direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C48⋊C2, C80⋊6S3, C240⋊8C2, C48⋊2C10, C15⋊9SD32, C40.76D6, C30.33D8, D24.1C10, C10.14D24, C20.40D12, C60.178D4, Dic12⋊1C10, C120.94C22, C16⋊2(C5×S3), C6.2(C5×D8), C3⋊1(C5×SD32), C2.4(C5×D24), C4.2(C5×D12), C8.14(S3×C10), (C5×D24).3C2, C12.25(C5×D4), C24.15(C2×C10), (C5×Dic12)⋊9C2, SmallGroup(480,119)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C48⋊C2
G = < a,b,c | a5=b48=c2=1, ab=ba, ac=ca, cbc=b23 >
Smallest permutation representation of C5×C48⋊C2
►On 240 points
Generators in S240
(1 105 210 170 94)(2 106 211 171 95)(3 107 212 172 96)(4 108 213 173 49)(5 109 214 174 50)(6 110 215 175 51)(7 111 216 176 52)(8 112 217 177 53)(9 113 218 178 54)(10 114 219 179 55)(11 115 220 180 56)(12 116 221 181 57)(13 117 222 182 58)(14 118 223 183 59)(15 119 224 184 60)(16 120 225 185 61)(17 121 226 186 62)(18 122 227 187 63)(19 123 228 188 64)(20 124 229 189 65)(21 125 230 190 66)(22 126 231 191 67)(23 127 232 192 68)(24 128 233 145 69)(25 129 234 146 70)(26 130 235 147 71)(27 131 236 148 72)(28 132 237 149 73)(29 133 238 150 74)(30 134 239 151 75)(31 135 240 152 76)(32 136 193 153 77)(33 137 194 154 78)(34 138 195 155 79)(35 139 196 156 80)(36 140 197 157 81)(37 141 198 158 82)(38 142 199 159 83)(39 143 200 160 84)(40 144 201 161 85)(41 97 202 162 86)(42 98 203 163 87)(43 99 204 164 88)(44 100 205 165 89)(45 101 206 166 90)(46 102 207 167 91)(47 103 208 168 92)(48 104 209 169 93)
(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)
(2 24)(3 47)(4 22)(5 45)(6 20)(7 43)(8 18)(9 41)(10 16)(11 39)(12 14)(13 37)(15 35)(17 33)(19 31)(21 29)(23 27)(26 48)(28 46)(30 44)(32 42)(34 40)(36 38)(49 67)(50 90)(51 65)(52 88)(53 63)(54 86)(55 61)(56 84)(57 59)(58 82)(60 80)(62 78)(64 76)(66 74)(68 72)(69 95)(71 93)(73 91)(75 89)(77 87)(79 85)(81 83)(92 96)(97 113)(98 136)(99 111)(100 134)(101 109)(102 132)(103 107)(104 130)(106 128)(108 126)(110 124)(112 122)(114 120)(115 143)(116 118)(117 141)(119 139)(121 137)(123 135)(125 133)(127 131)(138 144)(140 142)(145 171)(147 169)(148 192)(149 167)(150 190)(151 165)(152 188)(153 163)(154 186)(155 161)(156 184)(157 159)(158 182)(160 180)(162 178)(164 176)(166 174)(168 172)(173 191)(175 189)(177 187)(179 185)(181 183)(193 203)(194 226)(195 201)(196 224)(197 199)(198 222)(200 220)(202 218)(204 216)(205 239)(206 214)(207 237)(208 212)(209 235)(211 233)(213 231)(215 229)(217 227)(219 225)(221 223)(228 240)(230 238)(232 236)
G:=sub<Sym(240)| (1,105,210,170,94)(2,106,211,171,95)(3,107,212,172,96)(4,108,213,173,49)(5,109,214,174,50)(6,110,215,175,51)(7,111,216,176,52)(8,112,217,177,53)(9,113,218,178,54)(10,114,219,179,55)(11,115,220,180,56)(12,116,221,181,57)(13,117,222,182,58)(14,118,223,183,59)(15,119,224,184,60)(16,120,225,185,61)(17,121,226,186,62)(18,122,227,187,63)(19,123,228,188,64)(20,124,229,189,65)(21,125,230,190,66)(22,126,231,191,67)(23,127,232,192,68)(24,128,233,145,69)(25,129,234,146,70)(26,130,235,147,71)(27,131,236,148,72)(28,132,237,149,73)(29,133,238,150,74)(30,134,239,151,75)(31,135,240,152,76)(32,136,193,153,77)(33,137,194,154,78)(34,138,195,155,79)(35,139,196,156,80)(36,140,197,157,81)(37,141,198,158,82)(38,142,199,159,83)(39,143,200,160,84)(40,144,201,161,85)(41,97,202,162,86)(42,98,203,163,87)(43,99,204,164,88)(44,100,205,165,89)(45,101,206,166,90)(46,102,207,167,91)(47,103,208,168,92)(48,104,209,169,93), (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), (2,24)(3,47)(4,22)(5,45)(6,20)(7,43)(8,18)(9,41)(10,16)(11,39)(12,14)(13,37)(15,35)(17,33)(19,31)(21,29)(23,27)(26,48)(28,46)(30,44)(32,42)(34,40)(36,38)(49,67)(50,90)(51,65)(52,88)(53,63)(54,86)(55,61)(56,84)(57,59)(58,82)(60,80)(62,78)(64,76)(66,74)(68,72)(69,95)(71,93)(73,91)(75,89)(77,87)(79,85)(81,83)(92,96)(97,113)(98,136)(99,111)(100,134)(101,109)(102,132)(103,107)(104,130)(106,128)(108,126)(110,124)(112,122)(114,120)(115,143)(116,118)(117,141)(119,139)(121,137)(123,135)(125,133)(127,131)(138,144)(140,142)(145,171)(147,169)(148,192)(149,167)(150,190)(151,165)(152,188)(153,163)(154,186)(155,161)(156,184)(157,159)(158,182)(160,180)(162,178)(164,176)(166,174)(168,172)(173,191)(175,189)(177,187)(179,185)(181,183)(193,203)(194,226)(195,201)(196,224)(197,199)(198,222)(200,220)(202,218)(204,216)(205,239)(206,214)(207,237)(208,212)(209,235)(211,233)(213,231)(215,229)(217,227)(219,225)(221,223)(228,240)(230,238)(232,236)>;
G:=Group( (1,105,210,170,94)(2,106,211,171,95)(3,107,212,172,96)(4,108,213,173,49)(5,109,214,174,50)(6,110,215,175,51)(7,111,216,176,52)(8,112,217,177,53)(9,113,218,178,54)(10,114,219,179,55)(11,115,220,180,56)(12,116,221,181,57)(13,117,222,182,58)(14,118,223,183,59)(15,119,224,184,60)(16,120,225,185,61)(17,121,226,186,62)(18,122,227,187,63)(19,123,228,188,64)(20,124,229,189,65)(21,125,230,190,66)(22,126,231,191,67)(23,127,232,192,68)(24,128,233,145,69)(25,129,234,146,70)(26,130,235,147,71)(27,131,236,148,72)(28,132,237,149,73)(29,133,238,150,74)(30,134,239,151,75)(31,135,240,152,76)(32,136,193,153,77)(33,137,194,154,78)(34,138,195,155,79)(35,139,196,156,80)(36,140,197,157,81)(37,141,198,158,82)(38,142,199,159,83)(39,143,200,160,84)(40,144,201,161,85)(41,97,202,162,86)(42,98,203,163,87)(43,99,204,164,88)(44,100,205,165,89)(45,101,206,166,90)(46,102,207,167,91)(47,103,208,168,92)(48,104,209,169,93), (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), (2,24)(3,47)(4,22)(5,45)(6,20)(7,43)(8,18)(9,41)(10,16)(11,39)(12,14)(13,37)(15,35)(17,33)(19,31)(21,29)(23,27)(26,48)(28,46)(30,44)(32,42)(34,40)(36,38)(49,67)(50,90)(51,65)(52,88)(53,63)(54,86)(55,61)(56,84)(57,59)(58,82)(60,80)(62,78)(64,76)(66,74)(68,72)(69,95)(71,93)(73,91)(75,89)(77,87)(79,85)(81,83)(92,96)(97,113)(98,136)(99,111)(100,134)(101,109)(102,132)(103,107)(104,130)(106,128)(108,126)(110,124)(112,122)(114,120)(115,143)(116,118)(117,141)(119,139)(121,137)(123,135)(125,133)(127,131)(138,144)(140,142)(145,171)(147,169)(148,192)(149,167)(150,190)(151,165)(152,188)(153,163)(154,186)(155,161)(156,184)(157,159)(158,182)(160,180)(162,178)(164,176)(166,174)(168,172)(173,191)(175,189)(177,187)(179,185)(181,183)(193,203)(194,226)(195,201)(196,224)(197,199)(198,222)(200,220)(202,218)(204,216)(205,239)(206,214)(207,237)(208,212)(209,235)(211,233)(213,231)(215,229)(217,227)(219,225)(221,223)(228,240)(230,238)(232,236) );
G=PermutationGroup([[(1,105,210,170,94),(2,106,211,171,95),(3,107,212,172,96),(4,108,213,173,49),(5,109,214,174,50),(6,110,215,175,51),(7,111,216,176,52),(8,112,217,177,53),(9,113,218,178,54),(10,114,219,179,55),(11,115,220,180,56),(12,116,221,181,57),(13,117,222,182,58),(14,118,223,183,59),(15,119,224,184,60),(16,120,225,185,61),(17,121,226,186,62),(18,122,227,187,63),(19,123,228,188,64),(20,124,229,189,65),(21,125,230,190,66),(22,126,231,191,67),(23,127,232,192,68),(24,128,233,145,69),(25,129,234,146,70),(26,130,235,147,71),(27,131,236,148,72),(28,132,237,149,73),(29,133,238,150,74),(30,134,239,151,75),(31,135,240,152,76),(32,136,193,153,77),(33,137,194,154,78),(34,138,195,155,79),(35,139,196,156,80),(36,140,197,157,81),(37,141,198,158,82),(38,142,199,159,83),(39,143,200,160,84),(40,144,201,161,85),(41,97,202,162,86),(42,98,203,163,87),(43,99,204,164,88),(44,100,205,165,89),(45,101,206,166,90),(46,102,207,167,91),(47,103,208,168,92),(48,104,209,169,93)], [(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)], [(2,24),(3,47),(4,22),(5,45),(6,20),(7,43),(8,18),(9,41),(10,16),(11,39),(12,14),(13,37),(15,35),(17,33),(19,31),(21,29),(23,27),(26,48),(28,46),(30,44),(32,42),(34,40),(36,38),(49,67),(50,90),(51,65),(52,88),(53,63),(54,86),(55,61),(56,84),(57,59),(58,82),(60,80),(62,78),(64,76),(66,74),(68,72),(69,95),(71,93),(73,91),(75,89),(77,87),(79,85),(81,83),(92,96),(97,113),(98,136),(99,111),(100,134),(101,109),(102,132),(103,107),(104,130),(106,128),(108,126),(110,124),(112,122),(114,120),(115,143),(116,118),(117,141),(119,139),(121,137),(123,135),(125,133),(127,131),(138,144),(140,142),(145,171),(147,169),(148,192),(149,167),(150,190),(151,165),(152,188),(153,163),(154,186),(155,161),(156,184),(157,159),(158,182),(160,180),(162,178),(164,176),(166,174),(168,172),(173,191),(175,189),(177,187),(179,185),(181,183),(193,203),(194,226),(195,201),(196,224),(197,199),(198,222),(200,220),(202,218),(204,216),(205,239),(206,214),(207,237),(208,212),(209,235),(211,233),(213,231),(215,229),(217,227),(219,225),(221,223),(228,240),(230,238),(232,236)]])
135 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 5A | 5B | 5C | 5D | 6 | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 15A | 15B | 15C | 15D | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 40A | ··· | 40H | 48A | ··· | 48H | 60A | ··· | 60H | 80A | ··· | 80P | 120A | ··· | 120P | 240A | ··· | 240AF |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 48 | ··· | 48 | 60 | ··· | 60 | 80 | ··· | 80 | 120 | ··· | 120 | 240 | ··· | 240 |
| size | 1 | 1 | 24 | 2 | 2 | 24 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 24 | 24 | 24 | 24 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 24 | 24 | 24 | 24 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
135 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | S3 | D4 | D6 | D8 | D12 | C5×S3 | SD32 | C5×D4 | D24 | S3×C10 | C5×D8 | C48⋊C2 | C5×D12 | C5×SD32 | C5×D24 | C5×C48⋊C2 |
| kernel | C5×C48⋊C2 | C240 | C5×D24 | C5×Dic12 | C48⋊C2 | C48 | D24 | Dic12 | C80 | C60 | C40 | C30 | C20 | C16 | C15 | C12 | C10 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 16 | 16 | 32 |
Matrix representation of C5×C48⋊C2 ►in GL2(𝔽241) generated by
| 91 | 0 |
| 0 | 91 |
| 76 | 100 |
| 141 | 176 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(241))| [91,0,0,91],[76,141,100,176],[0,1,1,0] >;
C5×C48⋊C2 in GAP, Magma, Sage, TeX
C_5\times C_{48}\rtimes C_2 % in TeX
G:=Group("C5xC48:C2"); // GroupNames label
G:=SmallGroup(480,119);
// by ID
G=gap.SmallGroup(480,119);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,309,428,3923,80,4204,102,15686]);
// Polycyclic
G:=Group<a,b,c|a^5=b^48=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^23>;
// generators/relations
Export