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G = C3×Dic20order 240 = 24·3·5

Direct product of C3 and Dic20

Aliases: C3×Dic20, C155Q16, C40.1C6, C24.2D5, C120.2C2, C6.15D20, C30.25D4, C12.54D10, C60.61C22, Dic10.1C6, C8.(C3×D5), C51(C3×Q16), C4.10(C6×D5), C2.5(C3×D20), C10.3(C3×D4), C20.10(C2×C6), (C3×Dic10).3C2, SmallGroup(240,37)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×Dic20
 Chief series C1 — C5 — C10 — C20 — C60 — C3×Dic10 — C3×Dic20
 Lower central C5 — C10 — C20 — C3×Dic20
 Upper central C1 — C6 — C12 — C24

Generators and relations for C3×Dic20
G = < a,b,c | a3=b40=1, c2=b20, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C3×Dic20
Regular action on 240 points
Generators in S240
(1 151 58)(2 152 59)(3 153 60)(4 154 61)(5 155 62)(6 156 63)(7 157 64)(8 158 65)(9 159 66)(10 160 67)(11 121 68)(12 122 69)(13 123 70)(14 124 71)(15 125 72)(16 126 73)(17 127 74)(18 128 75)(19 129 76)(20 130 77)(21 131 78)(22 132 79)(23 133 80)(24 134 41)(25 135 42)(26 136 43)(27 137 44)(28 138 45)(29 139 46)(30 140 47)(31 141 48)(32 142 49)(33 143 50)(34 144 51)(35 145 52)(36 146 53)(37 147 54)(38 148 55)(39 149 56)(40 150 57)(81 174 211)(82 175 212)(83 176 213)(84 177 214)(85 178 215)(86 179 216)(87 180 217)(88 181 218)(89 182 219)(90 183 220)(91 184 221)(92 185 222)(93 186 223)(94 187 224)(95 188 225)(96 189 226)(97 190 227)(98 191 228)(99 192 229)(100 193 230)(101 194 231)(102 195 232)(103 196 233)(104 197 234)(105 198 235)(106 199 236)(107 200 237)(108 161 238)(109 162 239)(110 163 240)(111 164 201)(112 165 202)(113 166 203)(114 167 204)(115 168 205)(116 169 206)(117 170 207)(118 171 208)(119 172 209)(120 173 210)
(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)
(1 114 21 94)(2 113 22 93)(3 112 23 92)(4 111 24 91)(5 110 25 90)(6 109 26 89)(7 108 27 88)(8 107 28 87)(9 106 29 86)(10 105 30 85)(11 104 31 84)(12 103 32 83)(13 102 33 82)(14 101 34 81)(15 100 35 120)(16 99 36 119)(17 98 37 118)(18 97 38 117)(19 96 39 116)(20 95 40 115)(41 221 61 201)(42 220 62 240)(43 219 63 239)(44 218 64 238)(45 217 65 237)(46 216 66 236)(47 215 67 235)(48 214 68 234)(49 213 69 233)(50 212 70 232)(51 211 71 231)(52 210 72 230)(53 209 73 229)(54 208 74 228)(55 207 75 227)(56 206 76 226)(57 205 77 225)(58 204 78 224)(59 203 79 223)(60 202 80 222)(121 197 141 177)(122 196 142 176)(123 195 143 175)(124 194 144 174)(125 193 145 173)(126 192 146 172)(127 191 147 171)(128 190 148 170)(129 189 149 169)(130 188 150 168)(131 187 151 167)(132 186 152 166)(133 185 153 165)(134 184 154 164)(135 183 155 163)(136 182 156 162)(137 181 157 161)(138 180 158 200)(139 179 159 199)(140 178 160 198)

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

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

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

C3×Dic20 is a maximal subgroup of
C15⋊SD32  C24.D10  C15⋊Q32  C3⋊Dic40  Dic20⋊S3  Dic10.D6  D1205C2  D245D5  D30.4D4  C3×D5×Q16

69 conjugacy classes

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

69 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + + - image C1 C2 C2 C3 C6 C6 D4 D5 Q16 D10 C3×D4 C3×D5 D20 C3×Q16 C6×D5 Dic20 C3×D20 C3×Dic20 kernel C3×Dic20 C120 C3×Dic10 Dic20 C40 Dic10 C30 C24 C15 C12 C10 C8 C6 C5 C4 C3 C2 C1 # reps 1 1 2 2 2 4 1 2 2 2 2 4 4 4 4 8 8 16

Matrix representation of C3×Dic20 in GL2(𝔽241) generated by

 225 0 0 225
,
 227 13 34 20
,
 181 190 236 60
G:=sub<GL(2,GF(241))| [225,0,0,225],[227,34,13,20],[181,236,190,60] >;

C3×Dic20 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{20}
% in TeX

G:=Group("C3xDic20");
// GroupNames label

G:=SmallGroup(240,37);
// by ID

G=gap.SmallGroup(240,37);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-5,144,169,223,867,69,6917]);
// Polycyclic

G:=Group<a,b,c|a^3=b^40=1,c^2=b^20,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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