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G = C3×C5⋊2C16order 240 = 24·3·5

Direct product of C3 and C5⋊2C16

Aliases: C3×C52C16, C52C48, C154C16, C30.4C8, C40.2C6, C24.4D5, C20.5C12, C10.2C24, C120.5C2, C60.12C4, C12.5Dic5, C8.2(C3×D5), C6.2(C52C8), C4.2(C3×Dic5), C2.(C3×C52C8), SmallGroup(240,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C3×C5⋊2C16
 Chief series C1 — C5 — C10 — C20 — C40 — C120 — C3×C5⋊2C16
 Lower central C5 — C3×C5⋊2C16
 Upper central C1 — C24

Generators and relations for C3×C52C16
G = < a,b,c | a3=b5=c16=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C3×C52C16
Regular action on 240 points
Generators in S240
(1 171 67)(2 172 68)(3 173 69)(4 174 70)(5 175 71)(6 176 72)(7 161 73)(8 162 74)(9 163 75)(10 164 76)(11 165 77)(12 166 78)(13 167 79)(14 168 80)(15 169 65)(16 170 66)(17 38 138)(18 39 139)(19 40 140)(20 41 141)(21 42 142)(22 43 143)(23 44 144)(24 45 129)(25 46 130)(26 47 131)(27 48 132)(28 33 133)(29 34 134)(30 35 135)(31 36 136)(32 37 137)(49 205 96)(50 206 81)(51 207 82)(52 208 83)(53 193 84)(54 194 85)(55 195 86)(56 196 87)(57 197 88)(58 198 89)(59 199 90)(60 200 91)(61 201 92)(62 202 93)(63 203 94)(64 204 95)(97 236 149)(98 237 150)(99 238 151)(100 239 152)(101 240 153)(102 225 154)(103 226 155)(104 227 156)(105 228 157)(106 229 158)(107 230 159)(108 231 160)(109 232 145)(110 233 146)(111 234 147)(112 235 148)(113 217 177)(114 218 178)(115 219 179)(116 220 180)(117 221 181)(118 222 182)(119 223 183)(120 224 184)(121 209 185)(122 210 186)(123 211 187)(124 212 188)(125 213 189)(126 214 190)(127 215 191)(128 216 192)
(1 122 200 151 129)(2 130 152 201 123)(3 124 202 153 131)(4 132 154 203 125)(5 126 204 155 133)(6 134 156 205 127)(7 128 206 157 135)(8 136 158 207 113)(9 114 208 159 137)(10 138 160 193 115)(11 116 194 145 139)(12 140 146 195 117)(13 118 196 147 141)(14 142 148 197 119)(15 120 198 149 143)(16 144 150 199 121)(17 108 84 219 164)(18 165 220 85 109)(19 110 86 221 166)(20 167 222 87 111)(21 112 88 223 168)(22 169 224 89 97)(23 98 90 209 170)(24 171 210 91 99)(25 100 92 211 172)(26 173 212 93 101)(27 102 94 213 174)(28 175 214 95 103)(29 104 96 215 176)(30 161 216 81 105)(31 106 82 217 162)(32 163 218 83 107)(33 71 190 64 226)(34 227 49 191 72)(35 73 192 50 228)(36 229 51 177 74)(37 75 178 52 230)(38 231 53 179 76)(39 77 180 54 232)(40 233 55 181 78)(41 79 182 56 234)(42 235 57 183 80)(43 65 184 58 236)(44 237 59 185 66)(45 67 186 60 238)(46 239 61 187 68)(47 69 188 62 240)(48 225 63 189 70)
(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)| (1,171,67)(2,172,68)(3,173,69)(4,174,70)(5,175,71)(6,176,72)(7,161,73)(8,162,74)(9,163,75)(10,164,76)(11,165,77)(12,166,78)(13,167,79)(14,168,80)(15,169,65)(16,170,66)(17,38,138)(18,39,139)(19,40,140)(20,41,141)(21,42,142)(22,43,143)(23,44,144)(24,45,129)(25,46,130)(26,47,131)(27,48,132)(28,33,133)(29,34,134)(30,35,135)(31,36,136)(32,37,137)(49,205,96)(50,206,81)(51,207,82)(52,208,83)(53,193,84)(54,194,85)(55,195,86)(56,196,87)(57,197,88)(58,198,89)(59,199,90)(60,200,91)(61,201,92)(62,202,93)(63,203,94)(64,204,95)(97,236,149)(98,237,150)(99,238,151)(100,239,152)(101,240,153)(102,225,154)(103,226,155)(104,227,156)(105,228,157)(106,229,158)(107,230,159)(108,231,160)(109,232,145)(110,233,146)(111,234,147)(112,235,148)(113,217,177)(114,218,178)(115,219,179)(116,220,180)(117,221,181)(118,222,182)(119,223,183)(120,224,184)(121,209,185)(122,210,186)(123,211,187)(124,212,188)(125,213,189)(126,214,190)(127,215,191)(128,216,192), (1,122,200,151,129)(2,130,152,201,123)(3,124,202,153,131)(4,132,154,203,125)(5,126,204,155,133)(6,134,156,205,127)(7,128,206,157,135)(8,136,158,207,113)(9,114,208,159,137)(10,138,160,193,115)(11,116,194,145,139)(12,140,146,195,117)(13,118,196,147,141)(14,142,148,197,119)(15,120,198,149,143)(16,144,150,199,121)(17,108,84,219,164)(18,165,220,85,109)(19,110,86,221,166)(20,167,222,87,111)(21,112,88,223,168)(22,169,224,89,97)(23,98,90,209,170)(24,171,210,91,99)(25,100,92,211,172)(26,173,212,93,101)(27,102,94,213,174)(28,175,214,95,103)(29,104,96,215,176)(30,161,216,81,105)(31,106,82,217,162)(32,163,218,83,107)(33,71,190,64,226)(34,227,49,191,72)(35,73,192,50,228)(36,229,51,177,74)(37,75,178,52,230)(38,231,53,179,76)(39,77,180,54,232)(40,233,55,181,78)(41,79,182,56,234)(42,235,57,183,80)(43,65,184,58,236)(44,237,59,185,66)(45,67,186,60,238)(46,239,61,187,68)(47,69,188,62,240)(48,225,63,189,70), (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( (1,171,67)(2,172,68)(3,173,69)(4,174,70)(5,175,71)(6,176,72)(7,161,73)(8,162,74)(9,163,75)(10,164,76)(11,165,77)(12,166,78)(13,167,79)(14,168,80)(15,169,65)(16,170,66)(17,38,138)(18,39,139)(19,40,140)(20,41,141)(21,42,142)(22,43,143)(23,44,144)(24,45,129)(25,46,130)(26,47,131)(27,48,132)(28,33,133)(29,34,134)(30,35,135)(31,36,136)(32,37,137)(49,205,96)(50,206,81)(51,207,82)(52,208,83)(53,193,84)(54,194,85)(55,195,86)(56,196,87)(57,197,88)(58,198,89)(59,199,90)(60,200,91)(61,201,92)(62,202,93)(63,203,94)(64,204,95)(97,236,149)(98,237,150)(99,238,151)(100,239,152)(101,240,153)(102,225,154)(103,226,155)(104,227,156)(105,228,157)(106,229,158)(107,230,159)(108,231,160)(109,232,145)(110,233,146)(111,234,147)(112,235,148)(113,217,177)(114,218,178)(115,219,179)(116,220,180)(117,221,181)(118,222,182)(119,223,183)(120,224,184)(121,209,185)(122,210,186)(123,211,187)(124,212,188)(125,213,189)(126,214,190)(127,215,191)(128,216,192), (1,122,200,151,129)(2,130,152,201,123)(3,124,202,153,131)(4,132,154,203,125)(5,126,204,155,133)(6,134,156,205,127)(7,128,206,157,135)(8,136,158,207,113)(9,114,208,159,137)(10,138,160,193,115)(11,116,194,145,139)(12,140,146,195,117)(13,118,196,147,141)(14,142,148,197,119)(15,120,198,149,143)(16,144,150,199,121)(17,108,84,219,164)(18,165,220,85,109)(19,110,86,221,166)(20,167,222,87,111)(21,112,88,223,168)(22,169,224,89,97)(23,98,90,209,170)(24,171,210,91,99)(25,100,92,211,172)(26,173,212,93,101)(27,102,94,213,174)(28,175,214,95,103)(29,104,96,215,176)(30,161,216,81,105)(31,106,82,217,162)(32,163,218,83,107)(33,71,190,64,226)(34,227,49,191,72)(35,73,192,50,228)(36,229,51,177,74)(37,75,178,52,230)(38,231,53,179,76)(39,77,180,54,232)(40,233,55,181,78)(41,79,182,56,234)(42,235,57,183,80)(43,65,184,58,236)(44,237,59,185,66)(45,67,186,60,238)(46,239,61,187,68)(47,69,188,62,240)(48,225,63,189,70), (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([(1,171,67),(2,172,68),(3,173,69),(4,174,70),(5,175,71),(6,176,72),(7,161,73),(8,162,74),(9,163,75),(10,164,76),(11,165,77),(12,166,78),(13,167,79),(14,168,80),(15,169,65),(16,170,66),(17,38,138),(18,39,139),(19,40,140),(20,41,141),(21,42,142),(22,43,143),(23,44,144),(24,45,129),(25,46,130),(26,47,131),(27,48,132),(28,33,133),(29,34,134),(30,35,135),(31,36,136),(32,37,137),(49,205,96),(50,206,81),(51,207,82),(52,208,83),(53,193,84),(54,194,85),(55,195,86),(56,196,87),(57,197,88),(58,198,89),(59,199,90),(60,200,91),(61,201,92),(62,202,93),(63,203,94),(64,204,95),(97,236,149),(98,237,150),(99,238,151),(100,239,152),(101,240,153),(102,225,154),(103,226,155),(104,227,156),(105,228,157),(106,229,158),(107,230,159),(108,231,160),(109,232,145),(110,233,146),(111,234,147),(112,235,148),(113,217,177),(114,218,178),(115,219,179),(116,220,180),(117,221,181),(118,222,182),(119,223,183),(120,224,184),(121,209,185),(122,210,186),(123,211,187),(124,212,188),(125,213,189),(126,214,190),(127,215,191),(128,216,192)], [(1,122,200,151,129),(2,130,152,201,123),(3,124,202,153,131),(4,132,154,203,125),(5,126,204,155,133),(6,134,156,205,127),(7,128,206,157,135),(8,136,158,207,113),(9,114,208,159,137),(10,138,160,193,115),(11,116,194,145,139),(12,140,146,195,117),(13,118,196,147,141),(14,142,148,197,119),(15,120,198,149,143),(16,144,150,199,121),(17,108,84,219,164),(18,165,220,85,109),(19,110,86,221,166),(20,167,222,87,111),(21,112,88,223,168),(22,169,224,89,97),(23,98,90,209,170),(24,171,210,91,99),(25,100,92,211,172),(26,173,212,93,101),(27,102,94,213,174),(28,175,214,95,103),(29,104,96,215,176),(30,161,216,81,105),(31,106,82,217,162),(32,163,218,83,107),(33,71,190,64,226),(34,227,49,191,72),(35,73,192,50,228),(36,229,51,177,74),(37,75,178,52,230),(38,231,53,179,76),(39,77,180,54,232),(40,233,55,181,78),(41,79,182,56,234),(42,235,57,183,80),(43,65,184,58,236),(44,237,59,185,66),(45,67,186,60,238),(46,239,61,187,68),(47,69,188,62,240),(48,225,63,189,70)], [(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)])

C3×C52C16 is a maximal subgroup of
C15⋊C32  D152C16  C40.52D6  D30.5C8  C5⋊D48  D24.D5  Dic12⋊D5  C5⋊Dic24  D5×C48

96 conjugacy classes

 class 1 2 3A 3B 4A 4B 5A 5B 6A 6B 8A 8B 8C 8D 10A 10B 12A 12B 12C 12D 15A 15B 15C 15D 16A ··· 16H 20A 20B 20C 20D 24A ··· 24H 30A 30B 30C 30D 40A ··· 40H 48A ··· 48P 60A ··· 60H 120A ··· 120P order 1 2 3 3 4 4 5 5 6 6 8 8 8 8 10 10 12 12 12 12 15 15 15 15 16 ··· 16 20 20 20 20 24 ··· 24 30 30 30 30 40 ··· 40 48 ··· 48 60 ··· 60 120 ··· 120 size 1 1 1 1 1 1 2 2 1 1 1 1 1 1 2 2 1 1 1 1 2 2 2 2 5 ··· 5 2 2 2 2 1 ··· 1 2 2 2 2 2 ··· 2 5 ··· 5 2 ··· 2 2 ··· 2

96 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 C3 C4 C6 C8 C12 C16 C24 C48 D5 Dic5 C3×D5 C5⋊2C8 C3×Dic5 C5⋊2C16 C3×C5⋊2C8 C3×C5⋊2C16 kernel C3×C5⋊2C16 C120 C5⋊2C16 C60 C40 C30 C20 C15 C10 C5 C24 C12 C8 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 4 4 8 8 16 2 2 4 4 4 8 8 16

Matrix representation of C3×C52C16 in GL3(𝔽241) generated by

 225 0 0 0 15 0 0 0 15
,
 1 0 0 0 51 240 0 1 0
,
 111 0 0 0 160 102 0 68 81
G:=sub<GL(3,GF(241))| [225,0,0,0,15,0,0,0,15],[1,0,0,0,51,1,0,240,0],[111,0,0,0,160,68,0,102,81] >;

C3×C52C16 in GAP, Magma, Sage, TeX

C_3\times C_5\rtimes_2C_{16}
% in TeX

G:=Group("C3xC5:2C16");
// GroupNames label

G:=SmallGroup(240,2);
// by ID

G=gap.SmallGroup(240,2);
# by ID

G:=PCGroup([6,-2,-3,-2,-2,-2,-5,36,50,69,6917]);
// Polycyclic

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

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