direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C20.C8, C60.4C8, C20.1C24, C15⋊9M5(2), C5⋊C16⋊2C6, C12.4(C5⋊C8), (C2×C30).4C8, C5⋊2(C3×M5(2)), C4.18(C6×F5), C5⋊2C8.6C12, (C2×C60).18C4, (C2×C10).2C24, C10.6(C2×C24), C30.40(C2×C8), C60.71(C2×C4), (C2×C20).7C12, C12.71(C2×F5), (C2×C12).16F5, C20.18(C2×C12), C4.(C3×C5⋊C8), C2.3(C6×C5⋊C8), (C3×C5⋊C16)⋊6C2, C22.(C3×C5⋊C8), C6.12(C2×C5⋊C8), (C2×C6).1(C5⋊C8), (C2×C4).5(C3×F5), (C6×C5⋊2C8).25C2, (C3×C5⋊2C8).13C4, (C2×C5⋊2C8).11C6, C5⋊2C8.16(C2×C6), (C3×C5⋊2C8).60C22, SmallGroup(480,278)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C20.C8
G = < a,b,c | a3=b20=1, c8=b10, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of C3×C20.C8
►On 240 points
Generators in S240
(1 93 52)(2 94 53)(3 95 54)(4 96 55)(5 97 56)(6 98 57)(7 99 58)(8 100 59)(9 81 60)(10 82 41)(11 83 42)(12 84 43)(13 85 44)(14 86 45)(15 87 46)(16 88 47)(17 89 48)(18 90 49)(19 91 50)(20 92 51)(21 106 71)(22 107 72)(23 108 73)(24 109 74)(25 110 75)(26 111 76)(27 112 77)(28 113 78)(29 114 79)(30 115 80)(31 116 61)(32 117 62)(33 118 63)(34 119 64)(35 120 65)(36 101 66)(37 102 67)(38 103 68)(39 104 69)(40 105 70)(121 219 170)(122 220 171)(123 201 172)(124 202 173)(125 203 174)(126 204 175)(127 205 176)(128 206 177)(129 207 178)(130 208 179)(131 209 180)(132 210 161)(133 211 162)(134 212 163)(135 213 164)(136 214 165)(137 215 166)(138 216 167)(139 217 168)(140 218 169)(141 221 181)(142 222 182)(143 223 183)(144 224 184)(145 225 185)(146 226 186)(147 227 187)(148 228 188)(149 229 189)(150 230 190)(151 231 191)(152 232 192)(153 233 193)(154 234 194)(155 235 195)(156 236 196)(157 237 197)(158 238 198)(159 239 199)(160 240 200)
(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)
(1 142 21 136 6 157 26 131 11 152 31 126 16 147 36 121)(2 149 30 139 7 144 35 134 12 159 40 129 17 154 25 124)(3 156 39 122 8 151 24 137 13 146 29 132 18 141 34 127)(4 143 28 125 9 158 33 140 14 153 38 135 19 148 23 130)(5 150 37 128 10 145 22 123 15 160 27 138 20 155 32 133)(41 185 72 172 46 200 77 167 51 195 62 162 56 190 67 177)(42 192 61 175 47 187 66 170 52 182 71 165 57 197 76 180)(43 199 70 178 48 194 75 173 53 189 80 168 58 184 65 163)(44 186 79 161 49 181 64 176 54 196 69 171 59 191 74 166)(45 193 68 164 50 188 73 179 55 183 78 174 60 198 63 169)(81 238 118 218 86 233 103 213 91 228 108 208 96 223 113 203)(82 225 107 201 87 240 112 216 92 235 117 211 97 230 102 206)(83 232 116 204 88 227 101 219 93 222 106 214 98 237 111 209)(84 239 105 207 89 234 110 202 94 229 115 217 99 224 120 212)(85 226 114 210 90 221 119 205 95 236 104 220 100 231 109 215)
G:=sub<Sym(240)| (1,93,52)(2,94,53)(3,95,54)(4,96,55)(5,97,56)(6,98,57)(7,99,58)(8,100,59)(9,81,60)(10,82,41)(11,83,42)(12,84,43)(13,85,44)(14,86,45)(15,87,46)(16,88,47)(17,89,48)(18,90,49)(19,91,50)(20,92,51)(21,106,71)(22,107,72)(23,108,73)(24,109,74)(25,110,75)(26,111,76)(27,112,77)(28,113,78)(29,114,79)(30,115,80)(31,116,61)(32,117,62)(33,118,63)(34,119,64)(35,120,65)(36,101,66)(37,102,67)(38,103,68)(39,104,69)(40,105,70)(121,219,170)(122,220,171)(123,201,172)(124,202,173)(125,203,174)(126,204,175)(127,205,176)(128,206,177)(129,207,178)(130,208,179)(131,209,180)(132,210,161)(133,211,162)(134,212,163)(135,213,164)(136,214,165)(137,215,166)(138,216,167)(139,217,168)(140,218,169)(141,221,181)(142,222,182)(143,223,183)(144,224,184)(145,225,185)(146,226,186)(147,227,187)(148,228,188)(149,229,189)(150,230,190)(151,231,191)(152,232,192)(153,233,193)(154,234,194)(155,235,195)(156,236,196)(157,237,197)(158,238,198)(159,239,199)(160,240,200), (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), (1,142,21,136,6,157,26,131,11,152,31,126,16,147,36,121)(2,149,30,139,7,144,35,134,12,159,40,129,17,154,25,124)(3,156,39,122,8,151,24,137,13,146,29,132,18,141,34,127)(4,143,28,125,9,158,33,140,14,153,38,135,19,148,23,130)(5,150,37,128,10,145,22,123,15,160,27,138,20,155,32,133)(41,185,72,172,46,200,77,167,51,195,62,162,56,190,67,177)(42,192,61,175,47,187,66,170,52,182,71,165,57,197,76,180)(43,199,70,178,48,194,75,173,53,189,80,168,58,184,65,163)(44,186,79,161,49,181,64,176,54,196,69,171,59,191,74,166)(45,193,68,164,50,188,73,179,55,183,78,174,60,198,63,169)(81,238,118,218,86,233,103,213,91,228,108,208,96,223,113,203)(82,225,107,201,87,240,112,216,92,235,117,211,97,230,102,206)(83,232,116,204,88,227,101,219,93,222,106,214,98,237,111,209)(84,239,105,207,89,234,110,202,94,229,115,217,99,224,120,212)(85,226,114,210,90,221,119,205,95,236,104,220,100,231,109,215)>;
G:=Group( (1,93,52)(2,94,53)(3,95,54)(4,96,55)(5,97,56)(6,98,57)(7,99,58)(8,100,59)(9,81,60)(10,82,41)(11,83,42)(12,84,43)(13,85,44)(14,86,45)(15,87,46)(16,88,47)(17,89,48)(18,90,49)(19,91,50)(20,92,51)(21,106,71)(22,107,72)(23,108,73)(24,109,74)(25,110,75)(26,111,76)(27,112,77)(28,113,78)(29,114,79)(30,115,80)(31,116,61)(32,117,62)(33,118,63)(34,119,64)(35,120,65)(36,101,66)(37,102,67)(38,103,68)(39,104,69)(40,105,70)(121,219,170)(122,220,171)(123,201,172)(124,202,173)(125,203,174)(126,204,175)(127,205,176)(128,206,177)(129,207,178)(130,208,179)(131,209,180)(132,210,161)(133,211,162)(134,212,163)(135,213,164)(136,214,165)(137,215,166)(138,216,167)(139,217,168)(140,218,169)(141,221,181)(142,222,182)(143,223,183)(144,224,184)(145,225,185)(146,226,186)(147,227,187)(148,228,188)(149,229,189)(150,230,190)(151,231,191)(152,232,192)(153,233,193)(154,234,194)(155,235,195)(156,236,196)(157,237,197)(158,238,198)(159,239,199)(160,240,200), (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), (1,142,21,136,6,157,26,131,11,152,31,126,16,147,36,121)(2,149,30,139,7,144,35,134,12,159,40,129,17,154,25,124)(3,156,39,122,8,151,24,137,13,146,29,132,18,141,34,127)(4,143,28,125,9,158,33,140,14,153,38,135,19,148,23,130)(5,150,37,128,10,145,22,123,15,160,27,138,20,155,32,133)(41,185,72,172,46,200,77,167,51,195,62,162,56,190,67,177)(42,192,61,175,47,187,66,170,52,182,71,165,57,197,76,180)(43,199,70,178,48,194,75,173,53,189,80,168,58,184,65,163)(44,186,79,161,49,181,64,176,54,196,69,171,59,191,74,166)(45,193,68,164,50,188,73,179,55,183,78,174,60,198,63,169)(81,238,118,218,86,233,103,213,91,228,108,208,96,223,113,203)(82,225,107,201,87,240,112,216,92,235,117,211,97,230,102,206)(83,232,116,204,88,227,101,219,93,222,106,214,98,237,111,209)(84,239,105,207,89,234,110,202,94,229,115,217,99,224,120,212)(85,226,114,210,90,221,119,205,95,236,104,220,100,231,109,215) );
G=PermutationGroup([[(1,93,52),(2,94,53),(3,95,54),(4,96,55),(5,97,56),(6,98,57),(7,99,58),(8,100,59),(9,81,60),(10,82,41),(11,83,42),(12,84,43),(13,85,44),(14,86,45),(15,87,46),(16,88,47),(17,89,48),(18,90,49),(19,91,50),(20,92,51),(21,106,71),(22,107,72),(23,108,73),(24,109,74),(25,110,75),(26,111,76),(27,112,77),(28,113,78),(29,114,79),(30,115,80),(31,116,61),(32,117,62),(33,118,63),(34,119,64),(35,120,65),(36,101,66),(37,102,67),(38,103,68),(39,104,69),(40,105,70),(121,219,170),(122,220,171),(123,201,172),(124,202,173),(125,203,174),(126,204,175),(127,205,176),(128,206,177),(129,207,178),(130,208,179),(131,209,180),(132,210,161),(133,211,162),(134,212,163),(135,213,164),(136,214,165),(137,215,166),(138,216,167),(139,217,168),(140,218,169),(141,221,181),(142,222,182),(143,223,183),(144,224,184),(145,225,185),(146,226,186),(147,227,187),(148,228,188),(149,229,189),(150,230,190),(151,231,191),(152,232,192),(153,233,193),(154,234,194),(155,235,195),(156,236,196),(157,237,197),(158,238,198),(159,239,199),(160,240,200)], [(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)], [(1,142,21,136,6,157,26,131,11,152,31,126,16,147,36,121),(2,149,30,139,7,144,35,134,12,159,40,129,17,154,25,124),(3,156,39,122,8,151,24,137,13,146,29,132,18,141,34,127),(4,143,28,125,9,158,33,140,14,153,38,135,19,148,23,130),(5,150,37,128,10,145,22,123,15,160,27,138,20,155,32,133),(41,185,72,172,46,200,77,167,51,195,62,162,56,190,67,177),(42,192,61,175,47,187,66,170,52,182,71,165,57,197,76,180),(43,199,70,178,48,194,75,173,53,189,80,168,58,184,65,163),(44,186,79,161,49,181,64,176,54,196,69,171,59,191,74,166),(45,193,68,164,50,188,73,179,55,183,78,174,60,198,63,169),(81,238,118,218,86,233,103,213,91,228,108,208,96,223,113,203),(82,225,107,201,87,240,112,216,92,235,117,211,97,230,102,206),(83,232,116,204,88,227,101,219,93,222,106,214,98,237,111,209),(84,239,105,207,89,234,110,202,94,229,115,217,99,224,120,212),(85,226,114,210,90,221,119,205,95,236,104,220,100,231,109,215)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 5 | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 16A | ··· | 16H | 20A | 20B | 20C | 20D | 24A | ··· | 24H | 24I | 24J | 24K | 24L | 30A | ··· | 30F | 48A | ··· | 48P | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 24 | 24 | 24 | 24 | 30 | ··· | 30 | 48 | ··· | 48 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 4 | ··· | 4 | 10 | ··· | 10 | 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 | 4 | 4 |
| type | + | + | + | + | - | + | - | |||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C8 | C12 | C12 | C24 | C24 | M5(2) | C3×M5(2) | F5 | C5⋊C8 | C2×F5 | C5⋊C8 | C3×F5 | C3×C5⋊C8 | C6×F5 | C3×C5⋊C8 | C20.C8 | C3×C20.C8 |
| kernel | C3×C20.C8 | C3×C5⋊C16 | C6×C5⋊2C8 | C20.C8 | C3×C5⋊2C8 | C2×C60 | C5⋊C16 | C2×C5⋊2C8 | C60 | C2×C30 | C5⋊2C8 | C2×C20 | C20 | C2×C10 | C15 | C5 | C2×C12 | C12 | C12 | C2×C6 | C2×C4 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of C3×C20.C8 ►in GL6(𝔽241)
| 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 15 | 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 |
| 177 | 0 | 0 | 0 | 0 | 0 |
| 132 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 64 | 0 | 0 |
| 0 | 0 | 177 | 131 | 0 | 0 |
| 0 | 0 | 0 | 0 | 64 | 110 |
| 0 | 0 | 0 | 0 | 131 | 131 |
| 13 | 239 | 0 | 0 | 0 | 0 |
| 220 | 228 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 231 | 108 | 0 | 0 |
| 0 | 0 | 80 | 10 | 0 | 0 |
G:=sub<GL(6,GF(241))| [15,0,0,0,0,0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[177,132,0,0,0,0,0,64,0,0,0,0,0,0,0,177,0,0,0,0,64,131,0,0,0,0,0,0,64,131,0,0,0,0,110,131],[13,220,0,0,0,0,239,228,0,0,0,0,0,0,0,0,231,80,0,0,0,0,108,10,0,0,1,0,0,0,0,0,0,1,0,0] >;
C3×C20.C8 in GAP, Magma, Sage, TeX
C_3\times C_{20}.C_8 % in TeX
G:=Group("C3xC20.C8"); // GroupNames label
G:=SmallGroup(480,278);
// by ID
G=gap.SmallGroup(480,278);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,701,80,102,9414,1595]);
// Polycyclic
G:=Group<a,b,c|a^3=b^20=1,c^8=b^10,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
Export