metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C60.1C8, C15⋊8M5(2), C20.1(C3⋊C8), C12.1(C5⋊C8), C4.(C15⋊C8), (C2×C60).5C4, (C2×C30).2C8, C15⋊C16⋊6C2, C60.57(C2×C4), C30.35(C2×C8), C5⋊2C8.43D6, C12.59(C2×F5), (C2×C12).11F5, C5⋊2(C12.C8), C3⋊2(C20.C8), C22.(C15⋊C8), (C2×C20).7Dic3, C5⋊2C8.6Dic3, C20.19(C2×Dic3), C6.7(C2×C5⋊C8), C10.7(C2×C3⋊C8), C4.18(C2×C3⋊F5), (C2×C6).3(C5⋊C8), (C2×C4).5(C3⋊F5), (C3×C5⋊2C8).9C4, C2.3(C2×C15⋊C8), (C2×C10).2(C3⋊C8), (C6×C5⋊2C8).19C2, (C2×C5⋊2C8).11S3, (C3×C5⋊2C8).56C22, SmallGroup(480,303)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.C8
G = < a,b | a60=1, b8=a30, bab-1=a47 >
Smallest permutation representation of C60.C8
►On 240 points
Generators in S240
(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)
(1 183 91 136 46 198 76 151 31 213 61 166 16 228 106 121)(2 206 80 123 47 221 65 138 32 236 110 153 17 191 95 168)(3 229 69 170 48 184 114 125 33 199 99 140 18 214 84 155)(4 192 118 157 49 207 103 172 34 222 88 127 19 237 73 142)(5 215 107 144 50 230 92 159 35 185 77 174 20 200 62 129)(6 238 96 131 51 193 81 146 36 208 66 161 21 223 111 176)(7 201 85 178 52 216 70 133 37 231 115 148 22 186 100 163)(8 224 74 165 53 239 119 180 38 194 104 135 23 209 89 150)(9 187 63 152 54 202 108 167 39 217 93 122 24 232 78 137)(10 210 112 139 55 225 97 154 40 240 82 169 25 195 67 124)(11 233 101 126 56 188 86 141 41 203 71 156 26 218 116 171)(12 196 90 173 57 211 75 128 42 226 120 143 27 181 105 158)(13 219 79 160 58 234 64 175 43 189 109 130 28 204 94 145)(14 182 68 147 59 197 113 162 44 212 98 177 29 227 83 132)(15 205 117 134 60 220 102 149 45 235 87 164 30 190 72 179)
G:=sub<Sym(240)| (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), (1,183,91,136,46,198,76,151,31,213,61,166,16,228,106,121)(2,206,80,123,47,221,65,138,32,236,110,153,17,191,95,168)(3,229,69,170,48,184,114,125,33,199,99,140,18,214,84,155)(4,192,118,157,49,207,103,172,34,222,88,127,19,237,73,142)(5,215,107,144,50,230,92,159,35,185,77,174,20,200,62,129)(6,238,96,131,51,193,81,146,36,208,66,161,21,223,111,176)(7,201,85,178,52,216,70,133,37,231,115,148,22,186,100,163)(8,224,74,165,53,239,119,180,38,194,104,135,23,209,89,150)(9,187,63,152,54,202,108,167,39,217,93,122,24,232,78,137)(10,210,112,139,55,225,97,154,40,240,82,169,25,195,67,124)(11,233,101,126,56,188,86,141,41,203,71,156,26,218,116,171)(12,196,90,173,57,211,75,128,42,226,120,143,27,181,105,158)(13,219,79,160,58,234,64,175,43,189,109,130,28,204,94,145)(14,182,68,147,59,197,113,162,44,212,98,177,29,227,83,132)(15,205,117,134,60,220,102,149,45,235,87,164,30,190,72,179)>;
G:=Group( (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), (1,183,91,136,46,198,76,151,31,213,61,166,16,228,106,121)(2,206,80,123,47,221,65,138,32,236,110,153,17,191,95,168)(3,229,69,170,48,184,114,125,33,199,99,140,18,214,84,155)(4,192,118,157,49,207,103,172,34,222,88,127,19,237,73,142)(5,215,107,144,50,230,92,159,35,185,77,174,20,200,62,129)(6,238,96,131,51,193,81,146,36,208,66,161,21,223,111,176)(7,201,85,178,52,216,70,133,37,231,115,148,22,186,100,163)(8,224,74,165,53,239,119,180,38,194,104,135,23,209,89,150)(9,187,63,152,54,202,108,167,39,217,93,122,24,232,78,137)(10,210,112,139,55,225,97,154,40,240,82,169,25,195,67,124)(11,233,101,126,56,188,86,141,41,203,71,156,26,218,116,171)(12,196,90,173,57,211,75,128,42,226,120,143,27,181,105,158)(13,219,79,160,58,234,64,175,43,189,109,130,28,204,94,145)(14,182,68,147,59,197,113,162,44,212,98,177,29,227,83,132)(15,205,117,134,60,220,102,149,45,235,87,164,30,190,72,179) );
G=PermutationGroup([[(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)], [(1,183,91,136,46,198,76,151,31,213,61,166,16,228,106,121),(2,206,80,123,47,221,65,138,32,236,110,153,17,191,95,168),(3,229,69,170,48,184,114,125,33,199,99,140,18,214,84,155),(4,192,118,157,49,207,103,172,34,222,88,127,19,237,73,142),(5,215,107,144,50,230,92,159,35,185,77,174,20,200,62,129),(6,238,96,131,51,193,81,146,36,208,66,161,21,223,111,176),(7,201,85,178,52,216,70,133,37,231,115,148,22,186,100,163),(8,224,74,165,53,239,119,180,38,194,104,135,23,209,89,150),(9,187,63,152,54,202,108,167,39,217,93,122,24,232,78,137),(10,210,112,139,55,225,97,154,40,240,82,169,25,195,67,124),(11,233,101,126,56,188,86,141,41,203,71,156,26,218,116,171),(12,196,90,173,57,211,75,128,42,226,120,143,27,181,105,158),(13,219,79,160,58,234,64,175,43,189,109,130,28,204,94,145),(14,182,68,147,59,197,113,162,44,212,98,177,29,227,83,132),(15,205,117,134,60,220,102,149,45,235,87,164,30,190,72,179)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 5 | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 15A | 15B | 16A | ··· | 16H | 20A | 20B | 20C | 20D | 24A | ··· | 24H | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 30 | ··· | 30 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | + | - | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | S3 | Dic3 | D6 | Dic3 | C3⋊C8 | C3⋊C8 | M5(2) | C12.C8 | F5 | C5⋊C8 | C2×F5 | C5⋊C8 | C3⋊F5 | C15⋊C8 | C2×C3⋊F5 | C15⋊C8 | C20.C8 | C60.C8 |
| kernel | C60.C8 | C15⋊C16 | C6×C5⋊2C8 | C3×C5⋊2C8 | C2×C60 | C60 | C2×C30 | C2×C5⋊2C8 | C5⋊2C8 | 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 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of C60.C8 ►in GL4(𝔽241) generated by
| 228 | 228 | 95 | 214 |
| 13 | 181 | 183 | 146 |
| 0 | 0 | 208 | 4 |
| 0 | 0 | 237 | 0 |
| 96 | 109 | 47 | 16 |
| 29 | 194 | 163 | 234 |
| 160 | 86 | 31 | 4 |
| 201 | 81 | 33 | 161 |
G:=sub<GL(4,GF(241))| [228,13,0,0,228,181,0,0,95,183,208,237,214,146,4,0],[96,29,160,201,109,194,86,81,47,163,31,33,16,234,4,161] >;
C60.C8 in GAP, Magma, Sage, TeX
C_{60}.C_8 % in TeX
G:=Group("C60.C8"); // GroupNames label
G:=SmallGroup(480,303);
// by ID
G=gap.SmallGroup(480,303);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,58,80,2693,14118,4724]);
// Polycyclic
G:=Group<a,b|a^60=1,b^8=a^30,b*a*b^-1=a^47>;
// generators/relations
Export