Copied to
clipboard

G = C5×C3⋊C16order 240 = 24·3·5

Direct product of C5 and C3⋊C16

Aliases: C5×C3⋊C16, C3⋊C80, C6.C40, C155C16, C30.5C8, C40.4S3, C12.2C20, C120.7C2, C24.3C10, C60.14C4, C20.8Dic3, C8.2(C5×S3), C10.3(C3⋊C8), C4.2(C5×Dic3), C2.(C5×C3⋊C8), SmallGroup(240,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C5×C3⋊C16
 Chief series C1 — C3 — C6 — C12 — C24 — C120 — C5×C3⋊C16
 Lower central C3 — C5×C3⋊C16
 Upper central C1 — C40

Generators and relations for C5×C3⋊C16
G = < a,b,c | a5=b3=c16=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C5×C3⋊C16
Regular action on 240 points
Generators in S240
(1 40 29 174 146)(2 41 30 175 147)(3 42 31 176 148)(4 43 32 161 149)(5 44 17 162 150)(6 45 18 163 151)(7 46 19 164 152)(8 47 20 165 153)(9 48 21 166 154)(10 33 22 167 155)(11 34 23 168 156)(12 35 24 169 157)(13 36 25 170 158)(14 37 26 171 159)(15 38 27 172 160)(16 39 28 173 145)(49 127 129 69 230)(50 128 130 70 231)(51 113 131 71 232)(52 114 132 72 233)(53 115 133 73 234)(54 116 134 74 235)(55 117 135 75 236)(56 118 136 76 237)(57 119 137 77 238)(58 120 138 78 239)(59 121 139 79 240)(60 122 140 80 225)(61 123 141 65 226)(62 124 142 66 227)(63 125 143 67 228)(64 126 144 68 229)(81 201 222 183 102)(82 202 223 184 103)(83 203 224 185 104)(84 204 209 186 105)(85 205 210 187 106)(86 206 211 188 107)(87 207 212 189 108)(88 208 213 190 109)(89 193 214 191 110)(90 194 215 192 111)(91 195 216 177 112)(92 196 217 178 97)(93 197 218 179 98)(94 198 219 180 99)(95 199 220 181 100)(96 200 221 182 101)
(1 96 70)(2 71 81)(3 82 72)(4 73 83)(5 84 74)(6 75 85)(7 86 76)(8 77 87)(9 88 78)(10 79 89)(11 90 80)(12 65 91)(13 92 66)(14 67 93)(15 94 68)(16 69 95)(17 209 54)(18 55 210)(19 211 56)(20 57 212)(21 213 58)(22 59 214)(23 215 60)(24 61 216)(25 217 62)(26 63 218)(27 219 64)(28 49 220)(29 221 50)(30 51 222)(31 223 52)(32 53 224)(33 240 193)(34 194 225)(35 226 195)(36 196 227)(37 228 197)(38 198 229)(39 230 199)(40 200 231)(41 232 201)(42 202 233)(43 234 203)(44 204 235)(45 236 205)(46 206 237)(47 238 207)(48 208 239)(97 142 158)(98 159 143)(99 144 160)(100 145 129)(101 130 146)(102 147 131)(103 132 148)(104 149 133)(105 134 150)(106 151 135)(107 136 152)(108 153 137)(109 138 154)(110 155 139)(111 140 156)(112 157 141)(113 183 175)(114 176 184)(115 185 161)(116 162 186)(117 187 163)(118 164 188)(119 189 165)(120 166 190)(121 191 167)(122 168 192)(123 177 169)(124 170 178)(125 179 171)(126 172 180)(127 181 173)(128 174 182)
(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)

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

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

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

C5×C3⋊C16 is a maximal subgroup of
D152C16  C40.51D6  D30.5C8  C3⋊D80  D40.S3  C24.D10  C3⋊Dic40  S3×C80

120 conjugacy classes

 class 1 2 3 4A 4B 5A 5B 5C 5D 6 8A 8B 8C 8D 10A 10B 10C 10D 12A 12B 15A 15B 15C 15D 16A ··· 16H 20A ··· 20H 24A 24B 24C 24D 30A 30B 30C 30D 40A ··· 40P 60A ··· 60H 80A ··· 80AF 120A ··· 120P order 1 2 3 4 4 5 5 5 5 6 8 8 8 8 10 10 10 10 12 12 15 15 15 15 16 ··· 16 20 ··· 20 24 24 24 24 30 30 30 30 40 ··· 40 60 ··· 60 80 ··· 80 120 ··· 120 size 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 ··· 3 1 ··· 1 2 2 2 2 2 2 2 2 1 ··· 1 2 ··· 2 3 ··· 3 2 ··· 2

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + - image C1 C2 C4 C5 C8 C10 C16 C20 C40 C80 S3 Dic3 C3⋊C8 C5×S3 C3⋊C16 C5×Dic3 C5×C3⋊C8 C5×C3⋊C16 kernel C5×C3⋊C16 C120 C60 C3⋊C16 C30 C24 C15 C12 C6 C3 C40 C20 C10 C8 C5 C4 C2 C1 # reps 1 1 2 4 4 4 8 8 16 32 1 1 2 4 4 4 8 16

Matrix representation of C5×C3⋊C16 in GL2(𝔽41) generated by

 18 0 0 18
,
 3 15 21 37
,
 0 34 18 0
G:=sub<GL(2,GF(41))| [18,0,0,18],[3,21,15,37],[0,18,34,0] >;

C5×C3⋊C16 in GAP, Magma, Sage, TeX

C_5\times C_3\rtimes C_{16}
% in TeX

G:=Group("C5xC3:C16");
// GroupNames label

G:=SmallGroup(240,1);
// by ID

G=gap.SmallGroup(240,1);
# by ID

G:=PCGroup([6,-2,-5,-2,-2,-2,-3,60,50,69,5765]);
// Polycyclic

G:=Group<a,b,c|a^5=b^3=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

׿
×
𝔽