metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C24.6F5, C120.6C4, C15⋊3M5(2), C40.5Dic3, C8.3(C3⋊F5), (C8×D5).8S3, (C6×D5).5C8, C15⋊C16⋊5C2, C30.11(C2×C8), C60.53(C2×C4), C6.7(D5⋊C8), D10.3(C3⋊C8), C5⋊2C8.41D6, C12.55(C2×F5), C3⋊2(C8.F5), C5⋊1(C12.C8), (D5×C12).11C4, (D5×C24).13C2, Dic5.3(C3⋊C8), (C4×D5).7Dic3, (C3×Dic5).5C8, C20.15(C2×Dic3), C2.3(C60.C4), C10.2(C2×C3⋊C8), C4.14(C2×C3⋊F5), (C3×C5⋊2C8).54C22, SmallGroup(480,295)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C120.C4
G = < a,b | a120=1, b4=a90, bab-1=a77 >
Smallest permutation representation of C120.C4
►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)
(1 158 46 143 91 128 16 233 61 218 106 203 31 188 76 173)(2 211 95 220 92 181 65 190 62 151 35 160 32 121 5 130)(3 144 24 177 93 234 114 147 63 204 84 237 33 174 54 207)(4 197 73 134 94 167 43 224 64 137 13 194 34 227 103 164)(6 183 51 168 96 153 21 138 66 123 111 228 36 213 81 198)(7 236 100 125 97 206 70 215 67 176 40 185 37 146 10 155)(8 169 29 202 98 139 119 172 68 229 89 142 38 199 59 232)(9 222 78 159 99 192 48 129 69 162 18 219 39 132 108 189)(11 208 56 193 101 178 26 163 71 148 116 133 41 238 86 223)(12 141 105 150 102 231 75 240 72 201 45 210 42 171 15 180)(14 127 83 184 104 217 53 154 74 187 23 124 44 157 113 214)(17 166 110 175 107 136 80 145 77 226 50 235 47 196 20 205)(19 152 88 209 109 122 58 179 79 212 28 149 49 182 118 239)(22 191 115 200 112 161 85 170 82 131 55 140 52 221 25 230)(27 216 120 225 117 186 90 195 87 156 60 165 57 126 30 135)
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), (1,158,46,143,91,128,16,233,61,218,106,203,31,188,76,173)(2,211,95,220,92,181,65,190,62,151,35,160,32,121,5,130)(3,144,24,177,93,234,114,147,63,204,84,237,33,174,54,207)(4,197,73,134,94,167,43,224,64,137,13,194,34,227,103,164)(6,183,51,168,96,153,21,138,66,123,111,228,36,213,81,198)(7,236,100,125,97,206,70,215,67,176,40,185,37,146,10,155)(8,169,29,202,98,139,119,172,68,229,89,142,38,199,59,232)(9,222,78,159,99,192,48,129,69,162,18,219,39,132,108,189)(11,208,56,193,101,178,26,163,71,148,116,133,41,238,86,223)(12,141,105,150,102,231,75,240,72,201,45,210,42,171,15,180)(14,127,83,184,104,217,53,154,74,187,23,124,44,157,113,214)(17,166,110,175,107,136,80,145,77,226,50,235,47,196,20,205)(19,152,88,209,109,122,58,179,79,212,28,149,49,182,118,239)(22,191,115,200,112,161,85,170,82,131,55,140,52,221,25,230)(27,216,120,225,117,186,90,195,87,156,60,165,57,126,30,135)>;
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), (1,158,46,143,91,128,16,233,61,218,106,203,31,188,76,173)(2,211,95,220,92,181,65,190,62,151,35,160,32,121,5,130)(3,144,24,177,93,234,114,147,63,204,84,237,33,174,54,207)(4,197,73,134,94,167,43,224,64,137,13,194,34,227,103,164)(6,183,51,168,96,153,21,138,66,123,111,228,36,213,81,198)(7,236,100,125,97,206,70,215,67,176,40,185,37,146,10,155)(8,169,29,202,98,139,119,172,68,229,89,142,38,199,59,232)(9,222,78,159,99,192,48,129,69,162,18,219,39,132,108,189)(11,208,56,193,101,178,26,163,71,148,116,133,41,238,86,223)(12,141,105,150,102,231,75,240,72,201,45,210,42,171,15,180)(14,127,83,184,104,217,53,154,74,187,23,124,44,157,113,214)(17,166,110,175,107,136,80,145,77,226,50,235,47,196,20,205)(19,152,88,209,109,122,58,179,79,212,28,149,49,182,118,239)(22,191,115,200,112,161,85,170,82,131,55,140,52,221,25,230)(27,216,120,225,117,186,90,195,87,156,60,165,57,126,30,135) );
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)], [(1,158,46,143,91,128,16,233,61,218,106,203,31,188,76,173),(2,211,95,220,92,181,65,190,62,151,35,160,32,121,5,130),(3,144,24,177,93,234,114,147,63,204,84,237,33,174,54,207),(4,197,73,134,94,167,43,224,64,137,13,194,34,227,103,164),(6,183,51,168,96,153,21,138,66,123,111,228,36,213,81,198),(7,236,100,125,97,206,70,215,67,176,40,185,37,146,10,155),(8,169,29,202,98,139,119,172,68,229,89,142,38,199,59,232),(9,222,78,159,99,192,48,129,69,162,18,219,39,132,108,189),(11,208,56,193,101,178,26,163,71,148,116,133,41,238,86,223),(12,141,105,150,102,231,75,240,72,201,45,210,42,171,15,180),(14,127,83,184,104,217,53,154,74,187,23,124,44,157,113,214),(17,166,110,175,107,136,80,145,77,226,50,235,47,196,20,205),(19,152,88,209,109,122,58,179,79,212,28,149,49,182,118,239),(22,191,115,200,112,161,85,170,82,131,55,140,52,221,25,230),(27,216,120,225,117,186,90,195,87,156,60,165,57,126,30,135)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 5 | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 10 | 12A | 12B | 12C | 12D | 15A | 15B | 16A | ··· | 16H | 20A | 20B | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | 30A | 30B | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 16 | ··· | 16 | 20 | 20 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 10 | 2 | 1 | 1 | 10 | 4 | 2 | 10 | 10 | 2 | 2 | 5 | 5 | 5 | 5 | 4 | 2 | 2 | 10 | 10 | 4 | 4 | 30 | ··· | 30 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 |
| type | + | + | + | + | + | - | - | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | S3 | D6 | Dic3 | Dic3 | C3⋊C8 | C3⋊C8 | M5(2) | C12.C8 | F5 | C2×F5 | C3⋊F5 | D5⋊C8 | C2×C3⋊F5 | C8.F5 | C60.C4 | C120.C4 |
| kernel | C120.C4 | C15⋊C16 | D5×C24 | C120 | D5×C12 | C3×Dic5 | C6×D5 | C8×D5 | C5⋊2C8 | C40 | C4×D5 | Dic5 | D10 | C15 | C5 | C24 | C12 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C120.C4 ►in GL8(𝔽241)
| 233 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 231 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 226 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 69 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 240 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 240 | 1 |
| 209 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 63 | 32 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 198 | 68 | 0 | 0 | 0 | 0 |
| 0 | 0 | 228 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 157 | 119 | 122 |
| 0 | 0 | 0 | 0 | 122 | 0 | 119 | 38 |
| 0 | 0 | 0 | 0 | 38 | 119 | 0 | 122 |
| 0 | 0 | 0 | 0 | 122 | 119 | 157 | 0 |
G:=sub<GL(8,GF(241))| [233,231,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,226,69,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,1,1,1,1],[209,63,0,0,0,0,0,0,3,32,0,0,0,0,0,0,0,0,198,228,0,0,0,0,0,0,68,43,0,0,0,0,0,0,0,0,0,122,38,122,0,0,0,0,157,0,119,119,0,0,0,0,119,119,0,157,0,0,0,0,122,38,122,0] >;
C120.C4 in GAP, Magma, Sage, TeX
C_{120}.C_4 % in TeX
G:=Group("C120.C4"); // GroupNames label
G:=SmallGroup(480,295);
// by ID
G=gap.SmallGroup(480,295);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,64,58,80,2693,14118,4724]);
// Polycyclic
G:=Group<a,b|a^120=1,b^4=a^90,b*a*b^-1=a^77>;
// generators/relations
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