direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6×C4.Dic5, C30⋊14M4(2), C60.280C23, C5⋊6(C6×M4(2)), (C2×C60).44C4, C4.9(C6×Dic5), (C2×C20).21C12, C60.243(C2×C4), C20.58(C2×C12), C10⋊4(C3×M4(2)), C15⋊30(C2×M4(2)), (C2×C12).433D10, (C22×C60).20C2, (C22×C30).20C4, C20.40(C22×C6), (C22×C20).13C6, (C2×C12).15Dic5, C12.48(C2×Dic5), (C22×C12).14D5, C23.4(C3×Dic5), (C22×C6).4Dic5, C22.5(C6×Dic5), C30.220(C22×C4), (C22×C10).14C12, C10.35(C22×C12), (C2×C60).520C22, C12.247(C22×D5), C6.31(C22×Dic5), C4.41(D5×C2×C6), (C2×C5⋊2C8)⋊12C6, (C6×C5⋊2C8)⋊26C2, C5⋊2C8⋊12(C2×C6), C2.3(C2×C6×Dic5), (C2×C4).101(C6×D5), (C2×C4).6(C3×Dic5), (C22×C4).6(C3×D5), (C2×C20).103(C2×C6), (C2×C30).189(C2×C4), (C2×C10).53(C2×C12), (C3×C5⋊2C8)⋊45C22, (C2×C6).23(C2×Dic5), SmallGroup(480,714)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×C4.Dic5
G = < a,b,c,d | a6=b4=1, c10=b2, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c9 >
Subgroups: 240 in 136 conjugacy classes, 98 normal (38 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C5, C6, C6, C6, C8, C2×C4, C2×C4, C23, C10, C10, C10, C12, C12, C2×C6, C2×C6, C2×C6, C15, C2×C8, M4(2), C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C24, C2×C12, C2×C12, C22×C6, C30, C30, C30, C2×M4(2), C5⋊2C8, C2×C20, C2×C20, C22×C10, C2×C24, C3×M4(2), C22×C12, C60, C60, C2×C30, C2×C30, C2×C30, C2×C5⋊2C8, C4.Dic5, C22×C20, C6×M4(2), C3×C5⋊2C8, C2×C60, C2×C60, C22×C30, C2×C4.Dic5, C6×C5⋊2C8, C3×C4.Dic5, C22×C60, C6×C4.Dic5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, D5, C12, C2×C6, M4(2), C22×C4, Dic5, D10, C2×C12, C22×C6, C3×D5, C2×M4(2), C2×Dic5, C22×D5, C3×M4(2), C22×C12, C3×Dic5, C6×D5, C4.Dic5, C22×Dic5, C6×M4(2), C6×Dic5, D5×C2×C6, C2×C4.Dic5, C3×C4.Dic5, C2×C6×Dic5, C6×C4.Dic5
►On 240 points
Generators in S240
(1 86 46 71 31 101)(2 87 47 72 32 102)(3 88 48 73 33 103)(4 89 49 74 34 104)(5 90 50 75 35 105)(6 91 51 76 36 106)(7 92 52 77 37 107)(8 93 53 78 38 108)(9 94 54 79 39 109)(10 95 55 80 40 110)(11 96 56 61 21 111)(12 97 57 62 22 112)(13 98 58 63 23 113)(14 99 59 64 24 114)(15 100 60 65 25 115)(16 81 41 66 26 116)(17 82 42 67 27 117)(18 83 43 68 28 118)(19 84 44 69 29 119)(20 85 45 70 30 120)(121 209 163 197 143 237)(122 210 164 198 144 238)(123 211 165 199 145 239)(124 212 166 200 146 240)(125 213 167 181 147 221)(126 214 168 182 148 222)(127 215 169 183 149 223)(128 216 170 184 150 224)(129 217 171 185 151 225)(130 218 172 186 152 226)(131 219 173 187 153 227)(132 220 174 188 154 228)(133 201 175 189 155 229)(134 202 176 190 156 230)(135 203 177 191 157 231)(136 204 178 192 158 232)(137 205 179 193 159 233)(138 206 180 194 160 234)(139 207 161 195 141 235)(140 208 162 196 142 236)
(1 76 11 66)(2 77 12 67)(3 78 13 68)(4 79 14 69)(5 80 15 70)(6 61 16 71)(7 62 17 72)(8 63 18 73)(9 64 19 74)(10 65 20 75)(21 81 31 91)(22 82 32 92)(23 83 33 93)(24 84 34 94)(25 85 35 95)(26 86 36 96)(27 87 37 97)(28 88 38 98)(29 89 39 99)(30 90 40 100)(41 101 51 111)(42 102 52 112)(43 103 53 113)(44 104 54 114)(45 105 55 115)(46 106 56 116)(47 107 57 117)(48 108 58 118)(49 109 59 119)(50 110 60 120)(121 192 131 182)(122 193 132 183)(123 194 133 184)(124 195 134 185)(125 196 135 186)(126 197 136 187)(127 198 137 188)(128 199 138 189)(129 200 139 190)(130 181 140 191)(141 202 151 212)(142 203 152 213)(143 204 153 214)(144 205 154 215)(145 206 155 216)(146 207 156 217)(147 208 157 218)(148 209 158 219)(149 210 159 220)(150 211 160 201)(161 230 171 240)(162 231 172 221)(163 232 173 222)(164 233 174 223)(165 234 175 224)(166 235 176 225)(167 236 177 226)(168 237 178 227)(169 238 179 228)(170 239 180 229)
(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)
(1 133 6 138 11 123 16 128)(2 122 7 127 12 132 17 137)(3 131 8 136 13 121 18 126)(4 140 9 125 14 130 19 135)(5 129 10 134 15 139 20 124)(21 145 26 150 31 155 36 160)(22 154 27 159 32 144 37 149)(23 143 28 148 33 153 38 158)(24 152 29 157 34 142 39 147)(25 141 30 146 35 151 40 156)(41 170 46 175 51 180 56 165)(42 179 47 164 52 169 57 174)(43 168 48 173 53 178 58 163)(44 177 49 162 54 167 59 172)(45 166 50 171 55 176 60 161)(61 199 66 184 71 189 76 194)(62 188 67 193 72 198 77 183)(63 197 68 182 73 187 78 192)(64 186 69 191 74 196 79 181)(65 195 70 200 75 185 80 190)(81 216 86 201 91 206 96 211)(82 205 87 210 92 215 97 220)(83 214 88 219 93 204 98 209)(84 203 89 208 94 213 99 218)(85 212 90 217 95 202 100 207)(101 229 106 234 111 239 116 224)(102 238 107 223 112 228 117 233)(103 227 108 232 113 237 118 222)(104 236 109 221 114 226 119 231)(105 225 110 230 115 235 120 240)
G:=sub<Sym(240)| (1,86,46,71,31,101)(2,87,47,72,32,102)(3,88,48,73,33,103)(4,89,49,74,34,104)(5,90,50,75,35,105)(6,91,51,76,36,106)(7,92,52,77,37,107)(8,93,53,78,38,108)(9,94,54,79,39,109)(10,95,55,80,40,110)(11,96,56,61,21,111)(12,97,57,62,22,112)(13,98,58,63,23,113)(14,99,59,64,24,114)(15,100,60,65,25,115)(16,81,41,66,26,116)(17,82,42,67,27,117)(18,83,43,68,28,118)(19,84,44,69,29,119)(20,85,45,70,30,120)(121,209,163,197,143,237)(122,210,164,198,144,238)(123,211,165,199,145,239)(124,212,166,200,146,240)(125,213,167,181,147,221)(126,214,168,182,148,222)(127,215,169,183,149,223)(128,216,170,184,150,224)(129,217,171,185,151,225)(130,218,172,186,152,226)(131,219,173,187,153,227)(132,220,174,188,154,228)(133,201,175,189,155,229)(134,202,176,190,156,230)(135,203,177,191,157,231)(136,204,178,192,158,232)(137,205,179,193,159,233)(138,206,180,194,160,234)(139,207,161,195,141,235)(140,208,162,196,142,236), (1,76,11,66)(2,77,12,67)(3,78,13,68)(4,79,14,69)(5,80,15,70)(6,61,16,71)(7,62,17,72)(8,63,18,73)(9,64,19,74)(10,65,20,75)(21,81,31,91)(22,82,32,92)(23,83,33,93)(24,84,34,94)(25,85,35,95)(26,86,36,96)(27,87,37,97)(28,88,38,98)(29,89,39,99)(30,90,40,100)(41,101,51,111)(42,102,52,112)(43,103,53,113)(44,104,54,114)(45,105,55,115)(46,106,56,116)(47,107,57,117)(48,108,58,118)(49,109,59,119)(50,110,60,120)(121,192,131,182)(122,193,132,183)(123,194,133,184)(124,195,134,185)(125,196,135,186)(126,197,136,187)(127,198,137,188)(128,199,138,189)(129,200,139,190)(130,181,140,191)(141,202,151,212)(142,203,152,213)(143,204,153,214)(144,205,154,215)(145,206,155,216)(146,207,156,217)(147,208,157,218)(148,209,158,219)(149,210,159,220)(150,211,160,201)(161,230,171,240)(162,231,172,221)(163,232,173,222)(164,233,174,223)(165,234,175,224)(166,235,176,225)(167,236,177,226)(168,237,178,227)(169,238,179,228)(170,239,180,229), (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), (1,133,6,138,11,123,16,128)(2,122,7,127,12,132,17,137)(3,131,8,136,13,121,18,126)(4,140,9,125,14,130,19,135)(5,129,10,134,15,139,20,124)(21,145,26,150,31,155,36,160)(22,154,27,159,32,144,37,149)(23,143,28,148,33,153,38,158)(24,152,29,157,34,142,39,147)(25,141,30,146,35,151,40,156)(41,170,46,175,51,180,56,165)(42,179,47,164,52,169,57,174)(43,168,48,173,53,178,58,163)(44,177,49,162,54,167,59,172)(45,166,50,171,55,176,60,161)(61,199,66,184,71,189,76,194)(62,188,67,193,72,198,77,183)(63,197,68,182,73,187,78,192)(64,186,69,191,74,196,79,181)(65,195,70,200,75,185,80,190)(81,216,86,201,91,206,96,211)(82,205,87,210,92,215,97,220)(83,214,88,219,93,204,98,209)(84,203,89,208,94,213,99,218)(85,212,90,217,95,202,100,207)(101,229,106,234,111,239,116,224)(102,238,107,223,112,228,117,233)(103,227,108,232,113,237,118,222)(104,236,109,221,114,226,119,231)(105,225,110,230,115,235,120,240)>;
G:=Group( (1,86,46,71,31,101)(2,87,47,72,32,102)(3,88,48,73,33,103)(4,89,49,74,34,104)(5,90,50,75,35,105)(6,91,51,76,36,106)(7,92,52,77,37,107)(8,93,53,78,38,108)(9,94,54,79,39,109)(10,95,55,80,40,110)(11,96,56,61,21,111)(12,97,57,62,22,112)(13,98,58,63,23,113)(14,99,59,64,24,114)(15,100,60,65,25,115)(16,81,41,66,26,116)(17,82,42,67,27,117)(18,83,43,68,28,118)(19,84,44,69,29,119)(20,85,45,70,30,120)(121,209,163,197,143,237)(122,210,164,198,144,238)(123,211,165,199,145,239)(124,212,166,200,146,240)(125,213,167,181,147,221)(126,214,168,182,148,222)(127,215,169,183,149,223)(128,216,170,184,150,224)(129,217,171,185,151,225)(130,218,172,186,152,226)(131,219,173,187,153,227)(132,220,174,188,154,228)(133,201,175,189,155,229)(134,202,176,190,156,230)(135,203,177,191,157,231)(136,204,178,192,158,232)(137,205,179,193,159,233)(138,206,180,194,160,234)(139,207,161,195,141,235)(140,208,162,196,142,236), (1,76,11,66)(2,77,12,67)(3,78,13,68)(4,79,14,69)(5,80,15,70)(6,61,16,71)(7,62,17,72)(8,63,18,73)(9,64,19,74)(10,65,20,75)(21,81,31,91)(22,82,32,92)(23,83,33,93)(24,84,34,94)(25,85,35,95)(26,86,36,96)(27,87,37,97)(28,88,38,98)(29,89,39,99)(30,90,40,100)(41,101,51,111)(42,102,52,112)(43,103,53,113)(44,104,54,114)(45,105,55,115)(46,106,56,116)(47,107,57,117)(48,108,58,118)(49,109,59,119)(50,110,60,120)(121,192,131,182)(122,193,132,183)(123,194,133,184)(124,195,134,185)(125,196,135,186)(126,197,136,187)(127,198,137,188)(128,199,138,189)(129,200,139,190)(130,181,140,191)(141,202,151,212)(142,203,152,213)(143,204,153,214)(144,205,154,215)(145,206,155,216)(146,207,156,217)(147,208,157,218)(148,209,158,219)(149,210,159,220)(150,211,160,201)(161,230,171,240)(162,231,172,221)(163,232,173,222)(164,233,174,223)(165,234,175,224)(166,235,176,225)(167,236,177,226)(168,237,178,227)(169,238,179,228)(170,239,180,229), (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), (1,133,6,138,11,123,16,128)(2,122,7,127,12,132,17,137)(3,131,8,136,13,121,18,126)(4,140,9,125,14,130,19,135)(5,129,10,134,15,139,20,124)(21,145,26,150,31,155,36,160)(22,154,27,159,32,144,37,149)(23,143,28,148,33,153,38,158)(24,152,29,157,34,142,39,147)(25,141,30,146,35,151,40,156)(41,170,46,175,51,180,56,165)(42,179,47,164,52,169,57,174)(43,168,48,173,53,178,58,163)(44,177,49,162,54,167,59,172)(45,166,50,171,55,176,60,161)(61,199,66,184,71,189,76,194)(62,188,67,193,72,198,77,183)(63,197,68,182,73,187,78,192)(64,186,69,191,74,196,79,181)(65,195,70,200,75,185,80,190)(81,216,86,201,91,206,96,211)(82,205,87,210,92,215,97,220)(83,214,88,219,93,204,98,209)(84,203,89,208,94,213,99,218)(85,212,90,217,95,202,100,207)(101,229,106,234,111,239,116,224)(102,238,107,223,112,228,117,233)(103,227,108,232,113,237,118,222)(104,236,109,221,114,226,119,231)(105,225,110,230,115,235,120,240) );
G=PermutationGroup([[(1,86,46,71,31,101),(2,87,47,72,32,102),(3,88,48,73,33,103),(4,89,49,74,34,104),(5,90,50,75,35,105),(6,91,51,76,36,106),(7,92,52,77,37,107),(8,93,53,78,38,108),(9,94,54,79,39,109),(10,95,55,80,40,110),(11,96,56,61,21,111),(12,97,57,62,22,112),(13,98,58,63,23,113),(14,99,59,64,24,114),(15,100,60,65,25,115),(16,81,41,66,26,116),(17,82,42,67,27,117),(18,83,43,68,28,118),(19,84,44,69,29,119),(20,85,45,70,30,120),(121,209,163,197,143,237),(122,210,164,198,144,238),(123,211,165,199,145,239),(124,212,166,200,146,240),(125,213,167,181,147,221),(126,214,168,182,148,222),(127,215,169,183,149,223),(128,216,170,184,150,224),(129,217,171,185,151,225),(130,218,172,186,152,226),(131,219,173,187,153,227),(132,220,174,188,154,228),(133,201,175,189,155,229),(134,202,176,190,156,230),(135,203,177,191,157,231),(136,204,178,192,158,232),(137,205,179,193,159,233),(138,206,180,194,160,234),(139,207,161,195,141,235),(140,208,162,196,142,236)], [(1,76,11,66),(2,77,12,67),(3,78,13,68),(4,79,14,69),(5,80,15,70),(6,61,16,71),(7,62,17,72),(8,63,18,73),(9,64,19,74),(10,65,20,75),(21,81,31,91),(22,82,32,92),(23,83,33,93),(24,84,34,94),(25,85,35,95),(26,86,36,96),(27,87,37,97),(28,88,38,98),(29,89,39,99),(30,90,40,100),(41,101,51,111),(42,102,52,112),(43,103,53,113),(44,104,54,114),(45,105,55,115),(46,106,56,116),(47,107,57,117),(48,108,58,118),(49,109,59,119),(50,110,60,120),(121,192,131,182),(122,193,132,183),(123,194,133,184),(124,195,134,185),(125,196,135,186),(126,197,136,187),(127,198,137,188),(128,199,138,189),(129,200,139,190),(130,181,140,191),(141,202,151,212),(142,203,152,213),(143,204,153,214),(144,205,154,215),(145,206,155,216),(146,207,156,217),(147,208,157,218),(148,209,158,219),(149,210,159,220),(150,211,160,201),(161,230,171,240),(162,231,172,221),(163,232,173,222),(164,233,174,223),(165,234,175,224),(166,235,176,225),(167,236,177,226),(168,237,178,227),(169,238,179,228),(170,239,180,229)], [(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)], [(1,133,6,138,11,123,16,128),(2,122,7,127,12,132,17,137),(3,131,8,136,13,121,18,126),(4,140,9,125,14,130,19,135),(5,129,10,134,15,139,20,124),(21,145,26,150,31,155,36,160),(22,154,27,159,32,144,37,149),(23,143,28,148,33,153,38,158),(24,152,29,157,34,142,39,147),(25,141,30,146,35,151,40,156),(41,170,46,175,51,180,56,165),(42,179,47,164,52,169,57,174),(43,168,48,173,53,178,58,163),(44,177,49,162,54,167,59,172),(45,166,50,171,55,176,60,161),(61,199,66,184,71,189,76,194),(62,188,67,193,72,198,77,183),(63,197,68,182,73,187,78,192),(64,186,69,191,74,196,79,181),(65,195,70,200,75,185,80,190),(81,216,86,201,91,206,96,211),(82,205,87,210,92,215,97,220),(83,214,88,219,93,204,98,209),(84,203,89,208,94,213,99,218),(85,212,90,217,95,202,100,207),(101,229,106,234,111,239,116,224),(102,238,107,223,112,228,117,233),(103,227,108,232,113,237,118,222),(104,236,109,221,114,226,119,231),(105,225,110,230,115,235,120,240)]])
156 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | ··· | 8H | 10A | ··· | 10N | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 24A | ··· | 24P | 30A | ··· | 30AB | 60A | ··· | 60AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 10 | ··· | 10 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 |
156 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | D5 | M4(2) | Dic5 | D10 | Dic5 | C3×D5 | C3×M4(2) | C3×Dic5 | C6×D5 | C3×Dic5 | C4.Dic5 | C3×C4.Dic5 |
| kernel | C6×C4.Dic5 | C6×C5⋊2C8 | C3×C4.Dic5 | C22×C60 | C2×C4.Dic5 | C2×C60 | C22×C30 | C2×C5⋊2C8 | C4.Dic5 | C22×C20 | C2×C20 | C22×C10 | C22×C12 | C30 | C2×C12 | C2×C12 | C22×C6 | C22×C4 | C10 | C2×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 2 | 4 | 1 | 2 | 6 | 2 | 4 | 8 | 2 | 12 | 4 | 2 | 4 | 6 | 6 | 2 | 4 | 8 | 12 | 12 | 4 | 16 | 32 |
Matrix representation of C6×C4.Dic5 ►in GL6(𝔽241)
| 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 226 | 0 | 0 | 0 |
| 0 | 0 | 0 | 226 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 | 0 |
| 0 | 0 | 0 | 177 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 177 | 0 | 0 | 0 |
| 0 | 0 | 0 | 177 | 0 | 0 |
| 0 | 0 | 0 | 0 | 190 | 240 |
| 0 | 0 | 0 | 0 | 191 | 240 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 177 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 222 |
| 0 | 0 | 0 | 0 | 127 | 239 |
G:=sub<GL(6,GF(241))| [15,0,0,0,0,0,0,15,0,0,0,0,0,0,226,0,0,0,0,0,0,226,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[64,0,0,0,0,0,0,177,0,0,0,0,0,0,64,0,0,0,0,0,0,177,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[64,0,0,0,0,0,0,64,0,0,0,0,0,0,177,0,0,0,0,0,0,177,0,0,0,0,0,0,190,191,0,0,0,0,240,240],[0,64,0,0,0,0,1,0,0,0,0,0,0,0,0,177,0,0,0,0,1,0,0,0,0,0,0,0,2,127,0,0,0,0,222,239] >;
C6×C4.Dic5 in GAP, Magma, Sage, TeX
C_6\times C_4.{\rm Dic}_5 % in TeX
G:=Group("C6xC4.Dic5"); // GroupNames label
G:=SmallGroup(480,714);
// by ID
G=gap.SmallGroup(480,714);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,1094,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^4=1,c^10=b^2,d^2=c^5,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^9>;
// generators/relations