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G = C5×D48order 480 = 25·3·5

Direct product of C5 and D48

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C5×D48
 Chief series C1 — C3 — C6 — C12 — C24 — C120 — C5×D24 — C5×D48
 Lower central C3 — C6 — C12 — C24 — C5×D48
 Upper central C1 — C10 — C20 — C40 — C80

Generators and relations for C5×D48
G = < a,b,c | a5=b48=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C5×D48
On 240 points
Generators in S240
(1 99 51 182 214)(2 100 52 183 215)(3 101 53 184 216)(4 102 54 185 217)(5 103 55 186 218)(6 104 56 187 219)(7 105 57 188 220)(8 106 58 189 221)(9 107 59 190 222)(10 108 60 191 223)(11 109 61 192 224)(12 110 62 145 225)(13 111 63 146 226)(14 112 64 147 227)(15 113 65 148 228)(16 114 66 149 229)(17 115 67 150 230)(18 116 68 151 231)(19 117 69 152 232)(20 118 70 153 233)(21 119 71 154 234)(22 120 72 155 235)(23 121 73 156 236)(24 122 74 157 237)(25 123 75 158 238)(26 124 76 159 239)(27 125 77 160 240)(28 126 78 161 193)(29 127 79 162 194)(30 128 80 163 195)(31 129 81 164 196)(32 130 82 165 197)(33 131 83 166 198)(34 132 84 167 199)(35 133 85 168 200)(36 134 86 169 201)(37 135 87 170 202)(38 136 88 171 203)(39 137 89 172 204)(40 138 90 173 205)(41 139 91 174 206)(42 140 92 175 207)(43 141 93 176 208)(44 142 94 177 209)(45 143 95 178 210)(46 144 96 179 211)(47 97 49 180 212)(48 98 50 181 213)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(49 67)(50 66)(51 65)(52 64)(53 63)(54 62)(55 61)(56 60)(57 59)(68 96)(69 95)(70 94)(71 93)(72 92)(73 91)(74 90)(75 89)(76 88)(77 87)(78 86)(79 85)(80 84)(81 83)(97 115)(98 114)(99 113)(100 112)(101 111)(102 110)(103 109)(104 108)(105 107)(116 144)(117 143)(118 142)(119 141)(120 140)(121 139)(122 138)(123 137)(124 136)(125 135)(126 134)(127 133)(128 132)(129 131)(145 185)(146 184)(147 183)(148 182)(149 181)(150 180)(151 179)(152 178)(153 177)(154 176)(155 175)(156 174)(157 173)(158 172)(159 171)(160 170)(161 169)(162 168)(163 167)(164 166)(186 192)(187 191)(188 190)(193 201)(194 200)(195 199)(196 198)(202 240)(203 239)(204 238)(205 237)(206 236)(207 235)(208 234)(209 233)(210 232)(211 231)(212 230)(213 229)(214 228)(215 227)(216 226)(217 225)(218 224)(219 223)(220 222)

G:=sub<Sym(240)| (1,99,51,182,214)(2,100,52,183,215)(3,101,53,184,216)(4,102,54,185,217)(5,103,55,186,218)(6,104,56,187,219)(7,105,57,188,220)(8,106,58,189,221)(9,107,59,190,222)(10,108,60,191,223)(11,109,61,192,224)(12,110,62,145,225)(13,111,63,146,226)(14,112,64,147,227)(15,113,65,148,228)(16,114,66,149,229)(17,115,67,150,230)(18,116,68,151,231)(19,117,69,152,232)(20,118,70,153,233)(21,119,71,154,234)(22,120,72,155,235)(23,121,73,156,236)(24,122,74,157,237)(25,123,75,158,238)(26,124,76,159,239)(27,125,77,160,240)(28,126,78,161,193)(29,127,79,162,194)(30,128,80,163,195)(31,129,81,164,196)(32,130,82,165,197)(33,131,83,166,198)(34,132,84,167,199)(35,133,85,168,200)(36,134,86,169,201)(37,135,87,170,202)(38,136,88,171,203)(39,137,89,172,204)(40,138,90,173,205)(41,139,91,174,206)(42,140,92,175,207)(43,141,93,176,208)(44,142,94,177,209)(45,143,95,178,210)(46,144,96,179,211)(47,97,49,180,212)(48,98,50,181,213), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(49,67)(50,66)(51,65)(52,64)(53,63)(54,62)(55,61)(56,60)(57,59)(68,96)(69,95)(70,94)(71,93)(72,92)(73,91)(74,90)(75,89)(76,88)(77,87)(78,86)(79,85)(80,84)(81,83)(97,115)(98,114)(99,113)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)(116,144)(117,143)(118,142)(119,141)(120,140)(121,139)(122,138)(123,137)(124,136)(125,135)(126,134)(127,133)(128,132)(129,131)(145,185)(146,184)(147,183)(148,182)(149,181)(150,180)(151,179)(152,178)(153,177)(154,176)(155,175)(156,174)(157,173)(158,172)(159,171)(160,170)(161,169)(162,168)(163,167)(164,166)(186,192)(187,191)(188,190)(193,201)(194,200)(195,199)(196,198)(202,240)(203,239)(204,238)(205,237)(206,236)(207,235)(208,234)(209,233)(210,232)(211,231)(212,230)(213,229)(214,228)(215,227)(216,226)(217,225)(218,224)(219,223)(220,222)>;

G:=Group( (1,99,51,182,214)(2,100,52,183,215)(3,101,53,184,216)(4,102,54,185,217)(5,103,55,186,218)(6,104,56,187,219)(7,105,57,188,220)(8,106,58,189,221)(9,107,59,190,222)(10,108,60,191,223)(11,109,61,192,224)(12,110,62,145,225)(13,111,63,146,226)(14,112,64,147,227)(15,113,65,148,228)(16,114,66,149,229)(17,115,67,150,230)(18,116,68,151,231)(19,117,69,152,232)(20,118,70,153,233)(21,119,71,154,234)(22,120,72,155,235)(23,121,73,156,236)(24,122,74,157,237)(25,123,75,158,238)(26,124,76,159,239)(27,125,77,160,240)(28,126,78,161,193)(29,127,79,162,194)(30,128,80,163,195)(31,129,81,164,196)(32,130,82,165,197)(33,131,83,166,198)(34,132,84,167,199)(35,133,85,168,200)(36,134,86,169,201)(37,135,87,170,202)(38,136,88,171,203)(39,137,89,172,204)(40,138,90,173,205)(41,139,91,174,206)(42,140,92,175,207)(43,141,93,176,208)(44,142,94,177,209)(45,143,95,178,210)(46,144,96,179,211)(47,97,49,180,212)(48,98,50,181,213), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(49,67)(50,66)(51,65)(52,64)(53,63)(54,62)(55,61)(56,60)(57,59)(68,96)(69,95)(70,94)(71,93)(72,92)(73,91)(74,90)(75,89)(76,88)(77,87)(78,86)(79,85)(80,84)(81,83)(97,115)(98,114)(99,113)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)(116,144)(117,143)(118,142)(119,141)(120,140)(121,139)(122,138)(123,137)(124,136)(125,135)(126,134)(127,133)(128,132)(129,131)(145,185)(146,184)(147,183)(148,182)(149,181)(150,180)(151,179)(152,178)(153,177)(154,176)(155,175)(156,174)(157,173)(158,172)(159,171)(160,170)(161,169)(162,168)(163,167)(164,166)(186,192)(187,191)(188,190)(193,201)(194,200)(195,199)(196,198)(202,240)(203,239)(204,238)(205,237)(206,236)(207,235)(208,234)(209,233)(210,232)(211,231)(212,230)(213,229)(214,228)(215,227)(216,226)(217,225)(218,224)(219,223)(220,222) );

G=PermutationGroup([(1,99,51,182,214),(2,100,52,183,215),(3,101,53,184,216),(4,102,54,185,217),(5,103,55,186,218),(6,104,56,187,219),(7,105,57,188,220),(8,106,58,189,221),(9,107,59,190,222),(10,108,60,191,223),(11,109,61,192,224),(12,110,62,145,225),(13,111,63,146,226),(14,112,64,147,227),(15,113,65,148,228),(16,114,66,149,229),(17,115,67,150,230),(18,116,68,151,231),(19,117,69,152,232),(20,118,70,153,233),(21,119,71,154,234),(22,120,72,155,235),(23,121,73,156,236),(24,122,74,157,237),(25,123,75,158,238),(26,124,76,159,239),(27,125,77,160,240),(28,126,78,161,193),(29,127,79,162,194),(30,128,80,163,195),(31,129,81,164,196),(32,130,82,165,197),(33,131,83,166,198),(34,132,84,167,199),(35,133,85,168,200),(36,134,86,169,201),(37,135,87,170,202),(38,136,88,171,203),(39,137,89,172,204),(40,138,90,173,205),(41,139,91,174,206),(42,140,92,175,207),(43,141,93,176,208),(44,142,94,177,209),(45,143,95,178,210),(46,144,96,179,211),(47,97,49,180,212),(48,98,50,181,213)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,48),(17,47),(18,46),(19,45),(20,44),(21,43),(22,42),(23,41),(24,40),(25,39),(26,38),(27,37),(28,36),(29,35),(30,34),(31,33),(49,67),(50,66),(51,65),(52,64),(53,63),(54,62),(55,61),(56,60),(57,59),(68,96),(69,95),(70,94),(71,93),(72,92),(73,91),(74,90),(75,89),(76,88),(77,87),(78,86),(79,85),(80,84),(81,83),(97,115),(98,114),(99,113),(100,112),(101,111),(102,110),(103,109),(104,108),(105,107),(116,144),(117,143),(118,142),(119,141),(120,140),(121,139),(122,138),(123,137),(124,136),(125,135),(126,134),(127,133),(128,132),(129,131),(145,185),(146,184),(147,183),(148,182),(149,181),(150,180),(151,179),(152,178),(153,177),(154,176),(155,175),(156,174),(157,173),(158,172),(159,171),(160,170),(161,169),(162,168),(163,167),(164,166),(186,192),(187,191),(188,190),(193,201),(194,200),(195,199),(196,198),(202,240),(203,239),(204,238),(205,237),(206,236),(207,235),(208,234),(209,233),(210,232),(211,231),(212,230),(213,229),(214,228),(215,227),(216,226),(217,225),(218,224),(219,223),(220,222)])

135 conjugacy classes

 class 1 2A 2B 2C 3 4 5A 5B 5C 5D 6 8A 8B 10A 10B 10C 10D 10E ··· 10L 12A 12B 15A 15B 15C 15D 16A 16B 16C 16D 20A 20B 20C 20D 24A 24B 24C 24D 30A 30B 30C 30D 40A ··· 40H 48A ··· 48H 60A ··· 60H 80A ··· 80P 120A ··· 120P 240A ··· 240AF order 1 2 2 2 3 4 5 5 5 5 6 8 8 10 10 10 10 10 ··· 10 12 12 15 15 15 15 16 16 16 16 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 24 2 2 1 1 1 1 2 2 2 1 1 1 1 24 ··· 24 2 2 2 2 2 2 2 2 2 2 2 2 2 2 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 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C5 C10 C10 S3 D4 D6 D8 D12 C5×S3 D16 C5×D4 D24 S3×C10 C5×D8 D48 C5×D12 C5×D16 C5×D24 C5×D48 kernel C5×D48 C240 C5×D24 D48 C48 D24 C80 C60 C40 C30 C20 C16 C15 C12 C10 C8 C6 C5 C4 C3 C2 C1 # reps 1 1 2 4 4 8 1 1 1 2 2 4 4 4 4 4 8 8 8 16 16 32

Matrix representation of C5×D48 in GL2(𝔽241) generated by

 91 0 0 91
,
 197 107 134 63
,
 136 232 127 105
G:=sub<GL(2,GF(241))| [91,0,0,91],[197,134,107,63],[136,127,232,105] >;

C5×D48 in GAP, Magma, Sage, TeX

C_5\times D_{48}
% in TeX

G:=Group("C5xD48");
// GroupNames label

G:=SmallGroup(480,118);
// by ID

G=gap.SmallGroup(480,118);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,309,428,1683,192,4204,102,15686]);
// 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^-1>;
// generators/relations

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