131,198,188,135,52,104,37,116)(5,154,132,199,189,136,53,105,38,117)(6,155,121,200,190,137,54,106,39,118)(7,156,122,201,191,138,55,107,40,119)(8,145,123,202,192,139,56,108,41,120)(9,146,124,203,181,140,57,97,42,109)(10,147,125,204,182,141,58,98,43,110)(11,148,126,193,183,142,59,99,44,111)(12,149,127,194,184,143,60,100,45,112)(13,85,33,84,225,229,162,205,177,62)(14,86,34,73,226,230,163,206,178,63)(15,87,35,74,227,231,164,207,179,64)(16,88,36,75,228,232,165,208,180,65)(17,89,25,76,217,233,166,209,169,66)(18,90,26,77,218,234,167,210,170,67)(19,91,27,78,219,235,168,211,171,68)(20,92,28,79,220,236,157,212,172,69)(21,93,29,80,221,237,158,213,173,70)(22,94,30,81,222,238,159,214,174,71)(23,95,31,82,223,239,160,215,175,72)(24,96,32,83,224,240,161,216,176,61), (1,72,144,223)(2,61,133,224)(3,62,134,225)(4,63,135,226)(5,64,136,227)(6,65,137,228)(7,66,138,217)(8,67,139,218)(9,68,140,219)(10,69,141,220)(11,70,142,221)(12,71,143,222)(13,187,229,115)(14,188,230,116)(15,189,231,117)(16,190,232,118)(17,191,233,119)(18,192,234,120)(19,181,235,109)(20,182,236,110)(21,183,237,111)(22,184,238,112)(23,185,239,113)(24,186,240,114)(25,122,209,107)(26,123,210,108)(27,124,211,97)(28,125,212,98)(29,126,213,99)(30,127,214,100)(31,128,215,101)(32,129,216,102)(33,130,205,103)(34,131,206,104)(35,132,207,105)(36,121,208,106)(37,86,198,163)(38,87,199,164)(39,88,200,165)(40,89,201,166)(41,90,202,167)(42,91,203,168)(43,92,204,157)(44,93,193,158)(45,94,194,159)(46,95,195,160)(47,96,196,161)(48,85,197,162)(49,82,150,175)(50,83,151,176)(51,84,152,177)(52,73,153,178)(53,74,154,179)(54,75,155,180)(55,76,156,169)(56,77,145,170)(57,78,146,171)(58,79,147,172)(59,80,148,173)(60,81,149,174), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,23)(14,22)(15,21)(16,20)(17,19)(25,27)(28,36)(29,35)(30,34)(31,33)(37,45)(38,44)(39,43)(40,42)(46,48)(49,51)(52,60)(53,59)(54,58)(55,57)(62,72)(63,71)(64,70)(65,69)(66,68)(73,81)(74,80)(75,79)(76,78)(82,84)(85,95)(86,94)(87,93)(88,92)(89,91)(97,107)(98,106)(99,105)(100,104)(101,103)(109,119)(110,118)(111,117)(112,116)(113,115)(121,125)(122,124)(126,132)(127,131)(128,130)(134,144)(135,143)(136,142)(137,141)(138,140)(146,156)(147,155)(148,154)(149,153)(150,152)(157,165)(158,164)(159,163)(160,162)(166,168)(169,171)(172,180)(173,179)(174,178)(175,177)(181,191)(182,190)(183,189)(184,188)(185,187)(193,199)(194,198)(195,197)(200,204)(201,203)(205,215)(206,214)(207,213)(208,212)(209,211)(217,219)(220,228)(221,227)(222,226)(223,225)(229,239)(230,238)(231,237)(232,236)(233,235) );
G=PermutationGroup([[(1,150,128,195,185,144,49,101,46,113),(2,151,129,196,186,133,50,102,47,114),(3,152,130,197,187,134,51,103,48,115),(4,153,131,198,188,135,52,104,37,116),(5,154,132,199,189,136,53,105,38,117),(6,155,121,200,190,137,54,106,39,118),(7,156,122,201,191,138,55,107,40,119),(8,145,123,202,192,139,56,108,41,120),(9,146,124,203,181,140,57,97,42,109),(10,147,125,204,182,141,58,98,43,110),(11,148,126,193,183,142,59,99,44,111),(12,149,127,194,184,143,60,100,45,112),(13,85,33,84,225,229,162,205,177,62),(14,86,34,73,226,230,163,206,178,63),(15,87,35,74,227,231,164,207,179,64),(16,88,36,75,228,232,165,208,180,65),(17,89,25,76,217,233,166,209,169,66),(18,90,26,77,218,234,167,210,170,67),(19,91,27,78,219,235,168,211,171,68),(20,92,28,79,220,236,157,212,172,69),(21,93,29,80,221,237,158,213,173,70),(22,94,30,81,222,238,159,214,174,71),(23,95,31,82,223,239,160,215,175,72),(24,96,32,83,224,240,161,216,176,61)], [(1,72,144,223),(2,61,133,224),(3,62,134,225),(4,63,135,226),(5,64,136,227),(6,65,137,228),(7,66,138,217),(8,67,139,218),(9,68,140,219),(10,69,141,220),(11,70,142,221),(12,71,143,222),(13,187,229,115),(14,188,230,116),(15,189,231,117),(16,190,232,118),(17,191,233,119),(18,192,234,120),(19,181,235,109),(20,182,236,110),(21,183,237,111),(22,184,238,112),(23,185,239,113),(24,186,240,114),(25,122,209,107),(26,123,210,108),(27,124,211,97),(28,125,212,98),(29,126,213,99),(30,127,214,100),(31,128,215,101),(32,129,216,102),(33,130,205,103),(34,131,206,104),(35,132,207,105),(36,121,208,106),(37,86,198,163),(38,87,199,164),(39,88,200,165),(40,89,201,166),(41,90,202,167),(42,91,203,168),(43,92,204,157),(44,93,193,158),(45,94,194,159),(46,95,195,160),(47,96,196,161),(48,85,197,162),(49,82,150,175),(50,83,151,176),(51,84,152,177),(52,73,153,178),(53,74,154,179),(54,75,155,180),(55,76,156,169),(56,77,145,170),(57,78,146,171),(58,79,147,172),(59,80,148,173),(60,81,149,174)], [(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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,23),(14,22),(15,21),(16,20),(17,19),(25,27),(28,36),(29,35),(30,34),(31,33),(37,45),(38,44),(39,43),(40,42),(46,48),(49,51),(52,60),(53,59),(54,58),(55,57),(62,72),(63,71),(64,70),(65,69),(66,68),(73,81),(74,80),(75,79),(76,78),(82,84),(85,95),(86,94),(87,93),(88,92),(89,91),(97,107),(98,106),(99,105),(100,104),(101,103),(109,119),(110,118),(111,117),(112,116),(113,115),(121,125),(122,124),(126,132),(127,131),(128,130),(134,144),(135,143),(136,142),(137,141),(138,140),(146,156),(147,155),(148,154),(149,153),(150,152),(157,165),(158,164),(159,163),(160,162),(166,168),(169,171),(172,180),(173,179),(174,178),(175,177),(181,191),(182,190),(183,189),(184,188),(185,187),(193,199),(194,198),(195,197),(200,204),(201,203),(205,215),(206,214),(207,213),(208,212),(209,211),(217,219),(220,228),(221,227),(222,226),(223,225),(229,239),(230,238),(231,237),(232,236),(233,235)]])
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 | ··· | 10N | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 15A | 15B | 20A | 20B | 20C | 20D | 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 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | - | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D5 | D6 | D6 | C4○D4 | Dic5 | D10 | D10 | D12 | C4×S3 | C4○D12 | S3×D5 | D4×D5 | D4⋊2D5 | S3×Dic5 | C2×S3×D5 | D12⋊5D5 | D5×D12 |
| kernel | Dic5×D12 | D6⋊Dic5 | C12×Dic5 | C60⋊5C4 | C2×S3×Dic5 | C10×D12 | C5×D12 | C4×Dic5 | C3×Dic5 | C2×D12 | C2×Dic5 | C2×C20 | C30 | D12 | C2×C12 | C22×S3 | Dic5 | C20 | C10 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 1 | 2 | 2 | 2 | 1 | 2 | 8 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of Dic5×D12 ►in GL5(𝔽61)
| 60 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 43 | 1 |
| 0 | 0 | 0 | 60 | 0 |
| 50 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 47 | 6 |
| 0 | 0 | 0 | 59 | 14 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 38 | 23 | 0 | 0 |
| 0 | 38 | 15 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,43,60,0,0,0,1,0],[50,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,47,59,0,0,0,6,14],[1,0,0,0,0,0,38,38,0,0,0,23,15,0,0,0,0,0,60,0,0,0,0,0,60],[60,0,0,0,0,0,1,0,0,0,0,1,60,0,0,0,0,0,1,0,0,0,0,0,1] >;
Dic5×D12 in GAP, Magma, Sage, TeX
{\rm Dic}_5\times D_{12} % in TeX
G:=Group("Dic5xD12"); // GroupNames label
G:=SmallGroup(480,491);
// by ID
G=gap.SmallGroup(480,491);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,219,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^12=d^2=1,b^2=a^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations