metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5.4D12, (C2×Dic3).F5, C5⋊(C12.47D4), C15⋊(C4.10D4), C22.F5.S3, C22.6(S3×F5), C2.15(D6⋊F5), C10.15(D6⋊C4), (C10×Dic3).5C4, (C2×Dic15).6C4, (C3×Dic5).33D4, (C2×Dic5).72D6, C6.15(C22⋊F5), C30.15(C22⋊C4), Dic5.4(C3⋊D4), C3⋊1(Dic5.D4), C15⋊8M4(2).1C2, (C6×Dic5).139C22, (C2×C6).4(C2×F5), (C2×C30).9(C2×C4), (C2×C15⋊Q8).10C2, (C2×C10).11(C4×S3), (C3×C22.F5).1C2, SmallGroup(480,251)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5.4D12
G = < a,b,c,d | a10=1, b2=c12=a5, d2=cbc-1=a5b, bab-1=a-1, cac-1=dad-1=a3, bd=db, dcd-1=a5bc11 >
Subgroups: 404 in 76 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, C8, C2×C4, Q8, C10, C10, Dic3, C12, C2×C6, C15, M4(2), C2×Q8, Dic5, Dic5, C20, C2×C10, C3⋊C8, C24, Dic6, C2×Dic3, C2×Dic3, C2×C12, C30, C30, C4.10D4, C5⋊C8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C4.Dic3, C3×M4(2), C2×Dic6, C5×Dic3, C3×Dic5, Dic15, C2×C30, C22.F5, C22.F5, C2×Dic10, C12.47D4, C3×C5⋊C8, C15⋊C8, C15⋊Q8, C6×Dic5, C10×Dic3, C2×Dic15, Dic5.D4, C3×C22.F5, C15⋊8M4(2), C2×C15⋊Q8, Dic5.4D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, F5, C4×S3, D12, C3⋊D4, C4.10D4, C2×F5, D6⋊C4, C22⋊F5, C12.47D4, S3×F5, Dic5.D4, D6⋊F5, Dic5.4D12
►On 240 points
Generators in S240
(1 123 94 44 225 13 135 82 32 237)(2 45 136 238 95 14 33 124 226 83)(3 239 34 84 137 15 227 46 96 125)(4 85 228 126 35 16 73 240 138 47)(5 127 74 48 229 17 139 86 36 217)(6 25 140 218 75 18 37 128 230 87)(7 219 38 88 141 19 231 26 76 129)(8 89 232 130 39 20 77 220 142 27)(9 131 78 28 233 21 143 90 40 221)(10 29 144 222 79 22 41 132 234 91)(11 223 42 92 121 23 235 30 80 133)(12 93 236 134 43 24 81 224 122 31)(49 186 113 205 146 61 174 101 193 158)(50 206 175 159 114 62 194 187 147 102)(51 160 195 103 176 63 148 207 115 188)(52 104 149 189 196 64 116 161 177 208)(53 190 117 209 150 65 178 105 197 162)(54 210 179 163 118 66 198 191 151 106)(55 164 199 107 180 67 152 211 119 192)(56 108 153 169 200 68 120 165 181 212)(57 170 97 213 154 69 182 109 201 166)(58 214 183 167 98 70 202 171 155 110)(59 168 203 111 184 71 156 215 99 172)(60 112 157 173 204 72 100 145 185 216)
(1 19 13 7)(2 8 14 20)(3 21 15 9)(4 10 16 22)(5 23 17 11)(6 12 18 24)(25 31 37 43)(26 44 38 32)(27 33 39 45)(28 46 40 34)(29 35 41 47)(30 48 42 36)(49 55 61 67)(50 68 62 56)(51 57 63 69)(52 70 64 58)(53 59 65 71)(54 72 66 60)(73 79 85 91)(74 92 86 80)(75 81 87 93)(76 94 88 82)(77 83 89 95)(78 96 90 84)(97 103 109 115)(98 116 110 104)(99 105 111 117)(100 118 112 106)(101 107 113 119)(102 120 114 108)(121 139 133 127)(122 128 134 140)(123 141 135 129)(124 130 136 142)(125 143 137 131)(126 132 138 144)(145 163 157 151)(146 152 158 164)(147 165 159 153)(148 154 160 166)(149 167 161 155)(150 156 162 168)(169 187 181 175)(170 176 182 188)(171 189 183 177)(172 178 184 190)(173 191 185 179)(174 180 186 192)(193 199 205 211)(194 212 206 200)(195 201 207 213)(196 214 208 202)(197 203 209 215)(198 216 210 204)(217 235 229 223)(218 224 230 236)(219 237 231 225)(220 226 232 238)(221 239 233 227)(222 228 234 240)
(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)
(1 184 7 178 13 172 19 190)(2 177 20 183 14 189 8 171)(3 182 9 176 15 170 21 188)(4 175 22 181 16 187 10 169)(5 180 11 174 17 192 23 186)(6 173 24 179 18 185 12 191)(25 60 43 66 37 72 31 54)(26 65 32 59 38 53 44 71)(27 58 45 64 39 70 33 52)(28 63 34 57 40 51 46 69)(29 56 47 62 41 68 35 50)(30 61 36 55 42 49 48 67)(73 114 91 120 85 102 79 108)(74 119 80 113 86 107 92 101)(75 112 93 118 87 100 81 106)(76 117 82 111 88 105 94 99)(77 110 95 116 89 98 83 104)(78 115 84 109 90 103 96 97)(121 146 127 164 133 158 139 152)(122 163 140 145 134 151 128 157)(123 168 129 162 135 156 141 150)(124 161 142 167 136 149 130 155)(125 166 131 160 137 154 143 148)(126 159 144 165 138 147 132 153)(193 217 211 223 205 229 199 235)(194 222 200 240 206 234 212 228)(195 239 213 221 207 227 201 233)(196 220 202 238 208 232 214 226)(197 237 215 219 209 225 203 231)(198 218 204 236 210 230 216 224)
G:=sub<Sym(240)| (1,123,94,44,225,13,135,82,32,237)(2,45,136,238,95,14,33,124,226,83)(3,239,34,84,137,15,227,46,96,125)(4,85,228,126,35,16,73,240,138,47)(5,127,74,48,229,17,139,86,36,217)(6,25,140,218,75,18,37,128,230,87)(7,219,38,88,141,19,231,26,76,129)(8,89,232,130,39,20,77,220,142,27)(9,131,78,28,233,21,143,90,40,221)(10,29,144,222,79,22,41,132,234,91)(11,223,42,92,121,23,235,30,80,133)(12,93,236,134,43,24,81,224,122,31)(49,186,113,205,146,61,174,101,193,158)(50,206,175,159,114,62,194,187,147,102)(51,160,195,103,176,63,148,207,115,188)(52,104,149,189,196,64,116,161,177,208)(53,190,117,209,150,65,178,105,197,162)(54,210,179,163,118,66,198,191,151,106)(55,164,199,107,180,67,152,211,119,192)(56,108,153,169,200,68,120,165,181,212)(57,170,97,213,154,69,182,109,201,166)(58,214,183,167,98,70,202,171,155,110)(59,168,203,111,184,71,156,215,99,172)(60,112,157,173,204,72,100,145,185,216), (1,19,13,7)(2,8,14,20)(3,21,15,9)(4,10,16,22)(5,23,17,11)(6,12,18,24)(25,31,37,43)(26,44,38,32)(27,33,39,45)(28,46,40,34)(29,35,41,47)(30,48,42,36)(49,55,61,67)(50,68,62,56)(51,57,63,69)(52,70,64,58)(53,59,65,71)(54,72,66,60)(73,79,85,91)(74,92,86,80)(75,81,87,93)(76,94,88,82)(77,83,89,95)(78,96,90,84)(97,103,109,115)(98,116,110,104)(99,105,111,117)(100,118,112,106)(101,107,113,119)(102,120,114,108)(121,139,133,127)(122,128,134,140)(123,141,135,129)(124,130,136,142)(125,143,137,131)(126,132,138,144)(145,163,157,151)(146,152,158,164)(147,165,159,153)(148,154,160,166)(149,167,161,155)(150,156,162,168)(169,187,181,175)(170,176,182,188)(171,189,183,177)(172,178,184,190)(173,191,185,179)(174,180,186,192)(193,199,205,211)(194,212,206,200)(195,201,207,213)(196,214,208,202)(197,203,209,215)(198,216,210,204)(217,235,229,223)(218,224,230,236)(219,237,231,225)(220,226,232,238)(221,239,233,227)(222,228,234,240), (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), (1,184,7,178,13,172,19,190)(2,177,20,183,14,189,8,171)(3,182,9,176,15,170,21,188)(4,175,22,181,16,187,10,169)(5,180,11,174,17,192,23,186)(6,173,24,179,18,185,12,191)(25,60,43,66,37,72,31,54)(26,65,32,59,38,53,44,71)(27,58,45,64,39,70,33,52)(28,63,34,57,40,51,46,69)(29,56,47,62,41,68,35,50)(30,61,36,55,42,49,48,67)(73,114,91,120,85,102,79,108)(74,119,80,113,86,107,92,101)(75,112,93,118,87,100,81,106)(76,117,82,111,88,105,94,99)(77,110,95,116,89,98,83,104)(78,115,84,109,90,103,96,97)(121,146,127,164,133,158,139,152)(122,163,140,145,134,151,128,157)(123,168,129,162,135,156,141,150)(124,161,142,167,136,149,130,155)(125,166,131,160,137,154,143,148)(126,159,144,165,138,147,132,153)(193,217,211,223,205,229,199,235)(194,222,200,240,206,234,212,228)(195,239,213,221,207,227,201,233)(196,220,202,238,208,232,214,226)(197,237,215,219,209,225,203,231)(198,218,204,236,210,230,216,224)>;
G:=Group( (1,123,94,44,225,13,135,82,32,237)(2,45,136,238,95,14,33,124,226,83)(3,239,34,84,137,15,227,46,96,125)(4,85,228,126,35,16,73,240,138,47)(5,127,74,48,229,17,139,86,36,217)(6,25,140,218,75,18,37,128,230,87)(7,219,38,88,141,19,231,26,76,129)(8,89,232,130,39,20,77,220,142,27)(9,131,78,28,233,21,143,90,40,221)(10,29,144,222,79,22,41,132,234,91)(11,223,42,92,121,23,235,30,80,133)(12,93,236,134,43,24,81,224,122,31)(49,186,113,205,146,61,174,101,193,158)(50,206,175,159,114,62,194,187,147,102)(51,160,195,103,176,63,148,207,115,188)(52,104,149,189,196,64,116,161,177,208)(53,190,117,209,150,65,178,105,197,162)(54,210,179,163,118,66,198,191,151,106)(55,164,199,107,180,67,152,211,119,192)(56,108,153,169,200,68,120,165,181,212)(57,170,97,213,154,69,182,109,201,166)(58,214,183,167,98,70,202,171,155,110)(59,168,203,111,184,71,156,215,99,172)(60,112,157,173,204,72,100,145,185,216), (1,19,13,7)(2,8,14,20)(3,21,15,9)(4,10,16,22)(5,23,17,11)(6,12,18,24)(25,31,37,43)(26,44,38,32)(27,33,39,45)(28,46,40,34)(29,35,41,47)(30,48,42,36)(49,55,61,67)(50,68,62,56)(51,57,63,69)(52,70,64,58)(53,59,65,71)(54,72,66,60)(73,79,85,91)(74,92,86,80)(75,81,87,93)(76,94,88,82)(77,83,89,95)(78,96,90,84)(97,103,109,115)(98,116,110,104)(99,105,111,117)(100,118,112,106)(101,107,113,119)(102,120,114,108)(121,139,133,127)(122,128,134,140)(123,141,135,129)(124,130,136,142)(125,143,137,131)(126,132,138,144)(145,163,157,151)(146,152,158,164)(147,165,159,153)(148,154,160,166)(149,167,161,155)(150,156,162,168)(169,187,181,175)(170,176,182,188)(171,189,183,177)(172,178,184,190)(173,191,185,179)(174,180,186,192)(193,199,205,211)(194,212,206,200)(195,201,207,213)(196,214,208,202)(197,203,209,215)(198,216,210,204)(217,235,229,223)(218,224,230,236)(219,237,231,225)(220,226,232,238)(221,239,233,227)(222,228,234,240), (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), (1,184,7,178,13,172,19,190)(2,177,20,183,14,189,8,171)(3,182,9,176,15,170,21,188)(4,175,22,181,16,187,10,169)(5,180,11,174,17,192,23,186)(6,173,24,179,18,185,12,191)(25,60,43,66,37,72,31,54)(26,65,32,59,38,53,44,71)(27,58,45,64,39,70,33,52)(28,63,34,57,40,51,46,69)(29,56,47,62,41,68,35,50)(30,61,36,55,42,49,48,67)(73,114,91,120,85,102,79,108)(74,119,80,113,86,107,92,101)(75,112,93,118,87,100,81,106)(76,117,82,111,88,105,94,99)(77,110,95,116,89,98,83,104)(78,115,84,109,90,103,96,97)(121,146,127,164,133,158,139,152)(122,163,140,145,134,151,128,157)(123,168,129,162,135,156,141,150)(124,161,142,167,136,149,130,155)(125,166,131,160,137,154,143,148)(126,159,144,165,138,147,132,153)(193,217,211,223,205,229,199,235)(194,222,200,240,206,234,212,228)(195,239,213,221,207,227,201,233)(196,220,202,238,208,232,214,226)(197,237,215,219,209,225,203,231)(198,218,204,236,210,230,216,224) );
G=PermutationGroup([[(1,123,94,44,225,13,135,82,32,237),(2,45,136,238,95,14,33,124,226,83),(3,239,34,84,137,15,227,46,96,125),(4,85,228,126,35,16,73,240,138,47),(5,127,74,48,229,17,139,86,36,217),(6,25,140,218,75,18,37,128,230,87),(7,219,38,88,141,19,231,26,76,129),(8,89,232,130,39,20,77,220,142,27),(9,131,78,28,233,21,143,90,40,221),(10,29,144,222,79,22,41,132,234,91),(11,223,42,92,121,23,235,30,80,133),(12,93,236,134,43,24,81,224,122,31),(49,186,113,205,146,61,174,101,193,158),(50,206,175,159,114,62,194,187,147,102),(51,160,195,103,176,63,148,207,115,188),(52,104,149,189,196,64,116,161,177,208),(53,190,117,209,150,65,178,105,197,162),(54,210,179,163,118,66,198,191,151,106),(55,164,199,107,180,67,152,211,119,192),(56,108,153,169,200,68,120,165,181,212),(57,170,97,213,154,69,182,109,201,166),(58,214,183,167,98,70,202,171,155,110),(59,168,203,111,184,71,156,215,99,172),(60,112,157,173,204,72,100,145,185,216)], [(1,19,13,7),(2,8,14,20),(3,21,15,9),(4,10,16,22),(5,23,17,11),(6,12,18,24),(25,31,37,43),(26,44,38,32),(27,33,39,45),(28,46,40,34),(29,35,41,47),(30,48,42,36),(49,55,61,67),(50,68,62,56),(51,57,63,69),(52,70,64,58),(53,59,65,71),(54,72,66,60),(73,79,85,91),(74,92,86,80),(75,81,87,93),(76,94,88,82),(77,83,89,95),(78,96,90,84),(97,103,109,115),(98,116,110,104),(99,105,111,117),(100,118,112,106),(101,107,113,119),(102,120,114,108),(121,139,133,127),(122,128,134,140),(123,141,135,129),(124,130,136,142),(125,143,137,131),(126,132,138,144),(145,163,157,151),(146,152,158,164),(147,165,159,153),(148,154,160,166),(149,167,161,155),(150,156,162,168),(169,187,181,175),(170,176,182,188),(171,189,183,177),(172,178,184,190),(173,191,185,179),(174,180,186,192),(193,199,205,211),(194,212,206,200),(195,201,207,213),(196,214,208,202),(197,203,209,215),(198,216,210,204),(217,235,229,223),(218,224,230,236),(219,237,231,225),(220,226,232,238),(221,239,233,227),(222,228,234,240)], [(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)], [(1,184,7,178,13,172,19,190),(2,177,20,183,14,189,8,171),(3,182,9,176,15,170,21,188),(4,175,22,181,16,187,10,169),(5,180,11,174,17,192,23,186),(6,173,24,179,18,185,12,191),(25,60,43,66,37,72,31,54),(26,65,32,59,38,53,44,71),(27,58,45,64,39,70,33,52),(28,63,34,57,40,51,46,69),(29,56,47,62,41,68,35,50),(30,61,36,55,42,49,48,67),(73,114,91,120,85,102,79,108),(74,119,80,113,86,107,92,101),(75,112,93,118,87,100,81,106),(76,117,82,111,88,105,94,99),(77,110,95,116,89,98,83,104),(78,115,84,109,90,103,96,97),(121,146,127,164,133,158,139,152),(122,163,140,145,134,151,128,157),(123,168,129,162,135,156,141,150),(124,161,142,167,136,149,130,155),(125,166,131,160,137,154,143,148),(126,159,144,165,138,147,132,153),(193,217,211,223,205,229,199,235),(194,222,200,240,206,234,212,228),(195,239,213,221,207,227,201,233),(196,220,202,238,208,232,214,226),(197,237,215,219,209,225,203,231),(198,218,204,236,210,230,216,224)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 5 | 6A | 6B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 12A | 12B | 12C | 15 | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30A | 30B | 30C |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 |
| size | 1 | 1 | 2 | 2 | 10 | 10 | 12 | 60 | 4 | 2 | 4 | 20 | 20 | 60 | 60 | 4 | 4 | 4 | 10 | 10 | 20 | 8 | 12 | 12 | 12 | 12 | 20 | 20 | 20 | 20 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - | - | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D6 | D12 | C3⋊D4 | C4×S3 | F5 | C4.10D4 | C2×F5 | C22⋊F5 | C12.47D4 | Dic5.D4 | S3×F5 | D6⋊F5 | Dic5.4D12 |
| kernel | Dic5.4D12 | C3×C22.F5 | C15⋊8M4(2) | C2×C15⋊Q8 | C10×Dic3 | C2×Dic15 | C22.F5 | C3×Dic5 | C2×Dic5 | Dic5 | Dic5 | C2×C10 | C2×Dic3 | C15 | C2×C6 | C6 | C5 | C3 | C22 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 2 |
Matrix representation of Dic5.4D12 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 240 | 0 | 0 |
| 0 | 0 | 191 | 51 | 0 | 0 |
| 0 | 0 | 110 | 177 | 1 | 52 |
| 0 | 0 | 128 | 131 | 189 | 189 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 91 | 106 | 0 | 0 |
| 0 | 0 | 31 | 150 | 0 | 0 |
| 0 | 0 | 81 | 50 | 135 | 228 |
| 0 | 0 | 137 | 14 | 197 | 106 |
| 105 | 120 | 0 | 0 | 0 | 0 |
| 120 | 105 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 1 |
| 0 | 0 | 195 | 64 | 239 | 189 |
| 0 | 0 | 192 | 212 | 177 | 0 |
| 0 | 0 | 42 | 77 | 177 | 0 |
| 105 | 121 | 0 | 0 | 0 | 0 |
| 120 | 136 | 0 | 0 | 0 | 0 |
| 0 | 0 | 54 | 103 | 126 | 126 |
| 0 | 0 | 96 | 76 | 0 | 207 |
| 0 | 0 | 126 | 187 | 214 | 138 |
| 0 | 0 | 49 | 54 | 214 | 138 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,191,110,128,0,0,240,51,177,131,0,0,0,0,1,189,0,0,0,0,52,189],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,91,31,81,137,0,0,106,150,50,14,0,0,0,0,135,197,0,0,0,0,228,106],[105,120,0,0,0,0,120,105,0,0,0,0,0,0,0,195,192,42,0,0,0,64,212,77,0,0,240,239,177,177,0,0,1,189,0,0],[105,120,0,0,0,0,121,136,0,0,0,0,0,0,54,96,126,49,0,0,103,76,187,54,0,0,126,0,214,214,0,0,126,207,138,138] >;
Dic5.4D12 in GAP, Magma, Sage, TeX
{\rm Dic}_5._4D_{12} % in TeX
G:=Group("Dic5.4D12"); // GroupNames label
G:=SmallGroup(480,251);
// by ID
G=gap.SmallGroup(480,251);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,120,219,100,675,1356,9414,4724]);
// Polycyclic
G:=Group<a,b,c,d|a^10=1,b^2=c^12=a^5,d^2=c*b*c^-1=a^5*b,b*a*b^-1=a^-1,c*a*c^-1=d*a*d^-1=a^3,b*d=d*b,d*c*d^-1=a^5*b*c^11>;
// generators/relations