direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C12.C8, C60.11C8, C12.1C40, C24.5C20, C40.83D6, C120.22C4, C15⋊15M5(2), C40.11Dic3, C120.110C22, C3⋊C16⋊5C10, C20.7(C3⋊C8), (C2×C6).3C40, C6.9(C2×C40), C3⋊2(C5×M5(2)), C8.21(S3×C10), (C2×C40).17S3, C30.71(C2×C8), (C2×C30).11C8, (C2×C60).45C4, C8.2(C5×Dic3), (C2×C24).13C10, (C2×C120).33C2, C60.254(C2×C4), C12.39(C2×C20), C24.26(C2×C10), (C2×C12).10C20, C20.72(C2×Dic3), (C2×C20).25Dic3, C4.11(C10×Dic3), C4.(C5×C3⋊C8), C22.(C5×C3⋊C8), C2.4(C10×C3⋊C8), (C5×C3⋊C16)⋊12C2, C10.23(C2×C3⋊C8), (C2×C8).7(C5×S3), (C2×C10).3(C3⋊C8), (C2×C4).5(C5×Dic3), SmallGroup(480,131)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C12.C8
G = < a,b,c | a5=b24=1, c4=b18, ab=ba, ac=ca, cbc-1=b5 >
Smallest permutation representation of C5×C12.C8
►On 240 points
Generators in S240
(1 111 83 50 33)(2 112 84 51 34)(3 113 85 52 35)(4 114 86 53 36)(5 115 87 54 37)(6 116 88 55 38)(7 117 89 56 39)(8 118 90 57 40)(9 119 91 58 41)(10 120 92 59 42)(11 97 93 60 43)(12 98 94 61 44)(13 99 95 62 45)(14 100 96 63 46)(15 101 73 64 47)(16 102 74 65 48)(17 103 75 66 25)(18 104 76 67 26)(19 105 77 68 27)(20 106 78 69 28)(21 107 79 70 29)(22 108 80 71 30)(23 109 81 72 31)(24 110 82 49 32)(121 217 193 187 145)(122 218 194 188 146)(123 219 195 189 147)(124 220 196 190 148)(125 221 197 191 149)(126 222 198 192 150)(127 223 199 169 151)(128 224 200 170 152)(129 225 201 171 153)(130 226 202 172 154)(131 227 203 173 155)(132 228 204 174 156)(133 229 205 175 157)(134 230 206 176 158)(135 231 207 177 159)(136 232 208 178 160)(137 233 209 179 161)(138 234 210 180 162)(139 235 211 181 163)(140 236 212 182 164)(141 237 213 183 165)(142 238 214 184 166)(143 239 215 185 167)(144 240 216 186 168)
(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 10 139 19 136 4 133 13 130 22 127 7 124 16 121)(2 123 11 144 20 141 5 138 14 135 23 132 8 129 17 126)(3 128 12 125 21 122 6 143 15 140 24 137 9 134 18 131)(25 150 34 147 43 168 28 165 37 162 46 159 31 156 40 153)(26 155 35 152 44 149 29 146 38 167 47 164 32 161 41 158)(27 160 36 157 45 154 30 151 39 148 48 145 33 166 42 163)(49 179 58 176 67 173 52 170 61 191 70 188 55 185 64 182)(50 184 59 181 68 178 53 175 62 172 71 169 56 190 65 187)(51 189 60 186 69 183 54 180 63 177 72 174 57 171 66 192)(73 212 82 209 91 206 76 203 85 200 94 197 79 194 88 215)(74 193 83 214 92 211 77 208 86 205 95 202 80 199 89 196)(75 198 84 195 93 216 78 213 87 210 96 207 81 204 90 201)(97 240 106 237 115 234 100 231 109 228 118 225 103 222 112 219)(98 221 107 218 116 239 101 236 110 233 119 230 104 227 113 224)(99 226 108 223 117 220 102 217 111 238 120 235 105 232 114 229)
G:=sub<Sym(240)| (1,111,83,50,33)(2,112,84,51,34)(3,113,85,52,35)(4,114,86,53,36)(5,115,87,54,37)(6,116,88,55,38)(7,117,89,56,39)(8,118,90,57,40)(9,119,91,58,41)(10,120,92,59,42)(11,97,93,60,43)(12,98,94,61,44)(13,99,95,62,45)(14,100,96,63,46)(15,101,73,64,47)(16,102,74,65,48)(17,103,75,66,25)(18,104,76,67,26)(19,105,77,68,27)(20,106,78,69,28)(21,107,79,70,29)(22,108,80,71,30)(23,109,81,72,31)(24,110,82,49,32)(121,217,193,187,145)(122,218,194,188,146)(123,219,195,189,147)(124,220,196,190,148)(125,221,197,191,149)(126,222,198,192,150)(127,223,199,169,151)(128,224,200,170,152)(129,225,201,171,153)(130,226,202,172,154)(131,227,203,173,155)(132,228,204,174,156)(133,229,205,175,157)(134,230,206,176,158)(135,231,207,177,159)(136,232,208,178,160)(137,233,209,179,161)(138,234,210,180,162)(139,235,211,181,163)(140,236,212,182,164)(141,237,213,183,165)(142,238,214,184,166)(143,239,215,185,167)(144,240,216,186,168), (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), (1,142,10,139,19,136,4,133,13,130,22,127,7,124,16,121)(2,123,11,144,20,141,5,138,14,135,23,132,8,129,17,126)(3,128,12,125,21,122,6,143,15,140,24,137,9,134,18,131)(25,150,34,147,43,168,28,165,37,162,46,159,31,156,40,153)(26,155,35,152,44,149,29,146,38,167,47,164,32,161,41,158)(27,160,36,157,45,154,30,151,39,148,48,145,33,166,42,163)(49,179,58,176,67,173,52,170,61,191,70,188,55,185,64,182)(50,184,59,181,68,178,53,175,62,172,71,169,56,190,65,187)(51,189,60,186,69,183,54,180,63,177,72,174,57,171,66,192)(73,212,82,209,91,206,76,203,85,200,94,197,79,194,88,215)(74,193,83,214,92,211,77,208,86,205,95,202,80,199,89,196)(75,198,84,195,93,216,78,213,87,210,96,207,81,204,90,201)(97,240,106,237,115,234,100,231,109,228,118,225,103,222,112,219)(98,221,107,218,116,239,101,236,110,233,119,230,104,227,113,224)(99,226,108,223,117,220,102,217,111,238,120,235,105,232,114,229)>;
G:=Group( (1,111,83,50,33)(2,112,84,51,34)(3,113,85,52,35)(4,114,86,53,36)(5,115,87,54,37)(6,116,88,55,38)(7,117,89,56,39)(8,118,90,57,40)(9,119,91,58,41)(10,120,92,59,42)(11,97,93,60,43)(12,98,94,61,44)(13,99,95,62,45)(14,100,96,63,46)(15,101,73,64,47)(16,102,74,65,48)(17,103,75,66,25)(18,104,76,67,26)(19,105,77,68,27)(20,106,78,69,28)(21,107,79,70,29)(22,108,80,71,30)(23,109,81,72,31)(24,110,82,49,32)(121,217,193,187,145)(122,218,194,188,146)(123,219,195,189,147)(124,220,196,190,148)(125,221,197,191,149)(126,222,198,192,150)(127,223,199,169,151)(128,224,200,170,152)(129,225,201,171,153)(130,226,202,172,154)(131,227,203,173,155)(132,228,204,174,156)(133,229,205,175,157)(134,230,206,176,158)(135,231,207,177,159)(136,232,208,178,160)(137,233,209,179,161)(138,234,210,180,162)(139,235,211,181,163)(140,236,212,182,164)(141,237,213,183,165)(142,238,214,184,166)(143,239,215,185,167)(144,240,216,186,168), (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), (1,142,10,139,19,136,4,133,13,130,22,127,7,124,16,121)(2,123,11,144,20,141,5,138,14,135,23,132,8,129,17,126)(3,128,12,125,21,122,6,143,15,140,24,137,9,134,18,131)(25,150,34,147,43,168,28,165,37,162,46,159,31,156,40,153)(26,155,35,152,44,149,29,146,38,167,47,164,32,161,41,158)(27,160,36,157,45,154,30,151,39,148,48,145,33,166,42,163)(49,179,58,176,67,173,52,170,61,191,70,188,55,185,64,182)(50,184,59,181,68,178,53,175,62,172,71,169,56,190,65,187)(51,189,60,186,69,183,54,180,63,177,72,174,57,171,66,192)(73,212,82,209,91,206,76,203,85,200,94,197,79,194,88,215)(74,193,83,214,92,211,77,208,86,205,95,202,80,199,89,196)(75,198,84,195,93,216,78,213,87,210,96,207,81,204,90,201)(97,240,106,237,115,234,100,231,109,228,118,225,103,222,112,219)(98,221,107,218,116,239,101,236,110,233,119,230,104,227,113,224)(99,226,108,223,117,220,102,217,111,238,120,235,105,232,114,229) );
G=PermutationGroup([[(1,111,83,50,33),(2,112,84,51,34),(3,113,85,52,35),(4,114,86,53,36),(5,115,87,54,37),(6,116,88,55,38),(7,117,89,56,39),(8,118,90,57,40),(9,119,91,58,41),(10,120,92,59,42),(11,97,93,60,43),(12,98,94,61,44),(13,99,95,62,45),(14,100,96,63,46),(15,101,73,64,47),(16,102,74,65,48),(17,103,75,66,25),(18,104,76,67,26),(19,105,77,68,27),(20,106,78,69,28),(21,107,79,70,29),(22,108,80,71,30),(23,109,81,72,31),(24,110,82,49,32),(121,217,193,187,145),(122,218,194,188,146),(123,219,195,189,147),(124,220,196,190,148),(125,221,197,191,149),(126,222,198,192,150),(127,223,199,169,151),(128,224,200,170,152),(129,225,201,171,153),(130,226,202,172,154),(131,227,203,173,155),(132,228,204,174,156),(133,229,205,175,157),(134,230,206,176,158),(135,231,207,177,159),(136,232,208,178,160),(137,233,209,179,161),(138,234,210,180,162),(139,235,211,181,163),(140,236,212,182,164),(141,237,213,183,165),(142,238,214,184,166),(143,239,215,185,167),(144,240,216,186,168)], [(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)], [(1,142,10,139,19,136,4,133,13,130,22,127,7,124,16,121),(2,123,11,144,20,141,5,138,14,135,23,132,8,129,17,126),(3,128,12,125,21,122,6,143,15,140,24,137,9,134,18,131),(25,150,34,147,43,168,28,165,37,162,46,159,31,156,40,153),(26,155,35,152,44,149,29,146,38,167,47,164,32,161,41,158),(27,160,36,157,45,154,30,151,39,148,48,145,33,166,42,163),(49,179,58,176,67,173,52,170,61,191,70,188,55,185,64,182),(50,184,59,181,68,178,53,175,62,172,71,169,56,190,65,187),(51,189,60,186,69,183,54,180,63,177,72,174,57,171,66,192),(73,212,82,209,91,206,76,203,85,200,94,197,79,194,88,215),(74,193,83,214,92,211,77,208,86,205,95,202,80,199,89,196),(75,198,84,195,93,216,78,213,87,210,96,207,81,204,90,201),(97,240,106,237,115,234,100,231,109,228,118,225,103,222,112,219),(98,221,107,218,116,239,101,236,110,233,119,230,104,227,113,224),(99,226,108,223,117,220,102,217,111,238,120,235,105,232,114,229)]])
180 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 16A | ··· | 16H | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 24A | ··· | 24H | 30A | ··· | 30L | 40A | ··· | 40P | 40Q | ··· | 40X | 60A | ··· | 60P | 80A | ··· | 80AF | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 16 | ··· | 16 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 40 | ··· | 40 | 60 | ··· | 60 | 80 | ··· | 80 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
180 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 | C4 | C4 | C5 | C8 | C8 | C10 | C10 | C20 | C20 | C40 | C40 | S3 | Dic3 | D6 | Dic3 | C3⋊C8 | C3⋊C8 | C5×S3 | M5(2) | C5×Dic3 | S3×C10 | C5×Dic3 | C12.C8 | C5×C3⋊C8 | C5×C3⋊C8 | C5×M5(2) | C5×C12.C8 |
| kernel | C5×C12.C8 | C5×C3⋊C16 | C2×C120 | C120 | C2×C60 | C12.C8 | C60 | C2×C30 | C3⋊C16 | C2×C24 | C24 | C2×C12 | C12 | C2×C6 | C2×C40 | C40 | C40 | C2×C20 | C20 | C2×C10 | C2×C8 | C15 | C8 | C8 | C2×C4 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 4 | 8 | 8 | 16 | 16 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 16 | 32 |
Matrix representation of C5×C12.C8 ►in GL2(𝔽241) generated by
| 98 | 0 |
| 0 | 98 |
| 239 | 0 |
| 0 | 209 |
| 0 | 1 |
| 211 | 0 |
G:=sub<GL(2,GF(241))| [98,0,0,98],[239,0,0,209],[0,211,1,0] >;
C5×C12.C8 in GAP, Magma, Sage, TeX
C_5\times C_{12}.C_8 % in TeX
G:=Group("C5xC12.C8"); // GroupNames label
G:=SmallGroup(480,131);
// by ID
G=gap.SmallGroup(480,131);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,140,1149,80,102,15686]);
// Polycyclic
G:=Group<a,b,c|a^5=b^24=1,c^4=b^18,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations
Export