direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C40.6C4, C40.6C12, C120.16C4, C12.70D20, C60.187D4, C24.2Dic5, (C2×C40).7C6, C20.34(C3×D4), C4.18(C3×D20), (C2×C24).11D5, C30.50(C4⋊C4), (C2×C30).16Q8, C8.1(C3×Dic5), C4.8(C6×Dic5), C60.242(C2×C4), (C2×C120).15C2, C20.57(C2×C12), C15⋊11(C8.C4), C4.Dic5.1C6, (C2×C12).424D10, C6.14(C4⋊Dic5), C12.47(C2×Dic5), (C2×C6).10Dic10, (C2×C60).517C22, C22.2(C3×Dic10), C5⋊3(C3×C8.C4), (C2×C8).5(C3×D5), C10.14(C3×C4⋊C4), (C2×C4).71(C6×D5), C2.5(C3×C4⋊Dic5), (C2×C10).3(C3×Q8), (C2×C20).100(C2×C6), (C3×C4.Dic5).5C2, SmallGroup(480,97)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C40.6C4
G = < a,b,c | a3=b40=1, c4=b20, ab=ba, ac=ca, cbc-1=b19 >
Subgroups: 128 in 60 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, C8, C8, C2×C4, C10, C10, C12, C2×C6, C15, C2×C8, M4(2), C20, C2×C10, C24, C24, C2×C12, C30, C30, C8.C4, C5⋊2C8, C40, C2×C20, C2×C24, C3×M4(2), C60, C2×C30, C4.Dic5, C2×C40, C3×C8.C4, C3×C5⋊2C8, C120, C2×C60, C40.6C4, C3×C4.Dic5, C2×C120, C3×C40.6C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, D5, C12, C2×C6, C4⋊C4, Dic5, D10, C2×C12, C3×D4, C3×Q8, C3×D5, C8.C4, Dic10, D20, C2×Dic5, C3×C4⋊C4, C3×Dic5, C6×D5, C4⋊Dic5, C3×C8.C4, C3×Dic10, C3×D20, C6×Dic5, C40.6C4, C3×C4⋊Dic5, C3×C40.6C4
►On 240 points
Generators in S240
(1 48 227)(2 49 228)(3 50 229)(4 51 230)(5 52 231)(6 53 232)(7 54 233)(8 55 234)(9 56 235)(10 57 236)(11 58 237)(12 59 238)(13 60 239)(14 61 240)(15 62 201)(16 63 202)(17 64 203)(18 65 204)(19 66 205)(20 67 206)(21 68 207)(22 69 208)(23 70 209)(24 71 210)(25 72 211)(26 73 212)(27 74 213)(28 75 214)(29 76 215)(30 77 216)(31 78 217)(32 79 218)(33 80 219)(34 41 220)(35 42 221)(36 43 222)(37 44 223)(38 45 224)(39 46 225)(40 47 226)(81 156 176)(82 157 177)(83 158 178)(84 159 179)(85 160 180)(86 121 181)(87 122 182)(88 123 183)(89 124 184)(90 125 185)(91 126 186)(92 127 187)(93 128 188)(94 129 189)(95 130 190)(96 131 191)(97 132 192)(98 133 193)(99 134 194)(100 135 195)(101 136 196)(102 137 197)(103 138 198)(104 139 199)(105 140 200)(106 141 161)(107 142 162)(108 143 163)(109 144 164)(110 145 165)(111 146 166)(112 147 167)(113 148 168)(114 149 169)(115 150 170)(116 151 171)(117 152 172)(118 153 173)(119 154 174)(120 155 175)
(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)
(1 164 31 174 21 184 11 194)(2 183 32 193 22 163 12 173)(3 162 33 172 23 182 13 192)(4 181 34 191 24 161 14 171)(5 200 35 170 25 180 15 190)(6 179 36 189 26 199 16 169)(7 198 37 168 27 178 17 188)(8 177 38 187 28 197 18 167)(9 196 39 166 29 176 19 186)(10 175 40 185 30 195 20 165)(41 96 71 106 61 116 51 86)(42 115 72 85 62 95 52 105)(43 94 73 104 63 114 53 84)(44 113 74 83 64 93 54 103)(45 92 75 102 65 112 55 82)(46 111 76 81 66 91 56 101)(47 90 77 100 67 110 57 120)(48 109 78 119 68 89 58 99)(49 88 79 98 69 108 59 118)(50 107 80 117 70 87 60 97)(121 220 131 210 141 240 151 230)(122 239 132 229 142 219 152 209)(123 218 133 208 143 238 153 228)(124 237 134 227 144 217 154 207)(125 216 135 206 145 236 155 226)(126 235 136 225 146 215 156 205)(127 214 137 204 147 234 157 224)(128 233 138 223 148 213 158 203)(129 212 139 202 149 232 159 222)(130 231 140 221 150 211 160 201)
G:=sub<Sym(240)| (1,48,227)(2,49,228)(3,50,229)(4,51,230)(5,52,231)(6,53,232)(7,54,233)(8,55,234)(9,56,235)(10,57,236)(11,58,237)(12,59,238)(13,60,239)(14,61,240)(15,62,201)(16,63,202)(17,64,203)(18,65,204)(19,66,205)(20,67,206)(21,68,207)(22,69,208)(23,70,209)(24,71,210)(25,72,211)(26,73,212)(27,74,213)(28,75,214)(29,76,215)(30,77,216)(31,78,217)(32,79,218)(33,80,219)(34,41,220)(35,42,221)(36,43,222)(37,44,223)(38,45,224)(39,46,225)(40,47,226)(81,156,176)(82,157,177)(83,158,178)(84,159,179)(85,160,180)(86,121,181)(87,122,182)(88,123,183)(89,124,184)(90,125,185)(91,126,186)(92,127,187)(93,128,188)(94,129,189)(95,130,190)(96,131,191)(97,132,192)(98,133,193)(99,134,194)(100,135,195)(101,136,196)(102,137,197)(103,138,198)(104,139,199)(105,140,200)(106,141,161)(107,142,162)(108,143,163)(109,144,164)(110,145,165)(111,146,166)(112,147,167)(113,148,168)(114,149,169)(115,150,170)(116,151,171)(117,152,172)(118,153,173)(119,154,174)(120,155,175), (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), (1,164,31,174,21,184,11,194)(2,183,32,193,22,163,12,173)(3,162,33,172,23,182,13,192)(4,181,34,191,24,161,14,171)(5,200,35,170,25,180,15,190)(6,179,36,189,26,199,16,169)(7,198,37,168,27,178,17,188)(8,177,38,187,28,197,18,167)(9,196,39,166,29,176,19,186)(10,175,40,185,30,195,20,165)(41,96,71,106,61,116,51,86)(42,115,72,85,62,95,52,105)(43,94,73,104,63,114,53,84)(44,113,74,83,64,93,54,103)(45,92,75,102,65,112,55,82)(46,111,76,81,66,91,56,101)(47,90,77,100,67,110,57,120)(48,109,78,119,68,89,58,99)(49,88,79,98,69,108,59,118)(50,107,80,117,70,87,60,97)(121,220,131,210,141,240,151,230)(122,239,132,229,142,219,152,209)(123,218,133,208,143,238,153,228)(124,237,134,227,144,217,154,207)(125,216,135,206,145,236,155,226)(126,235,136,225,146,215,156,205)(127,214,137,204,147,234,157,224)(128,233,138,223,148,213,158,203)(129,212,139,202,149,232,159,222)(130,231,140,221,150,211,160,201)>;
G:=Group( (1,48,227)(2,49,228)(3,50,229)(4,51,230)(5,52,231)(6,53,232)(7,54,233)(8,55,234)(9,56,235)(10,57,236)(11,58,237)(12,59,238)(13,60,239)(14,61,240)(15,62,201)(16,63,202)(17,64,203)(18,65,204)(19,66,205)(20,67,206)(21,68,207)(22,69,208)(23,70,209)(24,71,210)(25,72,211)(26,73,212)(27,74,213)(28,75,214)(29,76,215)(30,77,216)(31,78,217)(32,79,218)(33,80,219)(34,41,220)(35,42,221)(36,43,222)(37,44,223)(38,45,224)(39,46,225)(40,47,226)(81,156,176)(82,157,177)(83,158,178)(84,159,179)(85,160,180)(86,121,181)(87,122,182)(88,123,183)(89,124,184)(90,125,185)(91,126,186)(92,127,187)(93,128,188)(94,129,189)(95,130,190)(96,131,191)(97,132,192)(98,133,193)(99,134,194)(100,135,195)(101,136,196)(102,137,197)(103,138,198)(104,139,199)(105,140,200)(106,141,161)(107,142,162)(108,143,163)(109,144,164)(110,145,165)(111,146,166)(112,147,167)(113,148,168)(114,149,169)(115,150,170)(116,151,171)(117,152,172)(118,153,173)(119,154,174)(120,155,175), (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), (1,164,31,174,21,184,11,194)(2,183,32,193,22,163,12,173)(3,162,33,172,23,182,13,192)(4,181,34,191,24,161,14,171)(5,200,35,170,25,180,15,190)(6,179,36,189,26,199,16,169)(7,198,37,168,27,178,17,188)(8,177,38,187,28,197,18,167)(9,196,39,166,29,176,19,186)(10,175,40,185,30,195,20,165)(41,96,71,106,61,116,51,86)(42,115,72,85,62,95,52,105)(43,94,73,104,63,114,53,84)(44,113,74,83,64,93,54,103)(45,92,75,102,65,112,55,82)(46,111,76,81,66,91,56,101)(47,90,77,100,67,110,57,120)(48,109,78,119,68,89,58,99)(49,88,79,98,69,108,59,118)(50,107,80,117,70,87,60,97)(121,220,131,210,141,240,151,230)(122,239,132,229,142,219,152,209)(123,218,133,208,143,238,153,228)(124,237,134,227,144,217,154,207)(125,216,135,206,145,236,155,226)(126,235,136,225,146,215,156,205)(127,214,137,204,147,234,157,224)(128,233,138,223,148,213,158,203)(129,212,139,202,149,232,159,222)(130,231,140,221,150,211,160,201) );
G=PermutationGroup([[(1,48,227),(2,49,228),(3,50,229),(4,51,230),(5,52,231),(6,53,232),(7,54,233),(8,55,234),(9,56,235),(10,57,236),(11,58,237),(12,59,238),(13,60,239),(14,61,240),(15,62,201),(16,63,202),(17,64,203),(18,65,204),(19,66,205),(20,67,206),(21,68,207),(22,69,208),(23,70,209),(24,71,210),(25,72,211),(26,73,212),(27,74,213),(28,75,214),(29,76,215),(30,77,216),(31,78,217),(32,79,218),(33,80,219),(34,41,220),(35,42,221),(36,43,222),(37,44,223),(38,45,224),(39,46,225),(40,47,226),(81,156,176),(82,157,177),(83,158,178),(84,159,179),(85,160,180),(86,121,181),(87,122,182),(88,123,183),(89,124,184),(90,125,185),(91,126,186),(92,127,187),(93,128,188),(94,129,189),(95,130,190),(96,131,191),(97,132,192),(98,133,193),(99,134,194),(100,135,195),(101,136,196),(102,137,197),(103,138,198),(104,139,199),(105,140,200),(106,141,161),(107,142,162),(108,143,163),(109,144,164),(110,145,165),(111,146,166),(112,147,167),(113,148,168),(114,149,169),(115,150,170),(116,151,171),(117,152,172),(118,153,173),(119,154,174),(120,155,175)], [(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)], [(1,164,31,174,21,184,11,194),(2,183,32,193,22,163,12,173),(3,162,33,172,23,182,13,192),(4,181,34,191,24,161,14,171),(5,200,35,170,25,180,15,190),(6,179,36,189,26,199,16,169),(7,198,37,168,27,178,17,188),(8,177,38,187,28,197,18,167),(9,196,39,166,29,176,19,186),(10,175,40,185,30,195,20,165),(41,96,71,106,61,116,51,86),(42,115,72,85,62,95,52,105),(43,94,73,104,63,114,53,84),(44,113,74,83,64,93,54,103),(45,92,75,102,65,112,55,82),(46,111,76,81,66,91,56,101),(47,90,77,100,67,110,57,120),(48,109,78,119,68,89,58,99),(49,88,79,98,69,108,59,118),(50,107,80,117,70,87,60,97),(121,220,131,210,141,240,151,230),(122,239,132,229,142,219,152,209),(123,218,133,208,143,238,153,228),(124,237,134,227,144,217,154,207),(125,216,135,206,145,236,155,226),(126,235,136,225,146,215,156,205),(127,214,137,204,147,234,157,224),(128,233,138,223,148,213,158,203),(129,212,139,202,149,232,159,222),(130,231,140,221,150,211,160,201)]])
138 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 24A | ··· | 24H | 24I | ··· | 24P | 30A | ··· | 30L | 40A | ··· | 40P | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 24 | ··· | 24 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 20 | ··· | 20 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
138 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | + | - | ||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | D4 | Q8 | D5 | Dic5 | D10 | C3×D4 | C3×Q8 | C3×D5 | C8.C4 | D20 | Dic10 | C3×Dic5 | C6×D5 | C3×C8.C4 | C3×D20 | C3×Dic10 | C40.6C4 | C3×C40.6C4 |
| kernel | C3×C40.6C4 | C3×C4.Dic5 | C2×C120 | C40.6C4 | C120 | C4.Dic5 | C2×C40 | C40 | C60 | C2×C30 | C2×C24 | C24 | C2×C12 | C20 | C2×C10 | C2×C8 | C15 | C12 | C2×C6 | C8 | C2×C4 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 4 | 8 | 8 | 8 | 16 | 32 |
Matrix representation of C3×C40.6C4 ►in GL2(𝔽241) generated by
| 225 | 0 |
| 0 | 225 |
| 193 | 0 |
| 0 | 236 |
| 0 | 1 |
| 64 | 0 |
G:=sub<GL(2,GF(241))| [225,0,0,225],[193,0,0,236],[0,64,1,0] >;
C3×C40.6C4 in GAP, Magma, Sage, TeX
C_3\times C_{40}._6C_4 % in TeX
G:=Group("C3xC40.6C4"); // GroupNames label
G:=SmallGroup(480,97);
// by ID
G=gap.SmallGroup(480,97);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,176,136,2524,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^3=b^40=1,c^4=b^20,a*b=b*a,a*c=c*a,c*b*c^-1=b^19>;
// generators/relations