direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6×C4.F5, C30⋊4M4(2), C5⋊1(C6×M4(2)), C4.12(C6×F5), C60.65(C2×C4), (C2×C20).6C12, (C2×C60).17C4, (C4×D5).4C12, C12.65(C2×F5), (C2×C12).18F5, C20.12(C2×C12), (D5×C12).14C4, C10⋊1(C3×M4(2)), C15⋊13(C2×M4(2)), C6.46(C22×F5), C22.16(C6×F5), D10.13(C2×C12), C10.2(C22×C12), C30.84(C22×C4), (C22×D5).8C12, Dic5.15(C2×C12), (D5×C12).131C22, (C3×Dic5).70C23, Dic5.10(C22×C6), (C6×Dic5).277C22, (C2×C5⋊C8)⋊3C6, C5⋊C8⋊1(C2×C6), (C6×C5⋊C8)⋊10C2, C2.4(C2×C6×F5), (D5×C2×C6).18C4, (C2×C4×D5).13C6, (C2×C4).7(C3×F5), (C3×C5⋊C8)⋊11C22, (D5×C2×C12).34C2, (C2×C6).59(C2×F5), (C2×C30).57(C2×C4), (C6×D5).62(C2×C4), (C4×D5).29(C2×C6), (C2×C10).14(C2×C12), (C2×Dic5).54(C2×C6), (C3×Dic5).71(C2×C4), SmallGroup(480,1048)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×C4.F5
G = < a,b,c,d | a6=b4=c5=1, d4=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >
Subgroups: 392 in 136 conjugacy classes, 76 normal (28 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C8, C2×C4, C2×C4, C23, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C2×C8, M4(2), C22×C4, Dic5, C20, D10, D10, C2×C10, C24, C2×C12, C2×C12, C22×C6, C3×D5, C30, C30, C2×M4(2), C5⋊C8, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×C24, C3×M4(2), C22×C12, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, C4.F5, C2×C5⋊C8, C2×C4×D5, C6×M4(2), C3×C5⋊C8, D5×C12, C6×Dic5, C2×C60, D5×C2×C6, C2×C4.F5, C3×C4.F5, C6×C5⋊C8, D5×C2×C12, C6×C4.F5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, M4(2), C22×C4, F5, C2×C12, C22×C6, C2×M4(2), C2×F5, C3×M4(2), C22×C12, C3×F5, C4.F5, C22×F5, C6×M4(2), C6×F5, C2×C4.F5, C3×C4.F5, C2×C6×F5, C6×C4.F5
►On 240 points
Generators in S240
(1 239 221 51 25 65)(2 240 222 52 26 66)(3 233 223 53 27 67)(4 234 224 54 28 68)(5 235 217 55 29 69)(6 236 218 56 30 70)(7 237 219 49 31 71)(8 238 220 50 32 72)(9 167 140 21 103 207)(10 168 141 22 104 208)(11 161 142 23 97 201)(12 162 143 24 98 202)(13 163 144 17 99 203)(14 164 137 18 100 204)(15 165 138 19 101 205)(16 166 139 20 102 206)(33 183 153 229 85 125)(34 184 154 230 86 126)(35 177 155 231 87 127)(36 178 156 232 88 128)(37 179 157 225 81 121)(38 180 158 226 82 122)(39 181 159 227 83 123)(40 182 160 228 84 124)(41 145 185 79 119 215)(42 146 186 80 120 216)(43 147 187 73 113 209)(44 148 188 74 114 210)(45 149 189 75 115 211)(46 150 190 76 116 212)(47 151 191 77 117 213)(48 152 192 78 118 214)(57 199 109 95 135 175)(58 200 110 96 136 176)(59 193 111 89 129 169)(60 194 112 90 130 170)(61 195 105 91 131 171)(62 196 106 92 132 172)(63 197 107 93 133 173)(64 198 108 94 134 174)
(1 49 5 53)(2 54 6 50)(3 51 7 55)(4 56 8 52)(9 211 13 215)(10 216 14 212)(11 213 15 209)(12 210 16 214)(17 185 21 189)(18 190 22 186)(19 187 23 191)(20 192 24 188)(25 237 29 233)(26 234 30 238)(27 239 31 235)(28 236 32 240)(33 111 37 107)(34 108 38 112)(35 105 39 109)(36 110 40 106)(41 167 45 163)(42 164 46 168)(43 161 47 165)(44 166 48 162)(57 87 61 83)(58 84 62 88)(59 81 63 85)(60 86 64 82)(65 219 69 223)(66 224 70 220)(67 221 71 217)(68 218 72 222)(73 97 77 101)(74 102 78 98)(75 99 79 103)(76 104 80 100)(89 179 93 183)(90 184 94 180)(91 181 95 177)(92 178 96 182)(113 201 117 205)(114 206 118 202)(115 203 119 207)(116 208 120 204)(121 197 125 193)(122 194 126 198)(123 199 127 195)(124 196 128 200)(129 157 133 153)(130 154 134 158)(131 159 135 155)(132 156 136 160)(137 150 141 146)(138 147 142 151)(139 152 143 148)(140 149 144 145)(169 225 173 229)(170 230 174 226)(171 227 175 231)(172 232 176 228)
(1 175 167 77 37)(2 78 176 38 168)(3 39 79 161 169)(4 162 40 170 80)(5 171 163 73 33)(6 74 172 34 164)(7 35 75 165 173)(8 166 36 174 76)(9 191 121 65 135)(10 66 192 136 122)(11 129 67 123 185)(12 124 130 186 68)(13 187 125 69 131)(14 70 188 132 126)(15 133 71 127 189)(16 128 134 190 72)(17 209 153 217 195)(18 218 210 196 154)(19 197 219 155 211)(20 156 198 212 220)(21 213 157 221 199)(22 222 214 200 158)(23 193 223 159 215)(24 160 194 216 224)(25 95 207 151 81)(26 152 96 82 208)(27 83 145 201 89)(28 202 84 90 146)(29 91 203 147 85)(30 148 92 86 204)(31 87 149 205 93)(32 206 88 94 150)(41 97 111 53 227)(42 54 98 228 112)(43 229 55 105 99)(44 106 230 100 56)(45 101 107 49 231)(46 50 102 232 108)(47 225 51 109 103)(48 110 226 104 52)(57 140 117 179 239)(58 180 141 240 118)(59 233 181 119 142)(60 120 234 143 182)(61 144 113 183 235)(62 184 137 236 114)(63 237 177 115 138)(64 116 238 139 178)
(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)
G:=sub<Sym(240)| (1,239,221,51,25,65)(2,240,222,52,26,66)(3,233,223,53,27,67)(4,234,224,54,28,68)(5,235,217,55,29,69)(6,236,218,56,30,70)(7,237,219,49,31,71)(8,238,220,50,32,72)(9,167,140,21,103,207)(10,168,141,22,104,208)(11,161,142,23,97,201)(12,162,143,24,98,202)(13,163,144,17,99,203)(14,164,137,18,100,204)(15,165,138,19,101,205)(16,166,139,20,102,206)(33,183,153,229,85,125)(34,184,154,230,86,126)(35,177,155,231,87,127)(36,178,156,232,88,128)(37,179,157,225,81,121)(38,180,158,226,82,122)(39,181,159,227,83,123)(40,182,160,228,84,124)(41,145,185,79,119,215)(42,146,186,80,120,216)(43,147,187,73,113,209)(44,148,188,74,114,210)(45,149,189,75,115,211)(46,150,190,76,116,212)(47,151,191,77,117,213)(48,152,192,78,118,214)(57,199,109,95,135,175)(58,200,110,96,136,176)(59,193,111,89,129,169)(60,194,112,90,130,170)(61,195,105,91,131,171)(62,196,106,92,132,172)(63,197,107,93,133,173)(64,198,108,94,134,174), (1,49,5,53)(2,54,6,50)(3,51,7,55)(4,56,8,52)(9,211,13,215)(10,216,14,212)(11,213,15,209)(12,210,16,214)(17,185,21,189)(18,190,22,186)(19,187,23,191)(20,192,24,188)(25,237,29,233)(26,234,30,238)(27,239,31,235)(28,236,32,240)(33,111,37,107)(34,108,38,112)(35,105,39,109)(36,110,40,106)(41,167,45,163)(42,164,46,168)(43,161,47,165)(44,166,48,162)(57,87,61,83)(58,84,62,88)(59,81,63,85)(60,86,64,82)(65,219,69,223)(66,224,70,220)(67,221,71,217)(68,218,72,222)(73,97,77,101)(74,102,78,98)(75,99,79,103)(76,104,80,100)(89,179,93,183)(90,184,94,180)(91,181,95,177)(92,178,96,182)(113,201,117,205)(114,206,118,202)(115,203,119,207)(116,208,120,204)(121,197,125,193)(122,194,126,198)(123,199,127,195)(124,196,128,200)(129,157,133,153)(130,154,134,158)(131,159,135,155)(132,156,136,160)(137,150,141,146)(138,147,142,151)(139,152,143,148)(140,149,144,145)(169,225,173,229)(170,230,174,226)(171,227,175,231)(172,232,176,228), (1,175,167,77,37)(2,78,176,38,168)(3,39,79,161,169)(4,162,40,170,80)(5,171,163,73,33)(6,74,172,34,164)(7,35,75,165,173)(8,166,36,174,76)(9,191,121,65,135)(10,66,192,136,122)(11,129,67,123,185)(12,124,130,186,68)(13,187,125,69,131)(14,70,188,132,126)(15,133,71,127,189)(16,128,134,190,72)(17,209,153,217,195)(18,218,210,196,154)(19,197,219,155,211)(20,156,198,212,220)(21,213,157,221,199)(22,222,214,200,158)(23,193,223,159,215)(24,160,194,216,224)(25,95,207,151,81)(26,152,96,82,208)(27,83,145,201,89)(28,202,84,90,146)(29,91,203,147,85)(30,148,92,86,204)(31,87,149,205,93)(32,206,88,94,150)(41,97,111,53,227)(42,54,98,228,112)(43,229,55,105,99)(44,106,230,100,56)(45,101,107,49,231)(46,50,102,232,108)(47,225,51,109,103)(48,110,226,104,52)(57,140,117,179,239)(58,180,141,240,118)(59,233,181,119,142)(60,120,234,143,182)(61,144,113,183,235)(62,184,137,236,114)(63,237,177,115,138)(64,116,238,139,178), (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)>;
G:=Group( (1,239,221,51,25,65)(2,240,222,52,26,66)(3,233,223,53,27,67)(4,234,224,54,28,68)(5,235,217,55,29,69)(6,236,218,56,30,70)(7,237,219,49,31,71)(8,238,220,50,32,72)(9,167,140,21,103,207)(10,168,141,22,104,208)(11,161,142,23,97,201)(12,162,143,24,98,202)(13,163,144,17,99,203)(14,164,137,18,100,204)(15,165,138,19,101,205)(16,166,139,20,102,206)(33,183,153,229,85,125)(34,184,154,230,86,126)(35,177,155,231,87,127)(36,178,156,232,88,128)(37,179,157,225,81,121)(38,180,158,226,82,122)(39,181,159,227,83,123)(40,182,160,228,84,124)(41,145,185,79,119,215)(42,146,186,80,120,216)(43,147,187,73,113,209)(44,148,188,74,114,210)(45,149,189,75,115,211)(46,150,190,76,116,212)(47,151,191,77,117,213)(48,152,192,78,118,214)(57,199,109,95,135,175)(58,200,110,96,136,176)(59,193,111,89,129,169)(60,194,112,90,130,170)(61,195,105,91,131,171)(62,196,106,92,132,172)(63,197,107,93,133,173)(64,198,108,94,134,174), (1,49,5,53)(2,54,6,50)(3,51,7,55)(4,56,8,52)(9,211,13,215)(10,216,14,212)(11,213,15,209)(12,210,16,214)(17,185,21,189)(18,190,22,186)(19,187,23,191)(20,192,24,188)(25,237,29,233)(26,234,30,238)(27,239,31,235)(28,236,32,240)(33,111,37,107)(34,108,38,112)(35,105,39,109)(36,110,40,106)(41,167,45,163)(42,164,46,168)(43,161,47,165)(44,166,48,162)(57,87,61,83)(58,84,62,88)(59,81,63,85)(60,86,64,82)(65,219,69,223)(66,224,70,220)(67,221,71,217)(68,218,72,222)(73,97,77,101)(74,102,78,98)(75,99,79,103)(76,104,80,100)(89,179,93,183)(90,184,94,180)(91,181,95,177)(92,178,96,182)(113,201,117,205)(114,206,118,202)(115,203,119,207)(116,208,120,204)(121,197,125,193)(122,194,126,198)(123,199,127,195)(124,196,128,200)(129,157,133,153)(130,154,134,158)(131,159,135,155)(132,156,136,160)(137,150,141,146)(138,147,142,151)(139,152,143,148)(140,149,144,145)(169,225,173,229)(170,230,174,226)(171,227,175,231)(172,232,176,228), (1,175,167,77,37)(2,78,176,38,168)(3,39,79,161,169)(4,162,40,170,80)(5,171,163,73,33)(6,74,172,34,164)(7,35,75,165,173)(8,166,36,174,76)(9,191,121,65,135)(10,66,192,136,122)(11,129,67,123,185)(12,124,130,186,68)(13,187,125,69,131)(14,70,188,132,126)(15,133,71,127,189)(16,128,134,190,72)(17,209,153,217,195)(18,218,210,196,154)(19,197,219,155,211)(20,156,198,212,220)(21,213,157,221,199)(22,222,214,200,158)(23,193,223,159,215)(24,160,194,216,224)(25,95,207,151,81)(26,152,96,82,208)(27,83,145,201,89)(28,202,84,90,146)(29,91,203,147,85)(30,148,92,86,204)(31,87,149,205,93)(32,206,88,94,150)(41,97,111,53,227)(42,54,98,228,112)(43,229,55,105,99)(44,106,230,100,56)(45,101,107,49,231)(46,50,102,232,108)(47,225,51,109,103)(48,110,226,104,52)(57,140,117,179,239)(58,180,141,240,118)(59,233,181,119,142)(60,120,234,143,182)(61,144,113,183,235)(62,184,137,236,114)(63,237,177,115,138)(64,116,238,139,178), (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) );
G=PermutationGroup([[(1,239,221,51,25,65),(2,240,222,52,26,66),(3,233,223,53,27,67),(4,234,224,54,28,68),(5,235,217,55,29,69),(6,236,218,56,30,70),(7,237,219,49,31,71),(8,238,220,50,32,72),(9,167,140,21,103,207),(10,168,141,22,104,208),(11,161,142,23,97,201),(12,162,143,24,98,202),(13,163,144,17,99,203),(14,164,137,18,100,204),(15,165,138,19,101,205),(16,166,139,20,102,206),(33,183,153,229,85,125),(34,184,154,230,86,126),(35,177,155,231,87,127),(36,178,156,232,88,128),(37,179,157,225,81,121),(38,180,158,226,82,122),(39,181,159,227,83,123),(40,182,160,228,84,124),(41,145,185,79,119,215),(42,146,186,80,120,216),(43,147,187,73,113,209),(44,148,188,74,114,210),(45,149,189,75,115,211),(46,150,190,76,116,212),(47,151,191,77,117,213),(48,152,192,78,118,214),(57,199,109,95,135,175),(58,200,110,96,136,176),(59,193,111,89,129,169),(60,194,112,90,130,170),(61,195,105,91,131,171),(62,196,106,92,132,172),(63,197,107,93,133,173),(64,198,108,94,134,174)], [(1,49,5,53),(2,54,6,50),(3,51,7,55),(4,56,8,52),(9,211,13,215),(10,216,14,212),(11,213,15,209),(12,210,16,214),(17,185,21,189),(18,190,22,186),(19,187,23,191),(20,192,24,188),(25,237,29,233),(26,234,30,238),(27,239,31,235),(28,236,32,240),(33,111,37,107),(34,108,38,112),(35,105,39,109),(36,110,40,106),(41,167,45,163),(42,164,46,168),(43,161,47,165),(44,166,48,162),(57,87,61,83),(58,84,62,88),(59,81,63,85),(60,86,64,82),(65,219,69,223),(66,224,70,220),(67,221,71,217),(68,218,72,222),(73,97,77,101),(74,102,78,98),(75,99,79,103),(76,104,80,100),(89,179,93,183),(90,184,94,180),(91,181,95,177),(92,178,96,182),(113,201,117,205),(114,206,118,202),(115,203,119,207),(116,208,120,204),(121,197,125,193),(122,194,126,198),(123,199,127,195),(124,196,128,200),(129,157,133,153),(130,154,134,158),(131,159,135,155),(132,156,136,160),(137,150,141,146),(138,147,142,151),(139,152,143,148),(140,149,144,145),(169,225,173,229),(170,230,174,226),(171,227,175,231),(172,232,176,228)], [(1,175,167,77,37),(2,78,176,38,168),(3,39,79,161,169),(4,162,40,170,80),(5,171,163,73,33),(6,74,172,34,164),(7,35,75,165,173),(8,166,36,174,76),(9,191,121,65,135),(10,66,192,136,122),(11,129,67,123,185),(12,124,130,186,68),(13,187,125,69,131),(14,70,188,132,126),(15,133,71,127,189),(16,128,134,190,72),(17,209,153,217,195),(18,218,210,196,154),(19,197,219,155,211),(20,156,198,212,220),(21,213,157,221,199),(22,222,214,200,158),(23,193,223,159,215),(24,160,194,216,224),(25,95,207,151,81),(26,152,96,82,208),(27,83,145,201,89),(28,202,84,90,146),(29,91,203,147,85),(30,148,92,86,204),(31,87,149,205,93),(32,206,88,94,150),(41,97,111,53,227),(42,54,98,228,112),(43,229,55,105,99),(44,106,230,100,56),(45,101,107,49,231),(46,50,102,232,108),(47,225,51,109,103),(48,110,226,104,52),(57,140,117,179,239),(58,180,141,240,118),(59,233,181,119,142),(60,120,234,143,182),(61,144,113,183,235),(62,184,137,236,114),(63,237,177,115,138),(64,116,238,139,178)], [(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)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | ··· | 8H | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 15A | 15B | 20A | 20B | 20C | 20D | 24A | ··· | 24P | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 4 | 1 | ··· | 1 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 5 | ··· | 5 | 4 | 4 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | C12 | M4(2) | C3×M4(2) | F5 | C2×F5 | C2×F5 | C3×F5 | C4.F5 | C6×F5 | C6×F5 | C3×C4.F5 |
| kernel | C6×C4.F5 | C3×C4.F5 | C6×C5⋊C8 | D5×C2×C12 | C2×C4.F5 | D5×C12 | C2×C60 | D5×C2×C6 | C4.F5 | C2×C5⋊C8 | C2×C4×D5 | C4×D5 | C2×C20 | C22×D5 | C30 | C10 | C2×C12 | C12 | C2×C6 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 2 | 1 | 2 | 4 | 2 | 2 | 8 | 4 | 2 | 8 | 4 | 4 | 4 | 8 | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 |
Matrix representation of C6×C4.F5 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 177 | 0 | 0 | 0 | 0 | 0 |
| 162 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 | 1 | 0 |
| 0 | 0 | 240 | 0 | 0 | 1 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 5 | 239 | 0 | 0 | 0 | 0 |
| 101 | 236 | 0 | 0 | 0 | 0 |
| 0 | 0 | 207 | 34 | 220 | 0 |
| 0 | 0 | 186 | 34 | 0 | 207 |
| 0 | 0 | 207 | 0 | 34 | 186 |
| 0 | 0 | 0 | 220 | 34 | 207 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[177,162,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,240,240,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[5,101,0,0,0,0,239,236,0,0,0,0,0,0,207,186,207,0,0,0,34,34,0,220,0,0,220,0,34,34,0,0,0,207,186,207] >;
C6×C4.F5 in GAP, Magma, Sage, TeX
C_6\times C_4.F_5
% in TeX
G:=Group("C6xC4.F5"); // GroupNames label
G:=SmallGroup(480,1048);
// by ID
G=gap.SmallGroup(480,1048);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,1094,268,102,9414,818]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^4=c^5=1,d^4=b^2,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^3>;
// generators/relations