direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C20.4C8, C40.9C12, C60.10C8, C20.4C24, C120.19C4, C24.83D10, C15⋊14M5(2), C24.6Dic5, C120.101C22, C5⋊2C16⋊5C6, (C2×C30).9C8, C8.22(C6×D5), C5⋊4(C3×M5(2)), (C2×C60).43C4, (C2×C10).5C24, (C2×C40).10C6, C30.66(C2×C8), C40.22(C2×C6), (C2×C24).17D5, C8.2(C3×Dic5), C12.4(C5⋊2C8), C60.249(C2×C4), C20.61(C2×C12), (C2×C120).26C2, C10.18(C2×C24), (C2×C20).20C12, C4.11(C6×Dic5), (C2×C12).14Dic5, C12.50(C2×Dic5), C4.(C3×C5⋊2C8), C2.4(C6×C5⋊2C8), (C2×C8).7(C3×D5), C22.(C3×C5⋊2C8), C6.14(C2×C5⋊2C8), (C3×C5⋊2C16)⋊12C2, (C2×C6).1(C5⋊2C8), (C2×C4).5(C3×Dic5), SmallGroup(480,90)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C20.4C8
G = < a,b,c | a3=b40=1, c4=b10, ab=ba, ac=ca, cbc-1=b29 >
Smallest permutation representation of C3×C20.4C8
►On 240 points
Generators in S240
(1 83 59)(2 84 60)(3 85 61)(4 86 62)(5 87 63)(6 88 64)(7 89 65)(8 90 66)(9 91 67)(10 92 68)(11 93 69)(12 94 70)(13 95 71)(14 96 72)(15 97 73)(16 98 74)(17 99 75)(18 100 76)(19 101 77)(20 102 78)(21 103 79)(22 104 80)(23 105 41)(24 106 42)(25 107 43)(26 108 44)(27 109 45)(28 110 46)(29 111 47)(30 112 48)(31 113 49)(32 114 50)(33 115 51)(34 116 52)(35 117 53)(36 118 54)(37 119 55)(38 120 56)(39 81 57)(40 82 58)(121 207 188)(122 208 189)(123 209 190)(124 210 191)(125 211 192)(126 212 193)(127 213 194)(128 214 195)(129 215 196)(130 216 197)(131 217 198)(132 218 199)(133 219 200)(134 220 161)(135 221 162)(136 222 163)(137 223 164)(138 224 165)(139 225 166)(140 226 167)(141 227 168)(142 228 169)(143 229 170)(144 230 171)(145 231 172)(146 232 173)(147 233 174)(148 234 175)(149 235 176)(150 236 177)(151 237 178)(152 238 179)(153 239 180)(154 240 181)(155 201 182)(156 202 183)(157 203 184)(158 204 185)(159 205 186)(160 206 187)
(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)
(1 125 26 130 11 135 36 140 21 145 6 150 31 155 16 160)(2 154 27 159 12 124 37 129 22 134 7 139 32 144 17 149)(3 143 28 148 13 153 38 158 23 123 8 128 33 133 18 138)(4 132 29 137 14 142 39 147 24 152 9 157 34 122 19 127)(5 121 30 126 15 131 40 136 25 141 10 146 35 151 20 156)(41 190 66 195 51 200 76 165 61 170 46 175 71 180 56 185)(42 179 67 184 52 189 77 194 62 199 47 164 72 169 57 174)(43 168 68 173 53 178 78 183 63 188 48 193 73 198 58 163)(44 197 69 162 54 167 79 172 64 177 49 182 74 187 59 192)(45 186 70 191 55 196 80 161 65 166 50 171 75 176 60 181)(81 233 106 238 91 203 116 208 101 213 86 218 111 223 96 228)(82 222 107 227 92 232 117 237 102 202 87 207 112 212 97 217)(83 211 108 216 93 221 118 226 103 231 88 236 113 201 98 206)(84 240 109 205 94 210 119 215 104 220 89 225 114 230 99 235)(85 229 110 234 95 239 120 204 105 209 90 214 115 219 100 224)
G:=sub<Sym(240)| (1,83,59)(2,84,60)(3,85,61)(4,86,62)(5,87,63)(6,88,64)(7,89,65)(8,90,66)(9,91,67)(10,92,68)(11,93,69)(12,94,70)(13,95,71)(14,96,72)(15,97,73)(16,98,74)(17,99,75)(18,100,76)(19,101,77)(20,102,78)(21,103,79)(22,104,80)(23,105,41)(24,106,42)(25,107,43)(26,108,44)(27,109,45)(28,110,46)(29,111,47)(30,112,48)(31,113,49)(32,114,50)(33,115,51)(34,116,52)(35,117,53)(36,118,54)(37,119,55)(38,120,56)(39,81,57)(40,82,58)(121,207,188)(122,208,189)(123,209,190)(124,210,191)(125,211,192)(126,212,193)(127,213,194)(128,214,195)(129,215,196)(130,216,197)(131,217,198)(132,218,199)(133,219,200)(134,220,161)(135,221,162)(136,222,163)(137,223,164)(138,224,165)(139,225,166)(140,226,167)(141,227,168)(142,228,169)(143,229,170)(144,230,171)(145,231,172)(146,232,173)(147,233,174)(148,234,175)(149,235,176)(150,236,177)(151,237,178)(152,238,179)(153,239,180)(154,240,181)(155,201,182)(156,202,183)(157,203,184)(158,204,185)(159,205,186)(160,206,187), (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), (1,125,26,130,11,135,36,140,21,145,6,150,31,155,16,160)(2,154,27,159,12,124,37,129,22,134,7,139,32,144,17,149)(3,143,28,148,13,153,38,158,23,123,8,128,33,133,18,138)(4,132,29,137,14,142,39,147,24,152,9,157,34,122,19,127)(5,121,30,126,15,131,40,136,25,141,10,146,35,151,20,156)(41,190,66,195,51,200,76,165,61,170,46,175,71,180,56,185)(42,179,67,184,52,189,77,194,62,199,47,164,72,169,57,174)(43,168,68,173,53,178,78,183,63,188,48,193,73,198,58,163)(44,197,69,162,54,167,79,172,64,177,49,182,74,187,59,192)(45,186,70,191,55,196,80,161,65,166,50,171,75,176,60,181)(81,233,106,238,91,203,116,208,101,213,86,218,111,223,96,228)(82,222,107,227,92,232,117,237,102,202,87,207,112,212,97,217)(83,211,108,216,93,221,118,226,103,231,88,236,113,201,98,206)(84,240,109,205,94,210,119,215,104,220,89,225,114,230,99,235)(85,229,110,234,95,239,120,204,105,209,90,214,115,219,100,224)>;
G:=Group( (1,83,59)(2,84,60)(3,85,61)(4,86,62)(5,87,63)(6,88,64)(7,89,65)(8,90,66)(9,91,67)(10,92,68)(11,93,69)(12,94,70)(13,95,71)(14,96,72)(15,97,73)(16,98,74)(17,99,75)(18,100,76)(19,101,77)(20,102,78)(21,103,79)(22,104,80)(23,105,41)(24,106,42)(25,107,43)(26,108,44)(27,109,45)(28,110,46)(29,111,47)(30,112,48)(31,113,49)(32,114,50)(33,115,51)(34,116,52)(35,117,53)(36,118,54)(37,119,55)(38,120,56)(39,81,57)(40,82,58)(121,207,188)(122,208,189)(123,209,190)(124,210,191)(125,211,192)(126,212,193)(127,213,194)(128,214,195)(129,215,196)(130,216,197)(131,217,198)(132,218,199)(133,219,200)(134,220,161)(135,221,162)(136,222,163)(137,223,164)(138,224,165)(139,225,166)(140,226,167)(141,227,168)(142,228,169)(143,229,170)(144,230,171)(145,231,172)(146,232,173)(147,233,174)(148,234,175)(149,235,176)(150,236,177)(151,237,178)(152,238,179)(153,239,180)(154,240,181)(155,201,182)(156,202,183)(157,203,184)(158,204,185)(159,205,186)(160,206,187), (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), (1,125,26,130,11,135,36,140,21,145,6,150,31,155,16,160)(2,154,27,159,12,124,37,129,22,134,7,139,32,144,17,149)(3,143,28,148,13,153,38,158,23,123,8,128,33,133,18,138)(4,132,29,137,14,142,39,147,24,152,9,157,34,122,19,127)(5,121,30,126,15,131,40,136,25,141,10,146,35,151,20,156)(41,190,66,195,51,200,76,165,61,170,46,175,71,180,56,185)(42,179,67,184,52,189,77,194,62,199,47,164,72,169,57,174)(43,168,68,173,53,178,78,183,63,188,48,193,73,198,58,163)(44,197,69,162,54,167,79,172,64,177,49,182,74,187,59,192)(45,186,70,191,55,196,80,161,65,166,50,171,75,176,60,181)(81,233,106,238,91,203,116,208,101,213,86,218,111,223,96,228)(82,222,107,227,92,232,117,237,102,202,87,207,112,212,97,217)(83,211,108,216,93,221,118,226,103,231,88,236,113,201,98,206)(84,240,109,205,94,210,119,215,104,220,89,225,114,230,99,235)(85,229,110,234,95,239,120,204,105,209,90,214,115,219,100,224) );
G=PermutationGroup([[(1,83,59),(2,84,60),(3,85,61),(4,86,62),(5,87,63),(6,88,64),(7,89,65),(8,90,66),(9,91,67),(10,92,68),(11,93,69),(12,94,70),(13,95,71),(14,96,72),(15,97,73),(16,98,74),(17,99,75),(18,100,76),(19,101,77),(20,102,78),(21,103,79),(22,104,80),(23,105,41),(24,106,42),(25,107,43),(26,108,44),(27,109,45),(28,110,46),(29,111,47),(30,112,48),(31,113,49),(32,114,50),(33,115,51),(34,116,52),(35,117,53),(36,118,54),(37,119,55),(38,120,56),(39,81,57),(40,82,58),(121,207,188),(122,208,189),(123,209,190),(124,210,191),(125,211,192),(126,212,193),(127,213,194),(128,214,195),(129,215,196),(130,216,197),(131,217,198),(132,218,199),(133,219,200),(134,220,161),(135,221,162),(136,222,163),(137,223,164),(138,224,165),(139,225,166),(140,226,167),(141,227,168),(142,228,169),(143,229,170),(144,230,171),(145,231,172),(146,232,173),(147,233,174),(148,234,175),(149,235,176),(150,236,177),(151,237,178),(152,238,179),(153,239,180),(154,240,181),(155,201,182),(156,202,183),(157,203,184),(158,204,185),(159,205,186),(160,206,187)], [(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)], [(1,125,26,130,11,135,36,140,21,145,6,150,31,155,16,160),(2,154,27,159,12,124,37,129,22,134,7,139,32,144,17,149),(3,143,28,148,13,153,38,158,23,123,8,128,33,133,18,138),(4,132,29,137,14,142,39,147,24,152,9,157,34,122,19,127),(5,121,30,126,15,131,40,136,25,141,10,146,35,151,20,156),(41,190,66,195,51,200,76,165,61,170,46,175,71,180,56,185),(42,179,67,184,52,189,77,194,62,199,47,164,72,169,57,174),(43,168,68,173,53,178,78,183,63,188,48,193,73,198,58,163),(44,197,69,162,54,167,79,172,64,177,49,182,74,187,59,192),(45,186,70,191,55,196,80,161,65,166,50,171,75,176,60,181),(81,233,106,238,91,203,116,208,101,213,86,218,111,223,96,228),(82,222,107,227,92,232,117,237,102,202,87,207,112,212,97,217),(83,211,108,216,93,221,118,226,103,231,88,236,113,201,98,206),(84,240,109,205,94,210,119,215,104,220,89,225,114,230,99,235),(85,229,110,234,95,239,120,204,105,209,90,214,115,219,100,224)]])
156 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 15C | 15D | 16A | ··· | 16H | 20A | ··· | 20H | 24A | ··· | 24H | 24I | 24J | 24K | 24L | 30A | ··· | 30L | 40A | ··· | 40P | 48A | ··· | 48P | 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 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 16 | ··· | 16 | 20 | ··· | 20 | 24 | ··· | 24 | 24 | 24 | 24 | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 48 | ··· | 48 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 1 | ··· | 1 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | |||||||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C8 | C12 | C12 | C24 | C24 | D5 | Dic5 | D10 | Dic5 | C3×D5 | M5(2) | C5⋊2C8 | C5⋊2C8 | C3×Dic5 | C6×D5 | C3×Dic5 | C3×M5(2) | C3×C5⋊2C8 | C3×C5⋊2C8 | C20.4C8 | C3×C20.4C8 |
| kernel | C3×C20.4C8 | C3×C5⋊2C16 | C2×C120 | C20.4C8 | C120 | C2×C60 | C5⋊2C16 | C2×C40 | C60 | C2×C30 | C40 | C2×C20 | C20 | C2×C10 | C2×C24 | C24 | C24 | C2×C12 | C2×C8 | C15 | C12 | C2×C6 | C8 | C8 | C2×C4 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 16 | 32 |
Matrix representation of C3×C20.4C8 ►in GL2(𝔽241) generated by
| 15 | 0 |
| 0 | 15 |
| 116 | 0 |
| 124 | 200 |
| 64 | 90 |
| 238 | 177 |
G:=sub<GL(2,GF(241))| [15,0,0,15],[116,124,0,200],[64,238,90,177] >;
C3×C20.4C8 in GAP, Magma, Sage, TeX
C_3\times C_{20}._4C_8 % in TeX
G:=Group("C3xC20.4C8"); // GroupNames label
G:=SmallGroup(480,90);
// by ID
G=gap.SmallGroup(480,90);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,701,80,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^3=b^40=1,c^4=b^10,a*b=b*a,a*c=c*a,c*b*c^-1=b^29>;
// generators/relations
Export