direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×Dic20, C15⋊5Q16, C40.1C6, C24.2D5, C120.2C2, C6.15D20, C30.25D4, C12.54D10, C60.61C22, Dic10.1C6, C8.(C3×D5), C5⋊1(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
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 92 197)(2 93 198)(3 94 199)(4 95 200)(5 96 161)(6 97 162)(7 98 163)(8 99 164)(9 100 165)(10 101 166)(11 102 167)(12 103 168)(13 104 169)(14 105 170)(15 106 171)(16 107 172)(17 108 173)(18 109 174)(19 110 175)(20 111 176)(21 112 177)(22 113 178)(23 114 179)(24 115 180)(25 116 181)(26 117 182)(27 118 183)(28 119 184)(29 120 185)(30 81 186)(31 82 187)(32 83 188)(33 84 189)(34 85 190)(35 86 191)(36 87 192)(37 88 193)(38 89 194)(39 90 195)(40 91 196)(41 151 227)(42 152 228)(43 153 229)(44 154 230)(45 155 231)(46 156 232)(47 157 233)(48 158 234)(49 159 235)(50 160 236)(51 121 237)(52 122 238)(53 123 239)(54 124 240)(55 125 201)(56 126 202)(57 127 203)(58 128 204)(59 129 205)(60 130 206)(61 131 207)(62 132 208)(63 133 209)(64 134 210)(65 135 211)(66 136 212)(67 137 213)(68 138 214)(69 139 215)(70 140 216)(71 141 217)(72 142 218)(73 143 219)(74 144 220)(75 145 221)(76 146 222)(77 147 223)(78 148 224)(79 149 225)(80 150 226)
(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)
(1 232 21 212)(2 231 22 211)(3 230 23 210)(4 229 24 209)(5 228 25 208)(6 227 26 207)(7 226 27 206)(8 225 28 205)(9 224 29 204)(10 223 30 203)(11 222 31 202)(12 221 32 201)(13 220 33 240)(14 219 34 239)(15 218 35 238)(16 217 36 237)(17 216 37 236)(18 215 38 235)(19 214 39 234)(20 213 40 233)(41 117 61 97)(42 116 62 96)(43 115 63 95)(44 114 64 94)(45 113 65 93)(46 112 66 92)(47 111 67 91)(48 110 68 90)(49 109 69 89)(50 108 70 88)(51 107 71 87)(52 106 72 86)(53 105 73 85)(54 104 74 84)(55 103 75 83)(56 102 76 82)(57 101 77 81)(58 100 78 120)(59 99 79 119)(60 98 80 118)(121 172 141 192)(122 171 142 191)(123 170 143 190)(124 169 144 189)(125 168 145 188)(126 167 146 187)(127 166 147 186)(128 165 148 185)(129 164 149 184)(130 163 150 183)(131 162 151 182)(132 161 152 181)(133 200 153 180)(134 199 154 179)(135 198 155 178)(136 197 156 177)(137 196 157 176)(138 195 158 175)(139 194 159 174)(140 193 160 173)
G:=sub<Sym(240)| (1,92,197)(2,93,198)(3,94,199)(4,95,200)(5,96,161)(6,97,162)(7,98,163)(8,99,164)(9,100,165)(10,101,166)(11,102,167)(12,103,168)(13,104,169)(14,105,170)(15,106,171)(16,107,172)(17,108,173)(18,109,174)(19,110,175)(20,111,176)(21,112,177)(22,113,178)(23,114,179)(24,115,180)(25,116,181)(26,117,182)(27,118,183)(28,119,184)(29,120,185)(30,81,186)(31,82,187)(32,83,188)(33,84,189)(34,85,190)(35,86,191)(36,87,192)(37,88,193)(38,89,194)(39,90,195)(40,91,196)(41,151,227)(42,152,228)(43,153,229)(44,154,230)(45,155,231)(46,156,232)(47,157,233)(48,158,234)(49,159,235)(50,160,236)(51,121,237)(52,122,238)(53,123,239)(54,124,240)(55,125,201)(56,126,202)(57,127,203)(58,128,204)(59,129,205)(60,130,206)(61,131,207)(62,132,208)(63,133,209)(64,134,210)(65,135,211)(66,136,212)(67,137,213)(68,138,214)(69,139,215)(70,140,216)(71,141,217)(72,142,218)(73,143,219)(74,144,220)(75,145,221)(76,146,222)(77,147,223)(78,148,224)(79,149,225)(80,150,226), (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), (1,232,21,212)(2,231,22,211)(3,230,23,210)(4,229,24,209)(5,228,25,208)(6,227,26,207)(7,226,27,206)(8,225,28,205)(9,224,29,204)(10,223,30,203)(11,222,31,202)(12,221,32,201)(13,220,33,240)(14,219,34,239)(15,218,35,238)(16,217,36,237)(17,216,37,236)(18,215,38,235)(19,214,39,234)(20,213,40,233)(41,117,61,97)(42,116,62,96)(43,115,63,95)(44,114,64,94)(45,113,65,93)(46,112,66,92)(47,111,67,91)(48,110,68,90)(49,109,69,89)(50,108,70,88)(51,107,71,87)(52,106,72,86)(53,105,73,85)(54,104,74,84)(55,103,75,83)(56,102,76,82)(57,101,77,81)(58,100,78,120)(59,99,79,119)(60,98,80,118)(121,172,141,192)(122,171,142,191)(123,170,143,190)(124,169,144,189)(125,168,145,188)(126,167,146,187)(127,166,147,186)(128,165,148,185)(129,164,149,184)(130,163,150,183)(131,162,151,182)(132,161,152,181)(133,200,153,180)(134,199,154,179)(135,198,155,178)(136,197,156,177)(137,196,157,176)(138,195,158,175)(139,194,159,174)(140,193,160,173)>;
G:=Group( (1,92,197)(2,93,198)(3,94,199)(4,95,200)(5,96,161)(6,97,162)(7,98,163)(8,99,164)(9,100,165)(10,101,166)(11,102,167)(12,103,168)(13,104,169)(14,105,170)(15,106,171)(16,107,172)(17,108,173)(18,109,174)(19,110,175)(20,111,176)(21,112,177)(22,113,178)(23,114,179)(24,115,180)(25,116,181)(26,117,182)(27,118,183)(28,119,184)(29,120,185)(30,81,186)(31,82,187)(32,83,188)(33,84,189)(34,85,190)(35,86,191)(36,87,192)(37,88,193)(38,89,194)(39,90,195)(40,91,196)(41,151,227)(42,152,228)(43,153,229)(44,154,230)(45,155,231)(46,156,232)(47,157,233)(48,158,234)(49,159,235)(50,160,236)(51,121,237)(52,122,238)(53,123,239)(54,124,240)(55,125,201)(56,126,202)(57,127,203)(58,128,204)(59,129,205)(60,130,206)(61,131,207)(62,132,208)(63,133,209)(64,134,210)(65,135,211)(66,136,212)(67,137,213)(68,138,214)(69,139,215)(70,140,216)(71,141,217)(72,142,218)(73,143,219)(74,144,220)(75,145,221)(76,146,222)(77,147,223)(78,148,224)(79,149,225)(80,150,226), (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), (1,232,21,212)(2,231,22,211)(3,230,23,210)(4,229,24,209)(5,228,25,208)(6,227,26,207)(7,226,27,206)(8,225,28,205)(9,224,29,204)(10,223,30,203)(11,222,31,202)(12,221,32,201)(13,220,33,240)(14,219,34,239)(15,218,35,238)(16,217,36,237)(17,216,37,236)(18,215,38,235)(19,214,39,234)(20,213,40,233)(41,117,61,97)(42,116,62,96)(43,115,63,95)(44,114,64,94)(45,113,65,93)(46,112,66,92)(47,111,67,91)(48,110,68,90)(49,109,69,89)(50,108,70,88)(51,107,71,87)(52,106,72,86)(53,105,73,85)(54,104,74,84)(55,103,75,83)(56,102,76,82)(57,101,77,81)(58,100,78,120)(59,99,79,119)(60,98,80,118)(121,172,141,192)(122,171,142,191)(123,170,143,190)(124,169,144,189)(125,168,145,188)(126,167,146,187)(127,166,147,186)(128,165,148,185)(129,164,149,184)(130,163,150,183)(131,162,151,182)(132,161,152,181)(133,200,153,180)(134,199,154,179)(135,198,155,178)(136,197,156,177)(137,196,157,176)(138,195,158,175)(139,194,159,174)(140,193,160,173) );
G=PermutationGroup([[(1,92,197),(2,93,198),(3,94,199),(4,95,200),(5,96,161),(6,97,162),(7,98,163),(8,99,164),(9,100,165),(10,101,166),(11,102,167),(12,103,168),(13,104,169),(14,105,170),(15,106,171),(16,107,172),(17,108,173),(18,109,174),(19,110,175),(20,111,176),(21,112,177),(22,113,178),(23,114,179),(24,115,180),(25,116,181),(26,117,182),(27,118,183),(28,119,184),(29,120,185),(30,81,186),(31,82,187),(32,83,188),(33,84,189),(34,85,190),(35,86,191),(36,87,192),(37,88,193),(38,89,194),(39,90,195),(40,91,196),(41,151,227),(42,152,228),(43,153,229),(44,154,230),(45,155,231),(46,156,232),(47,157,233),(48,158,234),(49,159,235),(50,160,236),(51,121,237),(52,122,238),(53,123,239),(54,124,240),(55,125,201),(56,126,202),(57,127,203),(58,128,204),(59,129,205),(60,130,206),(61,131,207),(62,132,208),(63,133,209),(64,134,210),(65,135,211),(66,136,212),(67,137,213),(68,138,214),(69,139,215),(70,140,216),(71,141,217),(72,142,218),(73,143,219),(74,144,220),(75,145,221),(76,146,222),(77,147,223),(78,148,224),(79,149,225),(80,150,226)], [(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)], [(1,232,21,212),(2,231,22,211),(3,230,23,210),(4,229,24,209),(5,228,25,208),(6,227,26,207),(7,226,27,206),(8,225,28,205),(9,224,29,204),(10,223,30,203),(11,222,31,202),(12,221,32,201),(13,220,33,240),(14,219,34,239),(15,218,35,238),(16,217,36,237),(17,216,37,236),(18,215,38,235),(19,214,39,234),(20,213,40,233),(41,117,61,97),(42,116,62,96),(43,115,63,95),(44,114,64,94),(45,113,65,93),(46,112,66,92),(47,111,67,91),(48,110,68,90),(49,109,69,89),(50,108,70,88),(51,107,71,87),(52,106,72,86),(53,105,73,85),(54,104,74,84),(55,103,75,83),(56,102,76,82),(57,101,77,81),(58,100,78,120),(59,99,79,119),(60,98,80,118),(121,172,141,192),(122,171,142,191),(123,170,143,190),(124,169,144,189),(125,168,145,188),(126,167,146,187),(127,166,147,186),(128,165,148,185),(129,164,149,184),(130,163,150,183),(131,162,151,182),(132,161,152,181),(133,200,153,180),(134,199,154,179),(135,198,155,178),(136,197,156,177),(137,196,157,176),(138,195,158,175),(139,194,159,174),(140,193,160,173)]])
C3×Dic20 is a maximal subgroup of
C15⋊SD32 C24.D10 C15⋊Q32 C3⋊Dic40 Dic20⋊S3 Dic10.D6 D120⋊5C2 D24⋊5D5 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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