metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C60.7C8, C40.73D6, C8.22D30, C120.14C4, C24.78D10, C15⋊13M5(2), C40.9Dic3, C24.5Dic5, C8.2Dic15, C120.91C22, C20.4(C3⋊C8), C4.(C15⋊3C8), (C2×C30).7C8, C15⋊3C16⋊8C2, (C2×C8).7D15, C30.61(C2×C8), (C2×C60).37C4, (C2×C40).10S3, (C2×C24).13D5, C5⋊4(C12.C8), C3⋊2(C20.4C8), C12.1(C5⋊2C8), C60.235(C2×C4), (C2×C120).17C2, C22.(C15⋊3C8), (C2×C4).5Dic15, (C2×C12).8Dic5, C4.10(C2×Dic15), C20.61(C2×Dic3), C12.40(C2×Dic5), (C2×C20).19Dic3, C10.18(C2×C3⋊C8), C6.9(C2×C5⋊2C8), C2.4(C2×C15⋊3C8), (C2×C10).5(C3⋊C8), (C2×C6).3(C5⋊2C8), SmallGroup(480,172)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.7C8
G = < a,b | a60=1, b8=a30, bab-1=a-1 >
Smallest permutation representation of C60.7C8
►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)
(1 187 117 161 46 202 102 176 31 217 87 131 16 232 72 146)(2 186 118 160 47 201 103 175 32 216 88 130 17 231 73 145)(3 185 119 159 48 200 104 174 33 215 89 129 18 230 74 144)(4 184 120 158 49 199 105 173 34 214 90 128 19 229 75 143)(5 183 61 157 50 198 106 172 35 213 91 127 20 228 76 142)(6 182 62 156 51 197 107 171 36 212 92 126 21 227 77 141)(7 181 63 155 52 196 108 170 37 211 93 125 22 226 78 140)(8 240 64 154 53 195 109 169 38 210 94 124 23 225 79 139)(9 239 65 153 54 194 110 168 39 209 95 123 24 224 80 138)(10 238 66 152 55 193 111 167 40 208 96 122 25 223 81 137)(11 237 67 151 56 192 112 166 41 207 97 121 26 222 82 136)(12 236 68 150 57 191 113 165 42 206 98 180 27 221 83 135)(13 235 69 149 58 190 114 164 43 205 99 179 28 220 84 134)(14 234 70 148 59 189 115 163 44 204 100 178 29 219 85 133)(15 233 71 147 60 188 116 162 45 203 101 177 30 218 86 132)
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), (1,187,117,161,46,202,102,176,31,217,87,131,16,232,72,146)(2,186,118,160,47,201,103,175,32,216,88,130,17,231,73,145)(3,185,119,159,48,200,104,174,33,215,89,129,18,230,74,144)(4,184,120,158,49,199,105,173,34,214,90,128,19,229,75,143)(5,183,61,157,50,198,106,172,35,213,91,127,20,228,76,142)(6,182,62,156,51,197,107,171,36,212,92,126,21,227,77,141)(7,181,63,155,52,196,108,170,37,211,93,125,22,226,78,140)(8,240,64,154,53,195,109,169,38,210,94,124,23,225,79,139)(9,239,65,153,54,194,110,168,39,209,95,123,24,224,80,138)(10,238,66,152,55,193,111,167,40,208,96,122,25,223,81,137)(11,237,67,151,56,192,112,166,41,207,97,121,26,222,82,136)(12,236,68,150,57,191,113,165,42,206,98,180,27,221,83,135)(13,235,69,149,58,190,114,164,43,205,99,179,28,220,84,134)(14,234,70,148,59,189,115,163,44,204,100,178,29,219,85,133)(15,233,71,147,60,188,116,162,45,203,101,177,30,218,86,132)>;
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), (1,187,117,161,46,202,102,176,31,217,87,131,16,232,72,146)(2,186,118,160,47,201,103,175,32,216,88,130,17,231,73,145)(3,185,119,159,48,200,104,174,33,215,89,129,18,230,74,144)(4,184,120,158,49,199,105,173,34,214,90,128,19,229,75,143)(5,183,61,157,50,198,106,172,35,213,91,127,20,228,76,142)(6,182,62,156,51,197,107,171,36,212,92,126,21,227,77,141)(7,181,63,155,52,196,108,170,37,211,93,125,22,226,78,140)(8,240,64,154,53,195,109,169,38,210,94,124,23,225,79,139)(9,239,65,153,54,194,110,168,39,209,95,123,24,224,80,138)(10,238,66,152,55,193,111,167,40,208,96,122,25,223,81,137)(11,237,67,151,56,192,112,166,41,207,97,121,26,222,82,136)(12,236,68,150,57,191,113,165,42,206,98,180,27,221,83,135)(13,235,69,149,58,190,114,164,43,205,99,179,28,220,84,134)(14,234,70,148,59,189,115,163,44,204,100,178,29,219,85,133)(15,233,71,147,60,188,116,162,45,203,101,177,30,218,86,132) );
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)], [(1,187,117,161,46,202,102,176,31,217,87,131,16,232,72,146),(2,186,118,160,47,201,103,175,32,216,88,130,17,231,73,145),(3,185,119,159,48,200,104,174,33,215,89,129,18,230,74,144),(4,184,120,158,49,199,105,173,34,214,90,128,19,229,75,143),(5,183,61,157,50,198,106,172,35,213,91,127,20,228,76,142),(6,182,62,156,51,197,107,171,36,212,92,126,21,227,77,141),(7,181,63,155,52,196,108,170,37,211,93,125,22,226,78,140),(8,240,64,154,53,195,109,169,38,210,94,124,23,225,79,139),(9,239,65,153,54,194,110,168,39,209,95,123,24,224,80,138),(10,238,66,152,55,193,111,167,40,208,96,122,25,223,81,137),(11,237,67,151,56,192,112,166,41,207,97,121,26,222,82,136),(12,236,68,150,57,191,113,165,42,206,98,180,27,221,83,135),(13,235,69,149,58,190,114,164,43,205,99,179,28,220,84,134),(14,234,70,148,59,189,115,163,44,204,100,178,29,219,85,133),(15,233,71,147,60,188,116,162,45,203,101,177,30,218,86,132)]])
132 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 16A | ··· | 16H | 20A | ··· | 20H | 24A | ··· | 24H | 30A | ··· | 30L | 40A | ··· | 40P | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 16 | ··· | 16 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 30 | ··· | 30 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | - | + | - | + | - | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | S3 | D5 | Dic3 | D6 | Dic3 | Dic5 | D10 | Dic5 | C3⋊C8 | C3⋊C8 | D15 | M5(2) | C5⋊2C8 | C5⋊2C8 | Dic15 | D30 | Dic15 | C12.C8 | C15⋊3C8 | C15⋊3C8 | C20.4C8 | C60.7C8 |
| kernel | C60.7C8 | C15⋊3C16 | C2×C120 | C120 | C2×C60 | C60 | C2×C30 | C2×C40 | C2×C24 | C40 | C40 | C2×C20 | C24 | C24 | C2×C12 | C20 | C2×C10 | C2×C8 | C15 | C12 | C2×C6 | C8 | C8 | C2×C4 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 16 | 32 |
Matrix representation of C60.7C8 ►in GL2(𝔽241) generated by
| 158 | 0 |
| 0 | 90 |
| 0 | 1 |
| 211 | 0 |
G:=sub<GL(2,GF(241))| [158,0,0,90],[0,211,1,0] >;
C60.7C8 in GAP, Magma, Sage, TeX
C_{60}._7C_8 % in TeX
G:=Group("C60.7C8"); // GroupNames label
G:=SmallGroup(480,172);
// by ID
G=gap.SmallGroup(480,172);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,58,80,2693,18822]);
// Polycyclic
G:=Group<a,b|a^60=1,b^8=a^30,b*a*b^-1=a^-1>;
// generators/relations
Export