direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C40.9C4, 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)), C40.22(C2×C6), C30.66(C2×C8), (C2×C40).10C6, (C2×C10).5C24, (C2×C60).43C4, (C2×C24).17D5, C8.2(C3×Dic5), C12.4(C5⋊2C8), (C2×C20).20C12, C60.249(C2×C4), C20.61(C2×C12), (C2×C120).26C2, C10.18(C2×C24), C4.11(C6×Dic5), C12.50(C2×Dic5), (C2×C12).14Dic5, 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×C40.9C4
G = < a,b,c | a3=b40=1, c4=b10, ab=ba, ac=ca, cbc-1=b29 >
Smallest permutation representation of C3×C40.9C4
►On 240 points
Generators in S240
(1 84 79)(2 85 80)(3 86 41)(4 87 42)(5 88 43)(6 89 44)(7 90 45)(8 91 46)(9 92 47)(10 93 48)(11 94 49)(12 95 50)(13 96 51)(14 97 52)(15 98 53)(16 99 54)(17 100 55)(18 101 56)(19 102 57)(20 103 58)(21 104 59)(22 105 60)(23 106 61)(24 107 62)(25 108 63)(26 109 64)(27 110 65)(28 111 66)(29 112 67)(30 113 68)(31 114 69)(32 115 70)(33 116 71)(34 117 72)(35 118 73)(36 119 74)(37 120 75)(38 81 76)(39 82 77)(40 83 78)(121 233 199)(122 234 200)(123 235 161)(124 236 162)(125 237 163)(126 238 164)(127 239 165)(128 240 166)(129 201 167)(130 202 168)(131 203 169)(132 204 170)(133 205 171)(134 206 172)(135 207 173)(136 208 174)(137 209 175)(138 210 176)(139 211 177)(140 212 178)(141 213 179)(142 214 180)(143 215 181)(144 216 182)(145 217 183)(146 218 184)(147 219 185)(148 220 186)(149 221 187)(150 222 188)(151 223 189)(152 224 190)(153 225 191)(154 226 192)(155 227 193)(156 228 194)(157 229 195)(158 230 196)(159 231 197)(160 232 198)
(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)
(1 133 26 138 11 143 36 148 21 153 6 158 31 123 16 128)(2 122 27 127 12 132 37 137 22 142 7 147 32 152 17 157)(3 151 28 156 13 121 38 126 23 131 8 136 33 141 18 146)(4 140 29 145 14 150 39 155 24 160 9 125 34 130 19 135)(5 129 30 134 15 139 40 144 25 149 10 154 35 159 20 124)(41 189 66 194 51 199 76 164 61 169 46 174 71 179 56 184)(42 178 67 183 52 188 77 193 62 198 47 163 72 168 57 173)(43 167 68 172 53 177 78 182 63 187 48 192 73 197 58 162)(44 196 69 161 54 166 79 171 64 176 49 181 74 186 59 191)(45 185 70 190 55 195 80 200 65 165 50 170 75 175 60 180)(81 238 106 203 91 208 116 213 101 218 86 223 111 228 96 233)(82 227 107 232 92 237 117 202 102 207 87 212 112 217 97 222)(83 216 108 221 93 226 118 231 103 236 88 201 113 206 98 211)(84 205 109 210 94 215 119 220 104 225 89 230 114 235 99 240)(85 234 110 239 95 204 120 209 105 214 90 219 115 224 100 229)
G:=sub<Sym(240)| (1,84,79)(2,85,80)(3,86,41)(4,87,42)(5,88,43)(6,89,44)(7,90,45)(8,91,46)(9,92,47)(10,93,48)(11,94,49)(12,95,50)(13,96,51)(14,97,52)(15,98,53)(16,99,54)(17,100,55)(18,101,56)(19,102,57)(20,103,58)(21,104,59)(22,105,60)(23,106,61)(24,107,62)(25,108,63)(26,109,64)(27,110,65)(28,111,66)(29,112,67)(30,113,68)(31,114,69)(32,115,70)(33,116,71)(34,117,72)(35,118,73)(36,119,74)(37,120,75)(38,81,76)(39,82,77)(40,83,78)(121,233,199)(122,234,200)(123,235,161)(124,236,162)(125,237,163)(126,238,164)(127,239,165)(128,240,166)(129,201,167)(130,202,168)(131,203,169)(132,204,170)(133,205,171)(134,206,172)(135,207,173)(136,208,174)(137,209,175)(138,210,176)(139,211,177)(140,212,178)(141,213,179)(142,214,180)(143,215,181)(144,216,182)(145,217,183)(146,218,184)(147,219,185)(148,220,186)(149,221,187)(150,222,188)(151,223,189)(152,224,190)(153,225,191)(154,226,192)(155,227,193)(156,228,194)(157,229,195)(158,230,196)(159,231,197)(160,232,198), (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), (1,133,26,138,11,143,36,148,21,153,6,158,31,123,16,128)(2,122,27,127,12,132,37,137,22,142,7,147,32,152,17,157)(3,151,28,156,13,121,38,126,23,131,8,136,33,141,18,146)(4,140,29,145,14,150,39,155,24,160,9,125,34,130,19,135)(5,129,30,134,15,139,40,144,25,149,10,154,35,159,20,124)(41,189,66,194,51,199,76,164,61,169,46,174,71,179,56,184)(42,178,67,183,52,188,77,193,62,198,47,163,72,168,57,173)(43,167,68,172,53,177,78,182,63,187,48,192,73,197,58,162)(44,196,69,161,54,166,79,171,64,176,49,181,74,186,59,191)(45,185,70,190,55,195,80,200,65,165,50,170,75,175,60,180)(81,238,106,203,91,208,116,213,101,218,86,223,111,228,96,233)(82,227,107,232,92,237,117,202,102,207,87,212,112,217,97,222)(83,216,108,221,93,226,118,231,103,236,88,201,113,206,98,211)(84,205,109,210,94,215,119,220,104,225,89,230,114,235,99,240)(85,234,110,239,95,204,120,209,105,214,90,219,115,224,100,229)>;
G:=Group( (1,84,79)(2,85,80)(3,86,41)(4,87,42)(5,88,43)(6,89,44)(7,90,45)(8,91,46)(9,92,47)(10,93,48)(11,94,49)(12,95,50)(13,96,51)(14,97,52)(15,98,53)(16,99,54)(17,100,55)(18,101,56)(19,102,57)(20,103,58)(21,104,59)(22,105,60)(23,106,61)(24,107,62)(25,108,63)(26,109,64)(27,110,65)(28,111,66)(29,112,67)(30,113,68)(31,114,69)(32,115,70)(33,116,71)(34,117,72)(35,118,73)(36,119,74)(37,120,75)(38,81,76)(39,82,77)(40,83,78)(121,233,199)(122,234,200)(123,235,161)(124,236,162)(125,237,163)(126,238,164)(127,239,165)(128,240,166)(129,201,167)(130,202,168)(131,203,169)(132,204,170)(133,205,171)(134,206,172)(135,207,173)(136,208,174)(137,209,175)(138,210,176)(139,211,177)(140,212,178)(141,213,179)(142,214,180)(143,215,181)(144,216,182)(145,217,183)(146,218,184)(147,219,185)(148,220,186)(149,221,187)(150,222,188)(151,223,189)(152,224,190)(153,225,191)(154,226,192)(155,227,193)(156,228,194)(157,229,195)(158,230,196)(159,231,197)(160,232,198), (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), (1,133,26,138,11,143,36,148,21,153,6,158,31,123,16,128)(2,122,27,127,12,132,37,137,22,142,7,147,32,152,17,157)(3,151,28,156,13,121,38,126,23,131,8,136,33,141,18,146)(4,140,29,145,14,150,39,155,24,160,9,125,34,130,19,135)(5,129,30,134,15,139,40,144,25,149,10,154,35,159,20,124)(41,189,66,194,51,199,76,164,61,169,46,174,71,179,56,184)(42,178,67,183,52,188,77,193,62,198,47,163,72,168,57,173)(43,167,68,172,53,177,78,182,63,187,48,192,73,197,58,162)(44,196,69,161,54,166,79,171,64,176,49,181,74,186,59,191)(45,185,70,190,55,195,80,200,65,165,50,170,75,175,60,180)(81,238,106,203,91,208,116,213,101,218,86,223,111,228,96,233)(82,227,107,232,92,237,117,202,102,207,87,212,112,217,97,222)(83,216,108,221,93,226,118,231,103,236,88,201,113,206,98,211)(84,205,109,210,94,215,119,220,104,225,89,230,114,235,99,240)(85,234,110,239,95,204,120,209,105,214,90,219,115,224,100,229) );
G=PermutationGroup([(1,84,79),(2,85,80),(3,86,41),(4,87,42),(5,88,43),(6,89,44),(7,90,45),(8,91,46),(9,92,47),(10,93,48),(11,94,49),(12,95,50),(13,96,51),(14,97,52),(15,98,53),(16,99,54),(17,100,55),(18,101,56),(19,102,57),(20,103,58),(21,104,59),(22,105,60),(23,106,61),(24,107,62),(25,108,63),(26,109,64),(27,110,65),(28,111,66),(29,112,67),(30,113,68),(31,114,69),(32,115,70),(33,116,71),(34,117,72),(35,118,73),(36,119,74),(37,120,75),(38,81,76),(39,82,77),(40,83,78),(121,233,199),(122,234,200),(123,235,161),(124,236,162),(125,237,163),(126,238,164),(127,239,165),(128,240,166),(129,201,167),(130,202,168),(131,203,169),(132,204,170),(133,205,171),(134,206,172),(135,207,173),(136,208,174),(137,209,175),(138,210,176),(139,211,177),(140,212,178),(141,213,179),(142,214,180),(143,215,181),(144,216,182),(145,217,183),(146,218,184),(147,219,185),(148,220,186),(149,221,187),(150,222,188),(151,223,189),(152,224,190),(153,225,191),(154,226,192),(155,227,193),(156,228,194),(157,229,195),(158,230,196),(159,231,197),(160,232,198)], [(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)], [(1,133,26,138,11,143,36,148,21,153,6,158,31,123,16,128),(2,122,27,127,12,132,37,137,22,142,7,147,32,152,17,157),(3,151,28,156,13,121,38,126,23,131,8,136,33,141,18,146),(4,140,29,145,14,150,39,155,24,160,9,125,34,130,19,135),(5,129,30,134,15,139,40,144,25,149,10,154,35,159,20,124),(41,189,66,194,51,199,76,164,61,169,46,174,71,179,56,184),(42,178,67,183,52,188,77,193,62,198,47,163,72,168,57,173),(43,167,68,172,53,177,78,182,63,187,48,192,73,197,58,162),(44,196,69,161,54,166,79,171,64,176,49,181,74,186,59,191),(45,185,70,190,55,195,80,200,65,165,50,170,75,175,60,180),(81,238,106,203,91,208,116,213,101,218,86,223,111,228,96,233),(82,227,107,232,92,237,117,202,102,207,87,212,112,217,97,222),(83,216,108,221,93,226,118,231,103,236,88,201,113,206,98,211),(84,205,109,210,94,215,119,220,104,225,89,230,114,235,99,240),(85,234,110,239,95,204,120,209,105,214,90,219,115,224,100,229)])
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 | C40.9C4 | C3×C40.9C4 |
| kernel | C3×C40.9C4 | C3×C5⋊2C16 | C2×C120 | C40.9C4 | 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×C40.9C4 ►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×C40.9C4 in GAP, Magma, Sage, TeX
C_3\times C_{40}._9C_4 % in TeX
G:=Group("C3xC40.9C4"); // 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