metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6.F5, D60.1C4, C5⋊C8.2D6, C4.F5⋊4S3, D15⋊C8⋊3C2, C15⋊4(C8○D4), C60.7(C2×C4), C3⋊2(Q8.F5), C4.14(S3×F5), C20.14(C4×S3), C5⋊1(D12.C4), D10.4(C4×S3), D30.2(C2×C4), (C4×D5).36D6, C3⋊D20.2C4, C12.18(C2×F5), C60.C4⋊3C2, Dic3.F5⋊3C2, Dic3.4(C2×F5), (C5×Dic6).1C4, C6.11(C22×F5), C12.28D10.4C2, C30.11(C22×C4), C15⋊C8.5C22, (D5×C12).36C22, D30.C2.6C22, Dic5.31(C22×S3), (C3×Dic5).29C23, C2.15(C2×S3×F5), C10.11(S3×C2×C4), (C3×C4.F5)⋊4C2, (C3×C5⋊C8).2C22, (C6×D5).20(C2×C4), (C5×Dic3).4(C2×C4), SmallGroup(480,992)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6.F5
G = < a,b,c,d | a12=c5=1, b2=d4=a6, bab-1=a-1, ac=ca, dad-1=a7, bc=cb, bd=db, dcd-1=c3 >
Subgroups: 612 in 124 conjugacy classes, 46 normal (28 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, Q8, D5, C10, Dic3, C12, C12, D6, C2×C6, C15, C2×C8, M4(2), C4○D4, Dic5, C20, C20, D10, D10, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D5, D15, C30, C8○D4, C5⋊C8, C5⋊C8, C4×D5, C4×D5, D20, C5×Q8, S3×C8, C8⋊S3, C2×C3⋊C8, C3×M4(2), C4○D12, C5×Dic3, C3×Dic5, C60, C6×D5, D30, D5⋊C8, C4.F5, C4.F5, Q8⋊2D5, D12.C4, C3×C5⋊C8, C15⋊C8, D30.C2, C3⋊D20, D5×C12, C5×Dic6, D60, Q8.F5, D15⋊C8, Dic3.F5, C3×C4.F5, C60.C4, C12.28D10, Dic6.F5
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, F5, C4×S3, C22×S3, C8○D4, C2×F5, S3×C2×C4, C22×F5, D12.C4, S3×F5, Q8.F5, C2×S3×F5, Dic6.F5
►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 47 7 41)(2 46 8 40)(3 45 9 39)(4 44 10 38)(5 43 11 37)(6 42 12 48)(13 62 19 68)(14 61 20 67)(15 72 21 66)(16 71 22 65)(17 70 23 64)(18 69 24 63)(25 100 31 106)(26 99 32 105)(27 98 33 104)(28 97 34 103)(29 108 35 102)(30 107 36 101)(49 119 55 113)(50 118 56 112)(51 117 57 111)(52 116 58 110)(53 115 59 109)(54 114 60 120)(73 162 79 168)(74 161 80 167)(75 160 81 166)(76 159 82 165)(77 158 83 164)(78 157 84 163)(85 124 91 130)(86 123 92 129)(87 122 93 128)(88 121 94 127)(89 132 95 126)(90 131 96 125)(133 234 139 240)(134 233 140 239)(135 232 141 238)(136 231 142 237)(137 230 143 236)(138 229 144 235)(145 215 151 209)(146 214 152 208)(147 213 153 207)(148 212 154 206)(149 211 155 205)(150 210 156 216)(169 181 175 187)(170 192 176 186)(171 191 177 185)(172 190 178 184)(173 189 179 183)(174 188 180 182)(193 227 199 221)(194 226 200 220)(195 225 201 219)(196 224 202 218)(197 223 203 217)(198 222 204 228)
(1 238 224 207 188)(2 239 225 208 189)(3 240 226 209 190)(4 229 227 210 191)(5 230 228 211 192)(6 231 217 212 181)(7 232 218 213 182)(8 233 219 214 183)(9 234 220 215 184)(10 235 221 216 185)(11 236 222 205 186)(12 237 223 206 187)(13 35 80 125 116)(14 36 81 126 117)(15 25 82 127 118)(16 26 83 128 119)(17 27 84 129 120)(18 28 73 130 109)(19 29 74 131 110)(20 30 75 132 111)(21 31 76 121 112)(22 32 77 122 113)(23 33 78 123 114)(24 34 79 124 115)(37 137 204 149 170)(38 138 193 150 171)(39 139 194 151 172)(40 140 195 152 173)(41 141 196 153 174)(42 142 197 154 175)(43 143 198 155 176)(44 144 199 156 177)(45 133 200 145 178)(46 134 201 146 179)(47 135 202 147 180)(48 136 203 148 169)(49 65 105 158 93)(50 66 106 159 94)(51 67 107 160 95)(52 68 108 161 96)(53 69 97 162 85)(54 70 98 163 86)(55 71 99 164 87)(56 72 100 165 88)(57 61 101 166 89)(58 62 102 167 90)(59 63 103 168 91)(60 64 104 157 92)
(1 96 41 131 7 90 47 125)(2 91 42 126 8 85 48 132)(3 86 43 121 9 92 37 127)(4 93 44 128 10 87 38 122)(5 88 45 123 11 94 39 129)(6 95 46 130 12 89 40 124)(13 188 108 141 19 182 102 135)(14 183 97 136 20 189 103 142)(15 190 98 143 21 184 104 137)(16 185 99 138 22 191 105 144)(17 192 100 133 23 186 106 139)(18 187 101 140 24 181 107 134)(25 240 70 176 31 234 64 170)(26 235 71 171 32 229 65 177)(27 230 72 178 33 236 66 172)(28 237 61 173 34 231 67 179)(29 232 62 180 35 238 68 174)(30 239 63 175 36 233 69 169)(49 199 83 216 55 193 77 210)(50 194 84 211 56 200 78 205)(51 201 73 206 57 195 79 212)(52 196 74 213 58 202 80 207)(53 203 75 208 59 197 81 214)(54 198 76 215 60 204 82 209)(109 223 166 152 115 217 160 146)(110 218 167 147 116 224 161 153)(111 225 168 154 117 219 162 148)(112 220 157 149 118 226 163 155)(113 227 158 156 119 221 164 150)(114 222 159 151 120 228 165 145)
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), (1,47,7,41)(2,46,8,40)(3,45,9,39)(4,44,10,38)(5,43,11,37)(6,42,12,48)(13,62,19,68)(14,61,20,67)(15,72,21,66)(16,71,22,65)(17,70,23,64)(18,69,24,63)(25,100,31,106)(26,99,32,105)(27,98,33,104)(28,97,34,103)(29,108,35,102)(30,107,36,101)(49,119,55,113)(50,118,56,112)(51,117,57,111)(52,116,58,110)(53,115,59,109)(54,114,60,120)(73,162,79,168)(74,161,80,167)(75,160,81,166)(76,159,82,165)(77,158,83,164)(78,157,84,163)(85,124,91,130)(86,123,92,129)(87,122,93,128)(88,121,94,127)(89,132,95,126)(90,131,96,125)(133,234,139,240)(134,233,140,239)(135,232,141,238)(136,231,142,237)(137,230,143,236)(138,229,144,235)(145,215,151,209)(146,214,152,208)(147,213,153,207)(148,212,154,206)(149,211,155,205)(150,210,156,216)(169,181,175,187)(170,192,176,186)(171,191,177,185)(172,190,178,184)(173,189,179,183)(174,188,180,182)(193,227,199,221)(194,226,200,220)(195,225,201,219)(196,224,202,218)(197,223,203,217)(198,222,204,228), (1,238,224,207,188)(2,239,225,208,189)(3,240,226,209,190)(4,229,227,210,191)(5,230,228,211,192)(6,231,217,212,181)(7,232,218,213,182)(8,233,219,214,183)(9,234,220,215,184)(10,235,221,216,185)(11,236,222,205,186)(12,237,223,206,187)(13,35,80,125,116)(14,36,81,126,117)(15,25,82,127,118)(16,26,83,128,119)(17,27,84,129,120)(18,28,73,130,109)(19,29,74,131,110)(20,30,75,132,111)(21,31,76,121,112)(22,32,77,122,113)(23,33,78,123,114)(24,34,79,124,115)(37,137,204,149,170)(38,138,193,150,171)(39,139,194,151,172)(40,140,195,152,173)(41,141,196,153,174)(42,142,197,154,175)(43,143,198,155,176)(44,144,199,156,177)(45,133,200,145,178)(46,134,201,146,179)(47,135,202,147,180)(48,136,203,148,169)(49,65,105,158,93)(50,66,106,159,94)(51,67,107,160,95)(52,68,108,161,96)(53,69,97,162,85)(54,70,98,163,86)(55,71,99,164,87)(56,72,100,165,88)(57,61,101,166,89)(58,62,102,167,90)(59,63,103,168,91)(60,64,104,157,92), (1,96,41,131,7,90,47,125)(2,91,42,126,8,85,48,132)(3,86,43,121,9,92,37,127)(4,93,44,128,10,87,38,122)(5,88,45,123,11,94,39,129)(6,95,46,130,12,89,40,124)(13,188,108,141,19,182,102,135)(14,183,97,136,20,189,103,142)(15,190,98,143,21,184,104,137)(16,185,99,138,22,191,105,144)(17,192,100,133,23,186,106,139)(18,187,101,140,24,181,107,134)(25,240,70,176,31,234,64,170)(26,235,71,171,32,229,65,177)(27,230,72,178,33,236,66,172)(28,237,61,173,34,231,67,179)(29,232,62,180,35,238,68,174)(30,239,63,175,36,233,69,169)(49,199,83,216,55,193,77,210)(50,194,84,211,56,200,78,205)(51,201,73,206,57,195,79,212)(52,196,74,213,58,202,80,207)(53,203,75,208,59,197,81,214)(54,198,76,215,60,204,82,209)(109,223,166,152,115,217,160,146)(110,218,167,147,116,224,161,153)(111,225,168,154,117,219,162,148)(112,220,157,149,118,226,163,155)(113,227,158,156,119,221,164,150)(114,222,159,151,120,228,165,145)>;
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), (1,47,7,41)(2,46,8,40)(3,45,9,39)(4,44,10,38)(5,43,11,37)(6,42,12,48)(13,62,19,68)(14,61,20,67)(15,72,21,66)(16,71,22,65)(17,70,23,64)(18,69,24,63)(25,100,31,106)(26,99,32,105)(27,98,33,104)(28,97,34,103)(29,108,35,102)(30,107,36,101)(49,119,55,113)(50,118,56,112)(51,117,57,111)(52,116,58,110)(53,115,59,109)(54,114,60,120)(73,162,79,168)(74,161,80,167)(75,160,81,166)(76,159,82,165)(77,158,83,164)(78,157,84,163)(85,124,91,130)(86,123,92,129)(87,122,93,128)(88,121,94,127)(89,132,95,126)(90,131,96,125)(133,234,139,240)(134,233,140,239)(135,232,141,238)(136,231,142,237)(137,230,143,236)(138,229,144,235)(145,215,151,209)(146,214,152,208)(147,213,153,207)(148,212,154,206)(149,211,155,205)(150,210,156,216)(169,181,175,187)(170,192,176,186)(171,191,177,185)(172,190,178,184)(173,189,179,183)(174,188,180,182)(193,227,199,221)(194,226,200,220)(195,225,201,219)(196,224,202,218)(197,223,203,217)(198,222,204,228), (1,238,224,207,188)(2,239,225,208,189)(3,240,226,209,190)(4,229,227,210,191)(5,230,228,211,192)(6,231,217,212,181)(7,232,218,213,182)(8,233,219,214,183)(9,234,220,215,184)(10,235,221,216,185)(11,236,222,205,186)(12,237,223,206,187)(13,35,80,125,116)(14,36,81,126,117)(15,25,82,127,118)(16,26,83,128,119)(17,27,84,129,120)(18,28,73,130,109)(19,29,74,131,110)(20,30,75,132,111)(21,31,76,121,112)(22,32,77,122,113)(23,33,78,123,114)(24,34,79,124,115)(37,137,204,149,170)(38,138,193,150,171)(39,139,194,151,172)(40,140,195,152,173)(41,141,196,153,174)(42,142,197,154,175)(43,143,198,155,176)(44,144,199,156,177)(45,133,200,145,178)(46,134,201,146,179)(47,135,202,147,180)(48,136,203,148,169)(49,65,105,158,93)(50,66,106,159,94)(51,67,107,160,95)(52,68,108,161,96)(53,69,97,162,85)(54,70,98,163,86)(55,71,99,164,87)(56,72,100,165,88)(57,61,101,166,89)(58,62,102,167,90)(59,63,103,168,91)(60,64,104,157,92), (1,96,41,131,7,90,47,125)(2,91,42,126,8,85,48,132)(3,86,43,121,9,92,37,127)(4,93,44,128,10,87,38,122)(5,88,45,123,11,94,39,129)(6,95,46,130,12,89,40,124)(13,188,108,141,19,182,102,135)(14,183,97,136,20,189,103,142)(15,190,98,143,21,184,104,137)(16,185,99,138,22,191,105,144)(17,192,100,133,23,186,106,139)(18,187,101,140,24,181,107,134)(25,240,70,176,31,234,64,170)(26,235,71,171,32,229,65,177)(27,230,72,178,33,236,66,172)(28,237,61,173,34,231,67,179)(29,232,62,180,35,238,68,174)(30,239,63,175,36,233,69,169)(49,199,83,216,55,193,77,210)(50,194,84,211,56,200,78,205)(51,201,73,206,57,195,79,212)(52,196,74,213,58,202,80,207)(53,203,75,208,59,197,81,214)(54,198,76,215,60,204,82,209)(109,223,166,152,115,217,160,146)(110,218,167,147,116,224,161,153)(111,225,168,154,117,219,162,148)(112,220,157,149,118,226,163,155)(113,227,158,156,119,221,164,150)(114,222,159,151,120,228,165,145) );
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)], [(1,47,7,41),(2,46,8,40),(3,45,9,39),(4,44,10,38),(5,43,11,37),(6,42,12,48),(13,62,19,68),(14,61,20,67),(15,72,21,66),(16,71,22,65),(17,70,23,64),(18,69,24,63),(25,100,31,106),(26,99,32,105),(27,98,33,104),(28,97,34,103),(29,108,35,102),(30,107,36,101),(49,119,55,113),(50,118,56,112),(51,117,57,111),(52,116,58,110),(53,115,59,109),(54,114,60,120),(73,162,79,168),(74,161,80,167),(75,160,81,166),(76,159,82,165),(77,158,83,164),(78,157,84,163),(85,124,91,130),(86,123,92,129),(87,122,93,128),(88,121,94,127),(89,132,95,126),(90,131,96,125),(133,234,139,240),(134,233,140,239),(135,232,141,238),(136,231,142,237),(137,230,143,236),(138,229,144,235),(145,215,151,209),(146,214,152,208),(147,213,153,207),(148,212,154,206),(149,211,155,205),(150,210,156,216),(169,181,175,187),(170,192,176,186),(171,191,177,185),(172,190,178,184),(173,189,179,183),(174,188,180,182),(193,227,199,221),(194,226,200,220),(195,225,201,219),(196,224,202,218),(197,223,203,217),(198,222,204,228)], [(1,238,224,207,188),(2,239,225,208,189),(3,240,226,209,190),(4,229,227,210,191),(5,230,228,211,192),(6,231,217,212,181),(7,232,218,213,182),(8,233,219,214,183),(9,234,220,215,184),(10,235,221,216,185),(11,236,222,205,186),(12,237,223,206,187),(13,35,80,125,116),(14,36,81,126,117),(15,25,82,127,118),(16,26,83,128,119),(17,27,84,129,120),(18,28,73,130,109),(19,29,74,131,110),(20,30,75,132,111),(21,31,76,121,112),(22,32,77,122,113),(23,33,78,123,114),(24,34,79,124,115),(37,137,204,149,170),(38,138,193,150,171),(39,139,194,151,172),(40,140,195,152,173),(41,141,196,153,174),(42,142,197,154,175),(43,143,198,155,176),(44,144,199,156,177),(45,133,200,145,178),(46,134,201,146,179),(47,135,202,147,180),(48,136,203,148,169),(49,65,105,158,93),(50,66,106,159,94),(51,67,107,160,95),(52,68,108,161,96),(53,69,97,162,85),(54,70,98,163,86),(55,71,99,164,87),(56,72,100,165,88),(57,61,101,166,89),(58,62,102,167,90),(59,63,103,168,91),(60,64,104,157,92)], [(1,96,41,131,7,90,47,125),(2,91,42,126,8,85,48,132),(3,86,43,121,9,92,37,127),(4,93,44,128,10,87,38,122),(5,88,45,123,11,94,39,129),(6,95,46,130,12,89,40,124),(13,188,108,141,19,182,102,135),(14,183,97,136,20,189,103,142),(15,190,98,143,21,184,104,137),(16,185,99,138,22,191,105,144),(17,192,100,133,23,186,106,139),(18,187,101,140,24,181,107,134),(25,240,70,176,31,234,64,170),(26,235,71,171,32,229,65,177),(27,230,72,178,33,236,66,172),(28,237,61,173,34,231,67,179),(29,232,62,180,35,238,68,174),(30,239,63,175,36,233,69,169),(49,199,83,216,55,193,77,210),(50,194,84,211,56,200,78,205),(51,201,73,206,57,195,79,212),(52,196,74,213,58,202,80,207),(53,203,75,208,59,197,81,214),(54,198,76,215,60,204,82,209),(109,223,166,152,115,217,160,146),(110,218,167,147,116,224,161,153),(111,225,168,154,117,219,162,148),(112,220,157,149,118,226,163,155),(113,227,158,156,119,221,164,150),(114,222,159,151,120,228,165,145)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5 | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10 | 12A | 12B | 12C | 15 | 20A | 20B | 20C | 24A | 24B | 24C | 24D | 30 | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 12 | 12 | 12 | 15 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 60 | 60 |
| size | 1 | 1 | 10 | 30 | 30 | 2 | 2 | 5 | 5 | 6 | 6 | 4 | 2 | 20 | 10 | 10 | 10 | 10 | 15 | 15 | 15 | 15 | 30 | 30 | 4 | 4 | 10 | 10 | 8 | 8 | 24 | 24 | 20 | 20 | 20 | 20 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D6 | D6 | C4×S3 | C4×S3 | C8○D4 | F5 | C2×F5 | C2×F5 | D12.C4 | S3×F5 | Q8.F5 | C2×S3×F5 | Dic6.F5 |
| kernel | Dic6.F5 | D15⋊C8 | Dic3.F5 | C3×C4.F5 | C60.C4 | C12.28D10 | C3⋊D20 | C5×Dic6 | D60 | C4.F5 | C5⋊C8 | C4×D5 | C20 | D10 | C15 | Dic6 | Dic3 | C12 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 2 |
Matrix representation of Dic6.F5 ►in GL8(𝔽241)
| 1 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 160 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 240 |
| 64 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 118 | 177 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 240 |
| 1 | 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 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 240 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 240 |
| 211 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 20 | 30 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 163 | 78 | 94 | 0 |
| 0 | 0 | 0 | 0 | 16 | 78 | 0 | 163 |
| 0 | 0 | 0 | 0 | 163 | 0 | 78 | 16 |
| 0 | 0 | 0 | 0 | 0 | 94 | 78 | 163 |
G:=sub<GL(8,GF(241))| [1,160,0,0,0,0,0,0,3,240,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240],[64,118,0,0,0,0,0,0,0,177,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,240,240,240,240],[211,20,0,0,0,0,0,0,0,30,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,163,16,163,0,0,0,0,0,78,78,0,94,0,0,0,0,94,0,78,78,0,0,0,0,0,163,16,163] >;
Dic6.F5 in GAP, Magma, Sage, TeX
{\rm Dic}_6.F_5 % in TeX
G:=Group("Dic6.F5"); // GroupNames label
G:=SmallGroup(480,992);
// by ID
G=gap.SmallGroup(480,992);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,422,219,100,80,1356,9414,2379]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^5=1,b^2=d^4=a^6,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations