metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C24.1F5, C120.1C4, C40.1Dic3, D10.6Dic6, Dic5.11D12, C8.1(C3⋊F5), (C8×D5).4S3, C6.4(C4⋊F5), (C6×D5).8Q8, C60.50(C2×C4), (C4×D5).89D6, (D5×C24).6C2, C30.11(C4⋊C4), C12.50(C2×F5), C3⋊1(D10.Q8), C5⋊2(C24.C4), C15⋊4(C8.C4), C2.7(C60⋊C4), C5⋊2C8.4Dic3, C12.F5.4C2, C10.4(C4⋊Dic3), (C3×Dic5).58D4, C20.11(C2×Dic3), (D5×C12).114C22, C4.11(C2×C3⋊F5), (C3×C5⋊2C8).6C4, SmallGroup(480,301)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.1F5
G = < a,b,c | a24=b5=1, c4=a12, ab=ba, cac-1=a-1, cbc-1=b3 >
Subgroups: 268 in 60 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, D5, C10, C12, C12, C2×C6, C15, C2×C8, M4(2), Dic5, C20, D10, C3⋊C8, C24, C24, C2×C12, C3×D5, C30, C8.C4, C5⋊2C8, C40, C5⋊C8, C4×D5, C4.Dic3, C2×C24, C3×Dic5, C60, C6×D5, C8×D5, C4.F5, C24.C4, C3×C5⋊2C8, C120, C15⋊C8, D5×C12, D10.Q8, D5×C24, C12.F5, C24.1F5
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, F5, Dic6, D12, C2×Dic3, C8.C4, C2×F5, C4⋊Dic3, C3⋊F5, C4⋊F5, C24.C4, C2×C3⋊F5, D10.Q8, C60⋊C4, C24.1F5
►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 99 179 78 158)(2 100 180 79 159)(3 101 181 80 160)(4 102 182 81 161)(5 103 183 82 162)(6 104 184 83 163)(7 105 185 84 164)(8 106 186 85 165)(9 107 187 86 166)(10 108 188 87 167)(11 109 189 88 168)(12 110 190 89 145)(13 111 191 90 146)(14 112 192 91 147)(15 113 169 92 148)(16 114 170 93 149)(17 115 171 94 150)(18 116 172 95 151)(19 117 173 96 152)(20 118 174 73 153)(21 119 175 74 154)(22 120 176 75 155)(23 97 177 76 156)(24 98 178 77 157)(25 69 123 223 210)(26 70 124 224 211)(27 71 125 225 212)(28 72 126 226 213)(29 49 127 227 214)(30 50 128 228 215)(31 51 129 229 216)(32 52 130 230 193)(33 53 131 231 194)(34 54 132 232 195)(35 55 133 233 196)(36 56 134 234 197)(37 57 135 235 198)(38 58 136 236 199)(39 59 137 237 200)(40 60 138 238 201)(41 61 139 239 202)(42 62 140 240 203)(43 63 141 217 204)(44 64 142 218 205)(45 65 143 219 206)(46 66 144 220 207)(47 67 121 221 208)(48 68 122 222 209)
(1 42 7 36 13 30 19 48)(2 41 8 35 14 29 20 47)(3 40 9 34 15 28 21 46)(4 39 10 33 16 27 22 45)(5 38 11 32 17 26 23 44)(6 37 12 31 18 25 24 43)(49 174 208 79 61 186 196 91)(50 173 209 78 62 185 197 90)(51 172 210 77 63 184 198 89)(52 171 211 76 64 183 199 88)(53 170 212 75 65 182 200 87)(54 169 213 74 66 181 201 86)(55 192 214 73 67 180 202 85)(56 191 215 96 68 179 203 84)(57 190 216 95 69 178 204 83)(58 189 193 94 70 177 205 82)(59 188 194 93 71 176 206 81)(60 187 195 92 72 175 207 80)(97 142 162 236 109 130 150 224)(98 141 163 235 110 129 151 223)(99 140 164 234 111 128 152 222)(100 139 165 233 112 127 153 221)(101 138 166 232 113 126 154 220)(102 137 167 231 114 125 155 219)(103 136 168 230 115 124 156 218)(104 135 145 229 116 123 157 217)(105 134 146 228 117 122 158 240)(106 133 147 227 118 121 159 239)(107 132 148 226 119 144 160 238)(108 131 149 225 120 143 161 237)
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), (1,99,179,78,158)(2,100,180,79,159)(3,101,181,80,160)(4,102,182,81,161)(5,103,183,82,162)(6,104,184,83,163)(7,105,185,84,164)(8,106,186,85,165)(9,107,187,86,166)(10,108,188,87,167)(11,109,189,88,168)(12,110,190,89,145)(13,111,191,90,146)(14,112,192,91,147)(15,113,169,92,148)(16,114,170,93,149)(17,115,171,94,150)(18,116,172,95,151)(19,117,173,96,152)(20,118,174,73,153)(21,119,175,74,154)(22,120,176,75,155)(23,97,177,76,156)(24,98,178,77,157)(25,69,123,223,210)(26,70,124,224,211)(27,71,125,225,212)(28,72,126,226,213)(29,49,127,227,214)(30,50,128,228,215)(31,51,129,229,216)(32,52,130,230,193)(33,53,131,231,194)(34,54,132,232,195)(35,55,133,233,196)(36,56,134,234,197)(37,57,135,235,198)(38,58,136,236,199)(39,59,137,237,200)(40,60,138,238,201)(41,61,139,239,202)(42,62,140,240,203)(43,63,141,217,204)(44,64,142,218,205)(45,65,143,219,206)(46,66,144,220,207)(47,67,121,221,208)(48,68,122,222,209), (1,42,7,36,13,30,19,48)(2,41,8,35,14,29,20,47)(3,40,9,34,15,28,21,46)(4,39,10,33,16,27,22,45)(5,38,11,32,17,26,23,44)(6,37,12,31,18,25,24,43)(49,174,208,79,61,186,196,91)(50,173,209,78,62,185,197,90)(51,172,210,77,63,184,198,89)(52,171,211,76,64,183,199,88)(53,170,212,75,65,182,200,87)(54,169,213,74,66,181,201,86)(55,192,214,73,67,180,202,85)(56,191,215,96,68,179,203,84)(57,190,216,95,69,178,204,83)(58,189,193,94,70,177,205,82)(59,188,194,93,71,176,206,81)(60,187,195,92,72,175,207,80)(97,142,162,236,109,130,150,224)(98,141,163,235,110,129,151,223)(99,140,164,234,111,128,152,222)(100,139,165,233,112,127,153,221)(101,138,166,232,113,126,154,220)(102,137,167,231,114,125,155,219)(103,136,168,230,115,124,156,218)(104,135,145,229,116,123,157,217)(105,134,146,228,117,122,158,240)(106,133,147,227,118,121,159,239)(107,132,148,226,119,144,160,238)(108,131,149,225,120,143,161,237)>;
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), (1,99,179,78,158)(2,100,180,79,159)(3,101,181,80,160)(4,102,182,81,161)(5,103,183,82,162)(6,104,184,83,163)(7,105,185,84,164)(8,106,186,85,165)(9,107,187,86,166)(10,108,188,87,167)(11,109,189,88,168)(12,110,190,89,145)(13,111,191,90,146)(14,112,192,91,147)(15,113,169,92,148)(16,114,170,93,149)(17,115,171,94,150)(18,116,172,95,151)(19,117,173,96,152)(20,118,174,73,153)(21,119,175,74,154)(22,120,176,75,155)(23,97,177,76,156)(24,98,178,77,157)(25,69,123,223,210)(26,70,124,224,211)(27,71,125,225,212)(28,72,126,226,213)(29,49,127,227,214)(30,50,128,228,215)(31,51,129,229,216)(32,52,130,230,193)(33,53,131,231,194)(34,54,132,232,195)(35,55,133,233,196)(36,56,134,234,197)(37,57,135,235,198)(38,58,136,236,199)(39,59,137,237,200)(40,60,138,238,201)(41,61,139,239,202)(42,62,140,240,203)(43,63,141,217,204)(44,64,142,218,205)(45,65,143,219,206)(46,66,144,220,207)(47,67,121,221,208)(48,68,122,222,209), (1,42,7,36,13,30,19,48)(2,41,8,35,14,29,20,47)(3,40,9,34,15,28,21,46)(4,39,10,33,16,27,22,45)(5,38,11,32,17,26,23,44)(6,37,12,31,18,25,24,43)(49,174,208,79,61,186,196,91)(50,173,209,78,62,185,197,90)(51,172,210,77,63,184,198,89)(52,171,211,76,64,183,199,88)(53,170,212,75,65,182,200,87)(54,169,213,74,66,181,201,86)(55,192,214,73,67,180,202,85)(56,191,215,96,68,179,203,84)(57,190,216,95,69,178,204,83)(58,189,193,94,70,177,205,82)(59,188,194,93,71,176,206,81)(60,187,195,92,72,175,207,80)(97,142,162,236,109,130,150,224)(98,141,163,235,110,129,151,223)(99,140,164,234,111,128,152,222)(100,139,165,233,112,127,153,221)(101,138,166,232,113,126,154,220)(102,137,167,231,114,125,155,219)(103,136,168,230,115,124,156,218)(104,135,145,229,116,123,157,217)(105,134,146,228,117,122,158,240)(106,133,147,227,118,121,159,239)(107,132,148,226,119,144,160,238)(108,131,149,225,120,143,161,237) );
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)], [(1,99,179,78,158),(2,100,180,79,159),(3,101,181,80,160),(4,102,182,81,161),(5,103,183,82,162),(6,104,184,83,163),(7,105,185,84,164),(8,106,186,85,165),(9,107,187,86,166),(10,108,188,87,167),(11,109,189,88,168),(12,110,190,89,145),(13,111,191,90,146),(14,112,192,91,147),(15,113,169,92,148),(16,114,170,93,149),(17,115,171,94,150),(18,116,172,95,151),(19,117,173,96,152),(20,118,174,73,153),(21,119,175,74,154),(22,120,176,75,155),(23,97,177,76,156),(24,98,178,77,157),(25,69,123,223,210),(26,70,124,224,211),(27,71,125,225,212),(28,72,126,226,213),(29,49,127,227,214),(30,50,128,228,215),(31,51,129,229,216),(32,52,130,230,193),(33,53,131,231,194),(34,54,132,232,195),(35,55,133,233,196),(36,56,134,234,197),(37,57,135,235,198),(38,58,136,236,199),(39,59,137,237,200),(40,60,138,238,201),(41,61,139,239,202),(42,62,140,240,203),(43,63,141,217,204),(44,64,142,218,205),(45,65,143,219,206),(46,66,144,220,207),(47,67,121,221,208),(48,68,122,222,209)], [(1,42,7,36,13,30,19,48),(2,41,8,35,14,29,20,47),(3,40,9,34,15,28,21,46),(4,39,10,33,16,27,22,45),(5,38,11,32,17,26,23,44),(6,37,12,31,18,25,24,43),(49,174,208,79,61,186,196,91),(50,173,209,78,62,185,197,90),(51,172,210,77,63,184,198,89),(52,171,211,76,64,183,199,88),(53,170,212,75,65,182,200,87),(54,169,213,74,66,181,201,86),(55,192,214,73,67,180,202,85),(56,191,215,96,68,179,203,84),(57,190,216,95,69,178,204,83),(58,189,193,94,70,177,205,82),(59,188,194,93,71,176,206,81),(60,187,195,92,72,175,207,80),(97,142,162,236,109,130,150,224),(98,141,163,235,110,129,151,223),(99,140,164,234,111,128,152,222),(100,139,165,233,112,127,153,221),(101,138,166,232,113,126,154,220),(102,137,167,231,114,125,155,219),(103,136,168,230,115,124,156,218),(104,135,145,229,116,123,157,217),(105,134,146,228,117,122,158,240),(106,133,147,227,118,121,159,239),(107,132,148,226,119,144,160,238),(108,131,149,225,120,143,161,237)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 5 | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10 | 12A | 12B | 12C | 12D | 15A | 15B | 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 | 8 | 8 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 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 | 2 | 5 | 5 | 4 | 2 | 10 | 10 | 2 | 2 | 10 | 10 | 60 | 60 | 60 | 60 | 4 | 2 | 2 | 10 | 10 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | - | - | + | + | - | + | + | ||||||||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | D4 | Q8 | Dic3 | Dic3 | D6 | D12 | Dic6 | C8.C4 | C24.C4 | F5 | C2×F5 | C3⋊F5 | C4⋊F5 | C2×C3⋊F5 | D10.Q8 | C60⋊C4 | C24.1F5 |
| kernel | C24.1F5 | D5×C24 | C12.F5 | C3×C5⋊2C8 | C120 | C8×D5 | C3×Dic5 | C6×D5 | C5⋊2C8 | C40 | C4×D5 | Dic5 | D10 | C15 | C5 | C24 | C12 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C24.1F5 ►in GL8(𝔽241)
| 211 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 57 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 | 34 | 34 |
| 0 | 0 | 0 | 0 | 207 | 224 | 207 | 0 |
| 0 | 0 | 0 | 0 | 0 | 207 | 224 | 207 |
| 0 | 0 | 0 | 0 | 34 | 34 | 0 | 17 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 240 | 240 | 240 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 64 | 123 | 0 | 0 | 0 | 0 | 0 | 0 |
| 26 | 177 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 56 | 191 | 0 | 0 | 0 | 0 |
| 0 | 0 | 135 | 185 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 178 | 159 | 178 |
| 0 | 0 | 0 | 0 | 222 | 0 | 63 | 63 |
| 0 | 0 | 0 | 0 | 63 | 63 | 0 | 222 |
| 0 | 0 | 0 | 0 | 178 | 159 | 178 | 0 |
G:=sub<GL(8,GF(241))| [211,57,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,240,1,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,17,207,0,34,0,0,0,0,0,224,207,34,0,0,0,0,34,207,224,0,0,0,0,0,34,0,207,17],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240,1,0,0,0,0,0,0,240,0,1,0,0,0,0,0,240,0,0,1,0,0,0,0,240,0,0,0],[64,26,0,0,0,0,0,0,123,177,0,0,0,0,0,0,0,0,56,135,0,0,0,0,0,0,191,185,0,0,0,0,0,0,0,0,0,222,63,178,0,0,0,0,178,0,63,159,0,0,0,0,159,63,0,178,0,0,0,0,178,63,222,0] >;
C24.1F5 in GAP, Magma, Sage, TeX
C_{24}._1F_5 % in TeX
G:=Group("C24.1F5"); // GroupNames label
G:=SmallGroup(480,301);
// by ID
G=gap.SmallGroup(480,301);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,176,100,675,80,2693,14118,4724]);
// Polycyclic
G:=Group<a,b,c|a^24=b^5=1,c^4=a^12,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^3>;
// generators/relations