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G = C3×C5⋊C16order 240 = 24·3·5

Direct product of C3 and C5⋊C16

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C3×C5⋊C16, C5⋊C48, C152C16, C10.C24, C30.2C8, C60.5C4, C12.5F5, C20.2C12, C6.2(C5⋊C8), C4.2(C3×F5), C52C8.2C6, C2.(C3×C5⋊C8), (C3×C52C8).4C2, SmallGroup(240,5)

Series: Derived Chief Lower central Upper central

C1C5 — C3×C5⋊C16
C1C5C10C20C52C8C3×C52C8 — C3×C5⋊C16
C5 — C3×C5⋊C16
C1C12

Generators and relations for C3×C5⋊C16
 G = < a,b,c | a3=b5=c16=1, ab=ba, ac=ca, cbc-1=b3 >

5C8
5C16
5C24
5C48

Smallest permutation representation of C3×C5⋊C16
Regular action on 240 points
Generators in S240
(1 100 203)(2 101 204)(3 102 205)(4 103 206)(5 104 207)(6 105 208)(7 106 193)(8 107 194)(9 108 195)(10 109 196)(11 110 197)(12 111 198)(13 112 199)(14 97 200)(15 98 201)(16 99 202)(17 190 149)(18 191 150)(19 192 151)(20 177 152)(21 178 153)(22 179 154)(23 180 155)(24 181 156)(25 182 157)(26 183 158)(27 184 159)(28 185 160)(29 186 145)(30 187 146)(31 188 147)(32 189 148)(33 229 130)(34 230 131)(35 231 132)(36 232 133)(37 233 134)(38 234 135)(39 235 136)(40 236 137)(41 237 138)(42 238 139)(43 239 140)(44 240 141)(45 225 142)(46 226 143)(47 227 144)(48 228 129)(49 79 169)(50 80 170)(51 65 171)(52 66 172)(53 67 173)(54 68 174)(55 69 175)(56 70 176)(57 71 161)(58 72 162)(59 73 163)(60 74 164)(61 75 165)(62 76 166)(63 77 167)(64 78 168)(81 125 222)(82 126 223)(83 127 224)(84 128 209)(85 113 210)(86 114 211)(87 115 212)(88 116 213)(89 117 214)(90 118 215)(91 119 216)(92 120 217)(93 121 218)(94 122 219)(95 123 220)(96 124 221)
(1 30 81 136 65)(2 137 31 66 82)(3 67 138 83 32)(4 84 68 17 139)(5 18 85 140 69)(6 141 19 70 86)(7 71 142 87 20)(8 88 72 21 143)(9 22 89 144 73)(10 129 23 74 90)(11 75 130 91 24)(12 92 76 25 131)(13 26 93 132 77)(14 133 27 78 94)(15 79 134 95 28)(16 96 80 29 135)(33 119 181 110 165)(34 111 120 166 182)(35 167 112 183 121)(36 184 168 122 97)(37 123 185 98 169)(38 99 124 170 186)(39 171 100 187 125)(40 188 172 126 101)(41 127 189 102 173)(42 103 128 174 190)(43 175 104 191 113)(44 192 176 114 105)(45 115 177 106 161)(46 107 116 162 178)(47 163 108 179 117)(48 180 164 118 109)(49 233 220 160 201)(50 145 234 202 221)(51 203 146 222 235)(52 223 204 236 147)(53 237 224 148 205)(54 149 238 206 209)(55 207 150 210 239)(56 211 208 240 151)(57 225 212 152 193)(58 153 226 194 213)(59 195 154 214 227)(60 215 196 228 155)(61 229 216 156 197)(62 157 230 198 217)(63 199 158 218 231)(64 219 200 232 159)
(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)| (1,100,203)(2,101,204)(3,102,205)(4,103,206)(5,104,207)(6,105,208)(7,106,193)(8,107,194)(9,108,195)(10,109,196)(11,110,197)(12,111,198)(13,112,199)(14,97,200)(15,98,201)(16,99,202)(17,190,149)(18,191,150)(19,192,151)(20,177,152)(21,178,153)(22,179,154)(23,180,155)(24,181,156)(25,182,157)(26,183,158)(27,184,159)(28,185,160)(29,186,145)(30,187,146)(31,188,147)(32,189,148)(33,229,130)(34,230,131)(35,231,132)(36,232,133)(37,233,134)(38,234,135)(39,235,136)(40,236,137)(41,237,138)(42,238,139)(43,239,140)(44,240,141)(45,225,142)(46,226,143)(47,227,144)(48,228,129)(49,79,169)(50,80,170)(51,65,171)(52,66,172)(53,67,173)(54,68,174)(55,69,175)(56,70,176)(57,71,161)(58,72,162)(59,73,163)(60,74,164)(61,75,165)(62,76,166)(63,77,167)(64,78,168)(81,125,222)(82,126,223)(83,127,224)(84,128,209)(85,113,210)(86,114,211)(87,115,212)(88,116,213)(89,117,214)(90,118,215)(91,119,216)(92,120,217)(93,121,218)(94,122,219)(95,123,220)(96,124,221), (1,30,81,136,65)(2,137,31,66,82)(3,67,138,83,32)(4,84,68,17,139)(5,18,85,140,69)(6,141,19,70,86)(7,71,142,87,20)(8,88,72,21,143)(9,22,89,144,73)(10,129,23,74,90)(11,75,130,91,24)(12,92,76,25,131)(13,26,93,132,77)(14,133,27,78,94)(15,79,134,95,28)(16,96,80,29,135)(33,119,181,110,165)(34,111,120,166,182)(35,167,112,183,121)(36,184,168,122,97)(37,123,185,98,169)(38,99,124,170,186)(39,171,100,187,125)(40,188,172,126,101)(41,127,189,102,173)(42,103,128,174,190)(43,175,104,191,113)(44,192,176,114,105)(45,115,177,106,161)(46,107,116,162,178)(47,163,108,179,117)(48,180,164,118,109)(49,233,220,160,201)(50,145,234,202,221)(51,203,146,222,235)(52,223,204,236,147)(53,237,224,148,205)(54,149,238,206,209)(55,207,150,210,239)(56,211,208,240,151)(57,225,212,152,193)(58,153,226,194,213)(59,195,154,214,227)(60,215,196,228,155)(61,229,216,156,197)(62,157,230,198,217)(63,199,158,218,231)(64,219,200,232,159), (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( (1,100,203)(2,101,204)(3,102,205)(4,103,206)(5,104,207)(6,105,208)(7,106,193)(8,107,194)(9,108,195)(10,109,196)(11,110,197)(12,111,198)(13,112,199)(14,97,200)(15,98,201)(16,99,202)(17,190,149)(18,191,150)(19,192,151)(20,177,152)(21,178,153)(22,179,154)(23,180,155)(24,181,156)(25,182,157)(26,183,158)(27,184,159)(28,185,160)(29,186,145)(30,187,146)(31,188,147)(32,189,148)(33,229,130)(34,230,131)(35,231,132)(36,232,133)(37,233,134)(38,234,135)(39,235,136)(40,236,137)(41,237,138)(42,238,139)(43,239,140)(44,240,141)(45,225,142)(46,226,143)(47,227,144)(48,228,129)(49,79,169)(50,80,170)(51,65,171)(52,66,172)(53,67,173)(54,68,174)(55,69,175)(56,70,176)(57,71,161)(58,72,162)(59,73,163)(60,74,164)(61,75,165)(62,76,166)(63,77,167)(64,78,168)(81,125,222)(82,126,223)(83,127,224)(84,128,209)(85,113,210)(86,114,211)(87,115,212)(88,116,213)(89,117,214)(90,118,215)(91,119,216)(92,120,217)(93,121,218)(94,122,219)(95,123,220)(96,124,221), (1,30,81,136,65)(2,137,31,66,82)(3,67,138,83,32)(4,84,68,17,139)(5,18,85,140,69)(6,141,19,70,86)(7,71,142,87,20)(8,88,72,21,143)(9,22,89,144,73)(10,129,23,74,90)(11,75,130,91,24)(12,92,76,25,131)(13,26,93,132,77)(14,133,27,78,94)(15,79,134,95,28)(16,96,80,29,135)(33,119,181,110,165)(34,111,120,166,182)(35,167,112,183,121)(36,184,168,122,97)(37,123,185,98,169)(38,99,124,170,186)(39,171,100,187,125)(40,188,172,126,101)(41,127,189,102,173)(42,103,128,174,190)(43,175,104,191,113)(44,192,176,114,105)(45,115,177,106,161)(46,107,116,162,178)(47,163,108,179,117)(48,180,164,118,109)(49,233,220,160,201)(50,145,234,202,221)(51,203,146,222,235)(52,223,204,236,147)(53,237,224,148,205)(54,149,238,206,209)(55,207,150,210,239)(56,211,208,240,151)(57,225,212,152,193)(58,153,226,194,213)(59,195,154,214,227)(60,215,196,228,155)(61,229,216,156,197)(62,157,230,198,217)(63,199,158,218,231)(64,219,200,232,159), (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([(1,100,203),(2,101,204),(3,102,205),(4,103,206),(5,104,207),(6,105,208),(7,106,193),(8,107,194),(9,108,195),(10,109,196),(11,110,197),(12,111,198),(13,112,199),(14,97,200),(15,98,201),(16,99,202),(17,190,149),(18,191,150),(19,192,151),(20,177,152),(21,178,153),(22,179,154),(23,180,155),(24,181,156),(25,182,157),(26,183,158),(27,184,159),(28,185,160),(29,186,145),(30,187,146),(31,188,147),(32,189,148),(33,229,130),(34,230,131),(35,231,132),(36,232,133),(37,233,134),(38,234,135),(39,235,136),(40,236,137),(41,237,138),(42,238,139),(43,239,140),(44,240,141),(45,225,142),(46,226,143),(47,227,144),(48,228,129),(49,79,169),(50,80,170),(51,65,171),(52,66,172),(53,67,173),(54,68,174),(55,69,175),(56,70,176),(57,71,161),(58,72,162),(59,73,163),(60,74,164),(61,75,165),(62,76,166),(63,77,167),(64,78,168),(81,125,222),(82,126,223),(83,127,224),(84,128,209),(85,113,210),(86,114,211),(87,115,212),(88,116,213),(89,117,214),(90,118,215),(91,119,216),(92,120,217),(93,121,218),(94,122,219),(95,123,220),(96,124,221)], [(1,30,81,136,65),(2,137,31,66,82),(3,67,138,83,32),(4,84,68,17,139),(5,18,85,140,69),(6,141,19,70,86),(7,71,142,87,20),(8,88,72,21,143),(9,22,89,144,73),(10,129,23,74,90),(11,75,130,91,24),(12,92,76,25,131),(13,26,93,132,77),(14,133,27,78,94),(15,79,134,95,28),(16,96,80,29,135),(33,119,181,110,165),(34,111,120,166,182),(35,167,112,183,121),(36,184,168,122,97),(37,123,185,98,169),(38,99,124,170,186),(39,171,100,187,125),(40,188,172,126,101),(41,127,189,102,173),(42,103,128,174,190),(43,175,104,191,113),(44,192,176,114,105),(45,115,177,106,161),(46,107,116,162,178),(47,163,108,179,117),(48,180,164,118,109),(49,233,220,160,201),(50,145,234,202,221),(51,203,146,222,235),(52,223,204,236,147),(53,237,224,148,205),(54,149,238,206,209),(55,207,150,210,239),(56,211,208,240,151),(57,225,212,152,193),(58,153,226,194,213),(59,195,154,214,227),(60,215,196,228,155),(61,229,216,156,197),(62,157,230,198,217),(63,199,158,218,231),(64,219,200,232,159)], [(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×C5⋊C16 is a maximal subgroup of   D15⋊C16  C15⋊M5(2)  D30.C8

60 conjugacy classes

class 1  2 3A3B4A4B 5 6A6B8A8B8C8D 10 12A12B12C12D15A15B16A···16H20A20B24A···24H30A30B48A···48P60A60B60C60D
order12334456688881012121212151516···16202024···24303048···4860606060
size111111411555541111445···5445···5445···54444

60 irreducible representations

dim1111111111444444
type+++-
imageC1C2C3C4C6C8C12C16C24C48F5C5⋊C8C3×F5C5⋊C16C3×C5⋊C8C3×C5⋊C16
kernelC3×C5⋊C16C3×C52C8C5⋊C16C60C52C8C30C20C15C10C5C12C6C4C3C2C1
# reps11222448816112224

Matrix representation of C3×C5⋊C16 in GL5(𝔽241)

2250000
01000
00100
00010
00001
,
10000
0000240
0100240
0010240
0001240
,
2400000
01186921353
0901221171
011924070143
0188212123172

G:=sub<GL(5,GF(241))| [225,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,240,240,240,240],[240,0,0,0,0,0,118,90,119,188,0,69,122,240,212,0,213,1,70,123,0,53,171,143,172] >;

C3×C5⋊C16 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(240,5);
// by ID

G=gap.SmallGroup(240,5);
# by ID

G:=PCGroup([6,-2,-3,-2,-2,-2,-5,36,50,69,3461,1169]);
// 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^3>;
// generators/relations

Export

Subgroup lattice of C3×C5⋊C16 in TeX

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