direct product, abelian, monomial, 2-elementary
Aliases: C4×C60, SmallGroup(240,81)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C4×C60 |
| C1 — C4×C60 |
| C1 — C4×C60 |
Generators and relations for C4×C60
G = < a,b | a4=b60=1, ab=ba >
Subgroups: 60, all normal (12 characteristic)
C1, C2, C3, C4, C22, C5, C6, C2×C4, C10, C12, C2×C6, C15, C42, C20, C2×C10, C2×C12, C30, C2×C20, C4×C12, C60, C2×C30, C4×C20, C2×C60, C4×C60
Quotients: C1, C2, C3, C4, C22, C5, C6, C2×C4, C10, C12, C2×C6, C15, C42, C20, C2×C10, C2×C12, C30, C2×C20, C4×C12, C60, C2×C30, C4×C20, C2×C60, C4×C60
►Regular action on 240 points
Generators in S240
(1 218 137 77)(2 219 138 78)(3 220 139 79)(4 221 140 80)(5 222 141 81)(6 223 142 82)(7 224 143 83)(8 225 144 84)(9 226 145 85)(10 227 146 86)(11 228 147 87)(12 229 148 88)(13 230 149 89)(14 231 150 90)(15 232 151 91)(16 233 152 92)(17 234 153 93)(18 235 154 94)(19 236 155 95)(20 237 156 96)(21 238 157 97)(22 239 158 98)(23 240 159 99)(24 181 160 100)(25 182 161 101)(26 183 162 102)(27 184 163 103)(28 185 164 104)(29 186 165 105)(30 187 166 106)(31 188 167 107)(32 189 168 108)(33 190 169 109)(34 191 170 110)(35 192 171 111)(36 193 172 112)(37 194 173 113)(38 195 174 114)(39 196 175 115)(40 197 176 116)(41 198 177 117)(42 199 178 118)(43 200 179 119)(44 201 180 120)(45 202 121 61)(46 203 122 62)(47 204 123 63)(48 205 124 64)(49 206 125 65)(50 207 126 66)(51 208 127 67)(52 209 128 68)(53 210 129 69)(54 211 130 70)(55 212 131 71)(56 213 132 72)(57 214 133 73)(58 215 134 74)(59 216 135 75)(60 217 136 76)
(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)
G:=sub<Sym(240)| (1,218,137,77)(2,219,138,78)(3,220,139,79)(4,221,140,80)(5,222,141,81)(6,223,142,82)(7,224,143,83)(8,225,144,84)(9,226,145,85)(10,227,146,86)(11,228,147,87)(12,229,148,88)(13,230,149,89)(14,231,150,90)(15,232,151,91)(16,233,152,92)(17,234,153,93)(18,235,154,94)(19,236,155,95)(20,237,156,96)(21,238,157,97)(22,239,158,98)(23,240,159,99)(24,181,160,100)(25,182,161,101)(26,183,162,102)(27,184,163,103)(28,185,164,104)(29,186,165,105)(30,187,166,106)(31,188,167,107)(32,189,168,108)(33,190,169,109)(34,191,170,110)(35,192,171,111)(36,193,172,112)(37,194,173,113)(38,195,174,114)(39,196,175,115)(40,197,176,116)(41,198,177,117)(42,199,178,118)(43,200,179,119)(44,201,180,120)(45,202,121,61)(46,203,122,62)(47,204,123,63)(48,205,124,64)(49,206,125,65)(50,207,126,66)(51,208,127,67)(52,209,128,68)(53,210,129,69)(54,211,130,70)(55,212,131,71)(56,213,132,72)(57,214,133,73)(58,215,134,74)(59,216,135,75)(60,217,136,76), (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)>;
G:=Group( (1,218,137,77)(2,219,138,78)(3,220,139,79)(4,221,140,80)(5,222,141,81)(6,223,142,82)(7,224,143,83)(8,225,144,84)(9,226,145,85)(10,227,146,86)(11,228,147,87)(12,229,148,88)(13,230,149,89)(14,231,150,90)(15,232,151,91)(16,233,152,92)(17,234,153,93)(18,235,154,94)(19,236,155,95)(20,237,156,96)(21,238,157,97)(22,239,158,98)(23,240,159,99)(24,181,160,100)(25,182,161,101)(26,183,162,102)(27,184,163,103)(28,185,164,104)(29,186,165,105)(30,187,166,106)(31,188,167,107)(32,189,168,108)(33,190,169,109)(34,191,170,110)(35,192,171,111)(36,193,172,112)(37,194,173,113)(38,195,174,114)(39,196,175,115)(40,197,176,116)(41,198,177,117)(42,199,178,118)(43,200,179,119)(44,201,180,120)(45,202,121,61)(46,203,122,62)(47,204,123,63)(48,205,124,64)(49,206,125,65)(50,207,126,66)(51,208,127,67)(52,209,128,68)(53,210,129,69)(54,211,130,70)(55,212,131,71)(56,213,132,72)(57,214,133,73)(58,215,134,74)(59,216,135,75)(60,217,136,76), (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) );
G=PermutationGroup([[(1,218,137,77),(2,219,138,78),(3,220,139,79),(4,221,140,80),(5,222,141,81),(6,223,142,82),(7,224,143,83),(8,225,144,84),(9,226,145,85),(10,227,146,86),(11,228,147,87),(12,229,148,88),(13,230,149,89),(14,231,150,90),(15,232,151,91),(16,233,152,92),(17,234,153,93),(18,235,154,94),(19,236,155,95),(20,237,156,96),(21,238,157,97),(22,239,158,98),(23,240,159,99),(24,181,160,100),(25,182,161,101),(26,183,162,102),(27,184,163,103),(28,185,164,104),(29,186,165,105),(30,187,166,106),(31,188,167,107),(32,189,168,108),(33,190,169,109),(34,191,170,110),(35,192,171,111),(36,193,172,112),(37,194,173,113),(38,195,174,114),(39,196,175,115),(40,197,176,116),(41,198,177,117),(42,199,178,118),(43,200,179,119),(44,201,180,120),(45,202,121,61),(46,203,122,62),(47,204,123,63),(48,205,124,64),(49,206,125,65),(50,207,126,66),(51,208,127,67),(52,209,128,68),(53,210,129,69),(54,211,130,70),(55,212,131,71),(56,213,132,72),(57,214,133,73),(58,215,134,74),(59,216,135,75),(60,217,136,76)], [(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)]])
C4×C60 is a maximal subgroup of
C42.D15 C60⋊5C8 D60⋊7C4 C60⋊8Q8 C60.24Q8 C42⋊2D15 C42⋊6D15 C42⋊7D15 C42⋊3D15
240 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | ··· | 4L | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 10A | ··· | 10L | 12A | ··· | 12X | 15A | ··· | 15H | 20A | ··· | 20AV | 30A | ··· | 30X | 60A | ··· | 60CR |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
240 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C4 | C5 | C6 | C10 | C12 | C15 | C20 | C30 | C60 |
| kernel | C4×C60 | C2×C60 | C4×C20 | C60 | C4×C12 | C2×C20 | C2×C12 | C20 | C42 | C12 | C2×C4 | C4 |
| # reps | 1 | 3 | 2 | 12 | 4 | 6 | 12 | 24 | 8 | 48 | 24 | 96 |
Matrix representation of C4×C60 ►in GL2(𝔽61) generated by
| 60 | 0 |
| 0 | 11 |
| 2 | 0 |
| 0 | 21 |
G:=sub<GL(2,GF(61))| [60,0,0,11],[2,0,0,21] >;
C4×C60 in GAP, Magma, Sage, TeX
C_4\times C_{60} % in TeX
G:=Group("C4xC60"); // GroupNames label
G:=SmallGroup(240,81);
// by ID
G=gap.SmallGroup(240,81);
# by ID
G:=PCGroup([6,-2,-2,-3,-5,-2,-2,360,727]);
// Polycyclic
G:=Group<a,b|a^4=b^60=1,a*b=b*a>;
// generators/relations