metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C24.5F5, C120.5C4, C40.4Dic3, D5⋊(C3⋊C16), C15⋊3(C2×C16), C8.5(C3⋊F5), C3⋊2(D5⋊C16), (C3×D5)⋊2C16, (C8×D5).7S3, (C6×D5).4C8, C15⋊C16⋊7C2, C30.10(C2×C8), C60.52(C2×C4), C6.6(D5⋊C8), D10.2(C3⋊C8), C5⋊2C8.40D6, C12.54(C2×F5), (D5×C12).10C4, (D5×C24).12C2, Dic5.2(C3⋊C8), (C4×D5).6Dic3, (C3×Dic5).4C8, C20.14(C2×Dic3), C2.1(C60.C4), C5⋊1(C2×C3⋊C16), C10.1(C2×C3⋊C8), C4.13(C2×C3⋊F5), (C3×C5⋊2C8).53C22, SmallGroup(480,294)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C24.F5 |
Generators and relations for C24.F5
G = < a,b,c | a24=b5=1, c4=a6, ab=ba, cac-1=a17, cbc-1=b3 >
Smallest permutation representation of C24.F5
►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)
(1 233 43 122 209)(2 234 44 123 210)(3 235 45 124 211)(4 236 46 125 212)(5 237 47 126 213)(6 238 48 127 214)(7 239 25 128 215)(8 240 26 129 216)(9 217 27 130 193)(10 218 28 131 194)(11 219 29 132 195)(12 220 30 133 196)(13 221 31 134 197)(14 222 32 135 198)(15 223 33 136 199)(16 224 34 137 200)(17 225 35 138 201)(18 226 36 139 202)(19 227 37 140 203)(20 228 38 141 204)(21 229 39 142 205)(22 230 40 143 206)(23 231 41 144 207)(24 232 42 121 208)(49 154 186 99 82)(50 155 187 100 83)(51 156 188 101 84)(52 157 189 102 85)(53 158 190 103 86)(54 159 191 104 87)(55 160 192 105 88)(56 161 169 106 89)(57 162 170 107 90)(58 163 171 108 91)(59 164 172 109 92)(60 165 173 110 93)(61 166 174 111 94)(62 167 175 112 95)(63 168 176 113 96)(64 145 177 114 73)(65 146 178 115 74)(66 147 179 116 75)(67 148 180 117 76)(68 149 181 118 77)(69 150 182 119 78)(70 151 183 120 79)(71 152 184 97 80)(72 153 185 98 81)
(1 55 16 70 7 61 22 52 13 67 4 58 19 49 10 64)(2 72 17 63 8 54 23 69 14 60 5 51 20 66 11 57)(3 65 18 56 9 71 24 62 15 53 6 68 21 59 12 50)(25 94 143 157 31 76 125 163 37 82 131 145 43 88 137 151)(26 87 144 150 32 93 126 156 38 75 132 162 44 81 138 168)(27 80 121 167 33 86 127 149 39 92 133 155 45 74 139 161)(28 73 122 160 34 79 128 166 40 85 134 148 46 91 140 154)(29 90 123 153 35 96 129 159 41 78 135 165 47 84 141 147)(30 83 124 146 36 89 130 152 42 95 136 158 48 77 142 164)(97 232 175 199 103 238 181 205 109 220 187 211 115 226 169 193)(98 225 176 216 104 231 182 198 110 237 188 204 116 219 170 210)(99 218 177 209 105 224 183 215 111 230 189 197 117 236 171 203)(100 235 178 202 106 217 184 208 112 223 190 214 118 229 172 196)(101 228 179 195 107 234 185 201 113 240 191 207 119 222 173 213)(102 221 180 212 108 227 186 194 114 233 192 200 120 239 174 206)
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), (1,233,43,122,209)(2,234,44,123,210)(3,235,45,124,211)(4,236,46,125,212)(5,237,47,126,213)(6,238,48,127,214)(7,239,25,128,215)(8,240,26,129,216)(9,217,27,130,193)(10,218,28,131,194)(11,219,29,132,195)(12,220,30,133,196)(13,221,31,134,197)(14,222,32,135,198)(15,223,33,136,199)(16,224,34,137,200)(17,225,35,138,201)(18,226,36,139,202)(19,227,37,140,203)(20,228,38,141,204)(21,229,39,142,205)(22,230,40,143,206)(23,231,41,144,207)(24,232,42,121,208)(49,154,186,99,82)(50,155,187,100,83)(51,156,188,101,84)(52,157,189,102,85)(53,158,190,103,86)(54,159,191,104,87)(55,160,192,105,88)(56,161,169,106,89)(57,162,170,107,90)(58,163,171,108,91)(59,164,172,109,92)(60,165,173,110,93)(61,166,174,111,94)(62,167,175,112,95)(63,168,176,113,96)(64,145,177,114,73)(65,146,178,115,74)(66,147,179,116,75)(67,148,180,117,76)(68,149,181,118,77)(69,150,182,119,78)(70,151,183,120,79)(71,152,184,97,80)(72,153,185,98,81), (1,55,16,70,7,61,22,52,13,67,4,58,19,49,10,64)(2,72,17,63,8,54,23,69,14,60,5,51,20,66,11,57)(3,65,18,56,9,71,24,62,15,53,6,68,21,59,12,50)(25,94,143,157,31,76,125,163,37,82,131,145,43,88,137,151)(26,87,144,150,32,93,126,156,38,75,132,162,44,81,138,168)(27,80,121,167,33,86,127,149,39,92,133,155,45,74,139,161)(28,73,122,160,34,79,128,166,40,85,134,148,46,91,140,154)(29,90,123,153,35,96,129,159,41,78,135,165,47,84,141,147)(30,83,124,146,36,89,130,152,42,95,136,158,48,77,142,164)(97,232,175,199,103,238,181,205,109,220,187,211,115,226,169,193)(98,225,176,216,104,231,182,198,110,237,188,204,116,219,170,210)(99,218,177,209,105,224,183,215,111,230,189,197,117,236,171,203)(100,235,178,202,106,217,184,208,112,223,190,214,118,229,172,196)(101,228,179,195,107,234,185,201,113,240,191,207,119,222,173,213)(102,221,180,212,108,227,186,194,114,233,192,200,120,239,174,206)>;
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), (1,233,43,122,209)(2,234,44,123,210)(3,235,45,124,211)(4,236,46,125,212)(5,237,47,126,213)(6,238,48,127,214)(7,239,25,128,215)(8,240,26,129,216)(9,217,27,130,193)(10,218,28,131,194)(11,219,29,132,195)(12,220,30,133,196)(13,221,31,134,197)(14,222,32,135,198)(15,223,33,136,199)(16,224,34,137,200)(17,225,35,138,201)(18,226,36,139,202)(19,227,37,140,203)(20,228,38,141,204)(21,229,39,142,205)(22,230,40,143,206)(23,231,41,144,207)(24,232,42,121,208)(49,154,186,99,82)(50,155,187,100,83)(51,156,188,101,84)(52,157,189,102,85)(53,158,190,103,86)(54,159,191,104,87)(55,160,192,105,88)(56,161,169,106,89)(57,162,170,107,90)(58,163,171,108,91)(59,164,172,109,92)(60,165,173,110,93)(61,166,174,111,94)(62,167,175,112,95)(63,168,176,113,96)(64,145,177,114,73)(65,146,178,115,74)(66,147,179,116,75)(67,148,180,117,76)(68,149,181,118,77)(69,150,182,119,78)(70,151,183,120,79)(71,152,184,97,80)(72,153,185,98,81), (1,55,16,70,7,61,22,52,13,67,4,58,19,49,10,64)(2,72,17,63,8,54,23,69,14,60,5,51,20,66,11,57)(3,65,18,56,9,71,24,62,15,53,6,68,21,59,12,50)(25,94,143,157,31,76,125,163,37,82,131,145,43,88,137,151)(26,87,144,150,32,93,126,156,38,75,132,162,44,81,138,168)(27,80,121,167,33,86,127,149,39,92,133,155,45,74,139,161)(28,73,122,160,34,79,128,166,40,85,134,148,46,91,140,154)(29,90,123,153,35,96,129,159,41,78,135,165,47,84,141,147)(30,83,124,146,36,89,130,152,42,95,136,158,48,77,142,164)(97,232,175,199,103,238,181,205,109,220,187,211,115,226,169,193)(98,225,176,216,104,231,182,198,110,237,188,204,116,219,170,210)(99,218,177,209,105,224,183,215,111,230,189,197,117,236,171,203)(100,235,178,202,106,217,184,208,112,223,190,214,118,229,172,196)(101,228,179,195,107,234,185,201,113,240,191,207,119,222,173,213)(102,221,180,212,108,227,186,194,114,233,192,200,120,239,174,206) );
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)], [(1,233,43,122,209),(2,234,44,123,210),(3,235,45,124,211),(4,236,46,125,212),(5,237,47,126,213),(6,238,48,127,214),(7,239,25,128,215),(8,240,26,129,216),(9,217,27,130,193),(10,218,28,131,194),(11,219,29,132,195),(12,220,30,133,196),(13,221,31,134,197),(14,222,32,135,198),(15,223,33,136,199),(16,224,34,137,200),(17,225,35,138,201),(18,226,36,139,202),(19,227,37,140,203),(20,228,38,141,204),(21,229,39,142,205),(22,230,40,143,206),(23,231,41,144,207),(24,232,42,121,208),(49,154,186,99,82),(50,155,187,100,83),(51,156,188,101,84),(52,157,189,102,85),(53,158,190,103,86),(54,159,191,104,87),(55,160,192,105,88),(56,161,169,106,89),(57,162,170,107,90),(58,163,171,108,91),(59,164,172,109,92),(60,165,173,110,93),(61,166,174,111,94),(62,167,175,112,95),(63,168,176,113,96),(64,145,177,114,73),(65,146,178,115,74),(66,147,179,116,75),(67,148,180,117,76),(68,149,181,118,77),(69,150,182,119,78),(70,151,183,120,79),(71,152,184,97,80),(72,153,185,98,81)], [(1,55,16,70,7,61,22,52,13,67,4,58,19,49,10,64),(2,72,17,63,8,54,23,69,14,60,5,51,20,66,11,57),(3,65,18,56,9,71,24,62,15,53,6,68,21,59,12,50),(25,94,143,157,31,76,125,163,37,82,131,145,43,88,137,151),(26,87,144,150,32,93,126,156,38,75,132,162,44,81,138,168),(27,80,121,167,33,86,127,149,39,92,133,155,45,74,139,161),(28,73,122,160,34,79,128,166,40,85,134,148,46,91,140,154),(29,90,123,153,35,96,129,159,41,78,135,165,47,84,141,147),(30,83,124,146,36,89,130,152,42,95,136,158,48,77,142,164),(97,232,175,199,103,238,181,205,109,220,187,211,115,226,169,193),(98,225,176,216,104,231,182,198,110,237,188,204,116,219,170,210),(99,218,177,209,105,224,183,215,111,230,189,197,117,236,171,203),(100,235,178,202,106,217,184,208,112,223,190,214,118,229,172,196),(101,228,179,195,107,234,185,201,113,240,191,207,119,222,173,213),(102,221,180,212,108,227,186,194,114,233,192,200,120,239,174,206)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5 | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10 | 12A | 12B | 12C | 12D | 15A | 15B | 16A | ··· | 16P | 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 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 8 | 8 | 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 | 5 | 5 | 2 | 1 | 1 | 5 | 5 | 4 | 2 | 10 | 10 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 4 | 2 | 2 | 10 | 10 | 4 | 4 | 15 | ··· | 15 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | - | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | S3 | D6 | Dic3 | Dic3 | C3⋊C8 | C3⋊C8 | C3⋊C16 | F5 | C2×F5 | C3⋊F5 | D5⋊C8 | C2×C3⋊F5 | D5⋊C16 | C60.C4 | C24.F5 |
| kernel | C24.F5 | C15⋊C16 | D5×C24 | C120 | D5×C12 | C3×Dic5 | C6×D5 | C3×D5 | C8×D5 | C5⋊2C8 | C40 | C4×D5 | Dic5 | D10 | D5 | C24 | C12 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 16 | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C24.F5 ►in GL6(𝔽241)
| 0 | 211 | 0 | 0 | 0 | 0 |
| 30 | 211 | 0 | 0 | 0 | 0 |
| 0 | 0 | 114 | 12 | 0 | 229 |
| 0 | 0 | 0 | 126 | 12 | 229 |
| 0 | 0 | 229 | 12 | 126 | 0 |
| 0 | 0 | 229 | 0 | 12 | 114 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 1 | 0 | 0 | 240 |
| 0 | 0 | 0 | 1 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 240 |
| 106 | 184 | 0 | 0 | 0 | 0 |
| 49 | 135 | 0 | 0 | 0 | 0 |
| 0 | 0 | 224 | 0 | 17 | 190 |
| 0 | 0 | 0 | 190 | 68 | 173 |
| 0 | 0 | 51 | 173 | 68 | 190 |
| 0 | 0 | 51 | 190 | 17 | 0 |
G:=sub<GL(6,GF(241))| [0,30,0,0,0,0,211,211,0,0,0,0,0,0,114,0,229,229,0,0,12,126,12,0,0,0,0,12,126,12,0,0,229,229,0,114],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,240,240,240,240],[106,49,0,0,0,0,184,135,0,0,0,0,0,0,224,0,51,51,0,0,0,190,173,190,0,0,17,68,68,17,0,0,190,173,190,0] >;
C24.F5 in GAP, Magma, Sage, TeX
C_{24}.F_5 % in TeX
G:=Group("C24.F5"); // GroupNames label
G:=SmallGroup(480,294);
// by ID
G=gap.SmallGroup(480,294);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,64,58,80,2693,14118,4724]);
// Polycyclic
G:=Group<a,b,c|a^24=b^5=1,c^4=a^6,a*b=b*a,c*a*c^-1=a^17,c*b*c^-1=b^3>;
// generators/relations
Export