metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C15⋊2Q16, C6.9D20, C3⋊2Dic20, C20.25D6, C30.11D4, C12.8D10, C60.11C22, Dic30.3C2, Dic10.1S3, C3⋊C8.D5, C4.4(S3×D5), C5⋊1(C3⋊Q16), C10.4(C3⋊D4), C2.7(C3⋊D20), (C3×Dic10).1C2, (C5×C3⋊C8).1C2, SmallGroup(240,23)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊Dic20
G = < a,b,c | a3=b40=1, c2=b20, bab-1=a-1, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3⋊Dic20
►Regular action on 240 points
Generators in S240
(1 224 76)(2 77 225)(3 226 78)(4 79 227)(5 228 80)(6 41 229)(7 230 42)(8 43 231)(9 232 44)(10 45 233)(11 234 46)(12 47 235)(13 236 48)(14 49 237)(15 238 50)(16 51 239)(17 240 52)(18 53 201)(19 202 54)(20 55 203)(21 204 56)(22 57 205)(23 206 58)(24 59 207)(25 208 60)(26 61 209)(27 210 62)(28 63 211)(29 212 64)(30 65 213)(31 214 66)(32 67 215)(33 216 68)(34 69 217)(35 218 70)(36 71 219)(37 220 72)(38 73 221)(39 222 74)(40 75 223)(81 170 142)(82 143 171)(83 172 144)(84 145 173)(85 174 146)(86 147 175)(87 176 148)(88 149 177)(89 178 150)(90 151 179)(91 180 152)(92 153 181)(93 182 154)(94 155 183)(95 184 156)(96 157 185)(97 186 158)(98 159 187)(99 188 160)(100 121 189)(101 190 122)(102 123 191)(103 192 124)(104 125 193)(105 194 126)(106 127 195)(107 196 128)(108 129 197)(109 198 130)(110 131 199)(111 200 132)(112 133 161)(113 162 134)(114 135 163)(115 164 136)(116 137 165)(117 166 138)(118 139 167)(119 168 140)(120 141 169)
(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)
(1 156 21 136)(2 155 22 135)(3 154 23 134)(4 153 24 133)(5 152 25 132)(6 151 26 131)(7 150 27 130)(8 149 28 129)(9 148 29 128)(10 147 30 127)(11 146 31 126)(12 145 32 125)(13 144 33 124)(14 143 34 123)(15 142 35 122)(16 141 36 121)(17 140 37 160)(18 139 38 159)(19 138 39 158)(20 137 40 157)(41 179 61 199)(42 178 62 198)(43 177 63 197)(44 176 64 196)(45 175 65 195)(46 174 66 194)(47 173 67 193)(48 172 68 192)(49 171 69 191)(50 170 70 190)(51 169 71 189)(52 168 72 188)(53 167 73 187)(54 166 74 186)(55 165 75 185)(56 164 76 184)(57 163 77 183)(58 162 78 182)(59 161 79 181)(60 200 80 180)(81 218 101 238)(82 217 102 237)(83 216 103 236)(84 215 104 235)(85 214 105 234)(86 213 106 233)(87 212 107 232)(88 211 108 231)(89 210 109 230)(90 209 110 229)(91 208 111 228)(92 207 112 227)(93 206 113 226)(94 205 114 225)(95 204 115 224)(96 203 116 223)(97 202 117 222)(98 201 118 221)(99 240 119 220)(100 239 120 219)
G:=sub<Sym(240)| (1,224,76)(2,77,225)(3,226,78)(4,79,227)(5,228,80)(6,41,229)(7,230,42)(8,43,231)(9,232,44)(10,45,233)(11,234,46)(12,47,235)(13,236,48)(14,49,237)(15,238,50)(16,51,239)(17,240,52)(18,53,201)(19,202,54)(20,55,203)(21,204,56)(22,57,205)(23,206,58)(24,59,207)(25,208,60)(26,61,209)(27,210,62)(28,63,211)(29,212,64)(30,65,213)(31,214,66)(32,67,215)(33,216,68)(34,69,217)(35,218,70)(36,71,219)(37,220,72)(38,73,221)(39,222,74)(40,75,223)(81,170,142)(82,143,171)(83,172,144)(84,145,173)(85,174,146)(86,147,175)(87,176,148)(88,149,177)(89,178,150)(90,151,179)(91,180,152)(92,153,181)(93,182,154)(94,155,183)(95,184,156)(96,157,185)(97,186,158)(98,159,187)(99,188,160)(100,121,189)(101,190,122)(102,123,191)(103,192,124)(104,125,193)(105,194,126)(106,127,195)(107,196,128)(108,129,197)(109,198,130)(110,131,199)(111,200,132)(112,133,161)(113,162,134)(114,135,163)(115,164,136)(116,137,165)(117,166,138)(118,139,167)(119,168,140)(120,141,169), (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), (1,156,21,136)(2,155,22,135)(3,154,23,134)(4,153,24,133)(5,152,25,132)(6,151,26,131)(7,150,27,130)(8,149,28,129)(9,148,29,128)(10,147,30,127)(11,146,31,126)(12,145,32,125)(13,144,33,124)(14,143,34,123)(15,142,35,122)(16,141,36,121)(17,140,37,160)(18,139,38,159)(19,138,39,158)(20,137,40,157)(41,179,61,199)(42,178,62,198)(43,177,63,197)(44,176,64,196)(45,175,65,195)(46,174,66,194)(47,173,67,193)(48,172,68,192)(49,171,69,191)(50,170,70,190)(51,169,71,189)(52,168,72,188)(53,167,73,187)(54,166,74,186)(55,165,75,185)(56,164,76,184)(57,163,77,183)(58,162,78,182)(59,161,79,181)(60,200,80,180)(81,218,101,238)(82,217,102,237)(83,216,103,236)(84,215,104,235)(85,214,105,234)(86,213,106,233)(87,212,107,232)(88,211,108,231)(89,210,109,230)(90,209,110,229)(91,208,111,228)(92,207,112,227)(93,206,113,226)(94,205,114,225)(95,204,115,224)(96,203,116,223)(97,202,117,222)(98,201,118,221)(99,240,119,220)(100,239,120,219)>;
G:=Group( (1,224,76)(2,77,225)(3,226,78)(4,79,227)(5,228,80)(6,41,229)(7,230,42)(8,43,231)(9,232,44)(10,45,233)(11,234,46)(12,47,235)(13,236,48)(14,49,237)(15,238,50)(16,51,239)(17,240,52)(18,53,201)(19,202,54)(20,55,203)(21,204,56)(22,57,205)(23,206,58)(24,59,207)(25,208,60)(26,61,209)(27,210,62)(28,63,211)(29,212,64)(30,65,213)(31,214,66)(32,67,215)(33,216,68)(34,69,217)(35,218,70)(36,71,219)(37,220,72)(38,73,221)(39,222,74)(40,75,223)(81,170,142)(82,143,171)(83,172,144)(84,145,173)(85,174,146)(86,147,175)(87,176,148)(88,149,177)(89,178,150)(90,151,179)(91,180,152)(92,153,181)(93,182,154)(94,155,183)(95,184,156)(96,157,185)(97,186,158)(98,159,187)(99,188,160)(100,121,189)(101,190,122)(102,123,191)(103,192,124)(104,125,193)(105,194,126)(106,127,195)(107,196,128)(108,129,197)(109,198,130)(110,131,199)(111,200,132)(112,133,161)(113,162,134)(114,135,163)(115,164,136)(116,137,165)(117,166,138)(118,139,167)(119,168,140)(120,141,169), (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), (1,156,21,136)(2,155,22,135)(3,154,23,134)(4,153,24,133)(5,152,25,132)(6,151,26,131)(7,150,27,130)(8,149,28,129)(9,148,29,128)(10,147,30,127)(11,146,31,126)(12,145,32,125)(13,144,33,124)(14,143,34,123)(15,142,35,122)(16,141,36,121)(17,140,37,160)(18,139,38,159)(19,138,39,158)(20,137,40,157)(41,179,61,199)(42,178,62,198)(43,177,63,197)(44,176,64,196)(45,175,65,195)(46,174,66,194)(47,173,67,193)(48,172,68,192)(49,171,69,191)(50,170,70,190)(51,169,71,189)(52,168,72,188)(53,167,73,187)(54,166,74,186)(55,165,75,185)(56,164,76,184)(57,163,77,183)(58,162,78,182)(59,161,79,181)(60,200,80,180)(81,218,101,238)(82,217,102,237)(83,216,103,236)(84,215,104,235)(85,214,105,234)(86,213,106,233)(87,212,107,232)(88,211,108,231)(89,210,109,230)(90,209,110,229)(91,208,111,228)(92,207,112,227)(93,206,113,226)(94,205,114,225)(95,204,115,224)(96,203,116,223)(97,202,117,222)(98,201,118,221)(99,240,119,220)(100,239,120,219) );
G=PermutationGroup([[(1,224,76),(2,77,225),(3,226,78),(4,79,227),(5,228,80),(6,41,229),(7,230,42),(8,43,231),(9,232,44),(10,45,233),(11,234,46),(12,47,235),(13,236,48),(14,49,237),(15,238,50),(16,51,239),(17,240,52),(18,53,201),(19,202,54),(20,55,203),(21,204,56),(22,57,205),(23,206,58),(24,59,207),(25,208,60),(26,61,209),(27,210,62),(28,63,211),(29,212,64),(30,65,213),(31,214,66),(32,67,215),(33,216,68),(34,69,217),(35,218,70),(36,71,219),(37,220,72),(38,73,221),(39,222,74),(40,75,223),(81,170,142),(82,143,171),(83,172,144),(84,145,173),(85,174,146),(86,147,175),(87,176,148),(88,149,177),(89,178,150),(90,151,179),(91,180,152),(92,153,181),(93,182,154),(94,155,183),(95,184,156),(96,157,185),(97,186,158),(98,159,187),(99,188,160),(100,121,189),(101,190,122),(102,123,191),(103,192,124),(104,125,193),(105,194,126),(106,127,195),(107,196,128),(108,129,197),(109,198,130),(110,131,199),(111,200,132),(112,133,161),(113,162,134),(114,135,163),(115,164,136),(116,137,165),(117,166,138),(118,139,167),(119,168,140),(120,141,169)], [(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)], [(1,156,21,136),(2,155,22,135),(3,154,23,134),(4,153,24,133),(5,152,25,132),(6,151,26,131),(7,150,27,130),(8,149,28,129),(9,148,29,128),(10,147,30,127),(11,146,31,126),(12,145,32,125),(13,144,33,124),(14,143,34,123),(15,142,35,122),(16,141,36,121),(17,140,37,160),(18,139,38,159),(19,138,39,158),(20,137,40,157),(41,179,61,199),(42,178,62,198),(43,177,63,197),(44,176,64,196),(45,175,65,195),(46,174,66,194),(47,173,67,193),(48,172,68,192),(49,171,69,191),(50,170,70,190),(51,169,71,189),(52,168,72,188),(53,167,73,187),(54,166,74,186),(55,165,75,185),(56,164,76,184),(57,163,77,183),(58,162,78,182),(59,161,79,181),(60,200,80,180),(81,218,101,238),(82,217,102,237),(83,216,103,236),(84,215,104,235),(85,214,105,234),(86,213,106,233),(87,212,107,232),(88,211,108,231),(89,210,109,230),(90,209,110,229),(91,208,111,228),(92,207,112,227),(93,206,113,226),(94,205,114,225),(95,204,115,224),(96,203,116,223),(97,202,117,222),(98,201,118,221),(99,240,119,220),(100,239,120,219)]])
C3⋊Dic20 is a maximal subgroup of
S3×Dic20 Dic20⋊S3 D6.1D20 C40.2D6 D20.31D6 C60.63D4 C12.D20 C60.8C23 D30.9D4 D12.24D10 D30.11D4 D5×C3⋊Q16 D15⋊Q16 D12.27D10 D30.44D4
C3⋊Dic20 is a maximal quotient of
C6.Dic20 Dic30⋊12C4 C60.5Q8
36 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 5A | 5B | 6 | 8A | 8B | 10A | 10B | 12A | 12B | 12C | 15A | 15B | 20A | 20B | 20C | 20D | 30A | 30B | 40A | ··· | 40H | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | 30 | 40 | ··· | 40 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 2 | 2 | 20 | 60 | 2 | 2 | 2 | 6 | 6 | 2 | 2 | 4 | 20 | 20 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | ··· | 6 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | - | - | + | + | - | |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | Q16 | D10 | C3⋊D4 | D20 | Dic20 | C3⋊Q16 | S3×D5 | C3⋊D20 | C3⋊Dic20 |
| kernel | C3⋊Dic20 | C5×C3⋊C8 | C3×Dic10 | Dic30 | Dic10 | C30 | C3⋊C8 | C20 | C15 | C12 | C10 | C6 | C3 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 8 | 1 | 2 | 2 | 4 |
Matrix representation of C3⋊Dic20 ►in GL4(𝔽241) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 1 |
| 0 | 0 | 240 | 0 |
| 33 | 232 | 0 | 0 |
| 9 | 56 | 0 | 0 |
| 0 | 0 | 129 | 114 |
| 0 | 0 | 2 | 112 |
| 227 | 114 | 0 | 0 |
| 123 | 14 | 0 | 0 |
| 0 | 0 | 171 | 140 |
| 0 | 0 | 101 | 70 |
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,240,240,0,0,1,0],[33,9,0,0,232,56,0,0,0,0,129,2,0,0,114,112],[227,123,0,0,114,14,0,0,0,0,171,101,0,0,140,70] >;
C3⋊Dic20 in GAP, Magma, Sage, TeX
C_3\rtimes {\rm Dic}_{20} % in TeX
G:=Group("C3:Dic20"); // GroupNames label
G:=SmallGroup(240,23);
// by ID
G=gap.SmallGroup(240,23);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,73,79,218,50,490,6917]);
// Polycyclic
G:=Group<a,b,c|a^3=b^40=1,c^2=b^20,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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