metabelian, supersoluble, monomial, A-group
Aliases: C3⋊Dic21, C42.3S3, C6.3D21, C21⋊1Dic3, C32⋊3Dic7, (C3×C21)⋊3C4, C7⋊(C3⋊Dic3), C14.(C3⋊S3), C2.(C3⋊D21), (C3×C6).2D7, (C3×C42).1C2, SmallGroup(252,24)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C21 — C3⋊Dic21 |
Generators and relations for C3⋊Dic21
G = < a,b,c | a3=b42=1, c2=b21, ab=ba, cac-1=a-1, cbc-1=b-1 >
Smallest permutation representation of C3⋊Dic21
►Regular action on 252 points
Generators in S252
(1 156 125)(2 157 126)(3 158 85)(4 159 86)(5 160 87)(6 161 88)(7 162 89)(8 163 90)(9 164 91)(10 165 92)(11 166 93)(12 167 94)(13 168 95)(14 127 96)(15 128 97)(16 129 98)(17 130 99)(18 131 100)(19 132 101)(20 133 102)(21 134 103)(22 135 104)(23 136 105)(24 137 106)(25 138 107)(26 139 108)(27 140 109)(28 141 110)(29 142 111)(30 143 112)(31 144 113)(32 145 114)(33 146 115)(34 147 116)(35 148 117)(36 149 118)(37 150 119)(38 151 120)(39 152 121)(40 153 122)(41 154 123)(42 155 124)(43 224 179)(44 225 180)(45 226 181)(46 227 182)(47 228 183)(48 229 184)(49 230 185)(50 231 186)(51 232 187)(52 233 188)(53 234 189)(54 235 190)(55 236 191)(56 237 192)(57 238 193)(58 239 194)(59 240 195)(60 241 196)(61 242 197)(62 243 198)(63 244 199)(64 245 200)(65 246 201)(66 247 202)(67 248 203)(68 249 204)(69 250 205)(70 251 206)(71 252 207)(72 211 208)(73 212 209)(74 213 210)(75 214 169)(76 215 170)(77 216 171)(78 217 172)(79 218 173)(80 219 174)(81 220 175)(82 221 176)(83 222 177)(84 223 178)
(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)(211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252)
(1 234 22 213)(2 233 23 212)(3 232 24 211)(4 231 25 252)(5 230 26 251)(6 229 27 250)(7 228 28 249)(8 227 29 248)(9 226 30 247)(10 225 31 246)(11 224 32 245)(12 223 33 244)(13 222 34 243)(14 221 35 242)(15 220 36 241)(16 219 37 240)(17 218 38 239)(18 217 39 238)(19 216 40 237)(20 215 41 236)(21 214 42 235)(43 145 64 166)(44 144 65 165)(45 143 66 164)(46 142 67 163)(47 141 68 162)(48 140 69 161)(49 139 70 160)(50 138 71 159)(51 137 72 158)(52 136 73 157)(53 135 74 156)(54 134 75 155)(55 133 76 154)(56 132 77 153)(57 131 78 152)(58 130 79 151)(59 129 80 150)(60 128 81 149)(61 127 82 148)(62 168 83 147)(63 167 84 146)(85 187 106 208)(86 186 107 207)(87 185 108 206)(88 184 109 205)(89 183 110 204)(90 182 111 203)(91 181 112 202)(92 180 113 201)(93 179 114 200)(94 178 115 199)(95 177 116 198)(96 176 117 197)(97 175 118 196)(98 174 119 195)(99 173 120 194)(100 172 121 193)(101 171 122 192)(102 170 123 191)(103 169 124 190)(104 210 125 189)(105 209 126 188)
G:=sub<Sym(252)| (1,156,125)(2,157,126)(3,158,85)(4,159,86)(5,160,87)(6,161,88)(7,162,89)(8,163,90)(9,164,91)(10,165,92)(11,166,93)(12,167,94)(13,168,95)(14,127,96)(15,128,97)(16,129,98)(17,130,99)(18,131,100)(19,132,101)(20,133,102)(21,134,103)(22,135,104)(23,136,105)(24,137,106)(25,138,107)(26,139,108)(27,140,109)(28,141,110)(29,142,111)(30,143,112)(31,144,113)(32,145,114)(33,146,115)(34,147,116)(35,148,117)(36,149,118)(37,150,119)(38,151,120)(39,152,121)(40,153,122)(41,154,123)(42,155,124)(43,224,179)(44,225,180)(45,226,181)(46,227,182)(47,228,183)(48,229,184)(49,230,185)(50,231,186)(51,232,187)(52,233,188)(53,234,189)(54,235,190)(55,236,191)(56,237,192)(57,238,193)(58,239,194)(59,240,195)(60,241,196)(61,242,197)(62,243,198)(63,244,199)(64,245,200)(65,246,201)(66,247,202)(67,248,203)(68,249,204)(69,250,205)(70,251,206)(71,252,207)(72,211,208)(73,212,209)(74,213,210)(75,214,169)(76,215,170)(77,216,171)(78,217,172)(79,218,173)(80,219,174)(81,220,175)(82,221,176)(83,222,177)(84,223,178), (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)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252), (1,234,22,213)(2,233,23,212)(3,232,24,211)(4,231,25,252)(5,230,26,251)(6,229,27,250)(7,228,28,249)(8,227,29,248)(9,226,30,247)(10,225,31,246)(11,224,32,245)(12,223,33,244)(13,222,34,243)(14,221,35,242)(15,220,36,241)(16,219,37,240)(17,218,38,239)(18,217,39,238)(19,216,40,237)(20,215,41,236)(21,214,42,235)(43,145,64,166)(44,144,65,165)(45,143,66,164)(46,142,67,163)(47,141,68,162)(48,140,69,161)(49,139,70,160)(50,138,71,159)(51,137,72,158)(52,136,73,157)(53,135,74,156)(54,134,75,155)(55,133,76,154)(56,132,77,153)(57,131,78,152)(58,130,79,151)(59,129,80,150)(60,128,81,149)(61,127,82,148)(62,168,83,147)(63,167,84,146)(85,187,106,208)(86,186,107,207)(87,185,108,206)(88,184,109,205)(89,183,110,204)(90,182,111,203)(91,181,112,202)(92,180,113,201)(93,179,114,200)(94,178,115,199)(95,177,116,198)(96,176,117,197)(97,175,118,196)(98,174,119,195)(99,173,120,194)(100,172,121,193)(101,171,122,192)(102,170,123,191)(103,169,124,190)(104,210,125,189)(105,209,126,188)>;
G:=Group( (1,156,125)(2,157,126)(3,158,85)(4,159,86)(5,160,87)(6,161,88)(7,162,89)(8,163,90)(9,164,91)(10,165,92)(11,166,93)(12,167,94)(13,168,95)(14,127,96)(15,128,97)(16,129,98)(17,130,99)(18,131,100)(19,132,101)(20,133,102)(21,134,103)(22,135,104)(23,136,105)(24,137,106)(25,138,107)(26,139,108)(27,140,109)(28,141,110)(29,142,111)(30,143,112)(31,144,113)(32,145,114)(33,146,115)(34,147,116)(35,148,117)(36,149,118)(37,150,119)(38,151,120)(39,152,121)(40,153,122)(41,154,123)(42,155,124)(43,224,179)(44,225,180)(45,226,181)(46,227,182)(47,228,183)(48,229,184)(49,230,185)(50,231,186)(51,232,187)(52,233,188)(53,234,189)(54,235,190)(55,236,191)(56,237,192)(57,238,193)(58,239,194)(59,240,195)(60,241,196)(61,242,197)(62,243,198)(63,244,199)(64,245,200)(65,246,201)(66,247,202)(67,248,203)(68,249,204)(69,250,205)(70,251,206)(71,252,207)(72,211,208)(73,212,209)(74,213,210)(75,214,169)(76,215,170)(77,216,171)(78,217,172)(79,218,173)(80,219,174)(81,220,175)(82,221,176)(83,222,177)(84,223,178), (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)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252), (1,234,22,213)(2,233,23,212)(3,232,24,211)(4,231,25,252)(5,230,26,251)(6,229,27,250)(7,228,28,249)(8,227,29,248)(9,226,30,247)(10,225,31,246)(11,224,32,245)(12,223,33,244)(13,222,34,243)(14,221,35,242)(15,220,36,241)(16,219,37,240)(17,218,38,239)(18,217,39,238)(19,216,40,237)(20,215,41,236)(21,214,42,235)(43,145,64,166)(44,144,65,165)(45,143,66,164)(46,142,67,163)(47,141,68,162)(48,140,69,161)(49,139,70,160)(50,138,71,159)(51,137,72,158)(52,136,73,157)(53,135,74,156)(54,134,75,155)(55,133,76,154)(56,132,77,153)(57,131,78,152)(58,130,79,151)(59,129,80,150)(60,128,81,149)(61,127,82,148)(62,168,83,147)(63,167,84,146)(85,187,106,208)(86,186,107,207)(87,185,108,206)(88,184,109,205)(89,183,110,204)(90,182,111,203)(91,181,112,202)(92,180,113,201)(93,179,114,200)(94,178,115,199)(95,177,116,198)(96,176,117,197)(97,175,118,196)(98,174,119,195)(99,173,120,194)(100,172,121,193)(101,171,122,192)(102,170,123,191)(103,169,124,190)(104,210,125,189)(105,209,126,188) );
G=PermutationGroup([[(1,156,125),(2,157,126),(3,158,85),(4,159,86),(5,160,87),(6,161,88),(7,162,89),(8,163,90),(9,164,91),(10,165,92),(11,166,93),(12,167,94),(13,168,95),(14,127,96),(15,128,97),(16,129,98),(17,130,99),(18,131,100),(19,132,101),(20,133,102),(21,134,103),(22,135,104),(23,136,105),(24,137,106),(25,138,107),(26,139,108),(27,140,109),(28,141,110),(29,142,111),(30,143,112),(31,144,113),(32,145,114),(33,146,115),(34,147,116),(35,148,117),(36,149,118),(37,150,119),(38,151,120),(39,152,121),(40,153,122),(41,154,123),(42,155,124),(43,224,179),(44,225,180),(45,226,181),(46,227,182),(47,228,183),(48,229,184),(49,230,185),(50,231,186),(51,232,187),(52,233,188),(53,234,189),(54,235,190),(55,236,191),(56,237,192),(57,238,193),(58,239,194),(59,240,195),(60,241,196),(61,242,197),(62,243,198),(63,244,199),(64,245,200),(65,246,201),(66,247,202),(67,248,203),(68,249,204),(69,250,205),(70,251,206),(71,252,207),(72,211,208),(73,212,209),(74,213,210),(75,214,169),(76,215,170),(77,216,171),(78,217,172),(79,218,173),(80,219,174),(81,220,175),(82,221,176),(83,222,177),(84,223,178)], [(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),(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252)], [(1,234,22,213),(2,233,23,212),(3,232,24,211),(4,231,25,252),(5,230,26,251),(6,229,27,250),(7,228,28,249),(8,227,29,248),(9,226,30,247),(10,225,31,246),(11,224,32,245),(12,223,33,244),(13,222,34,243),(14,221,35,242),(15,220,36,241),(16,219,37,240),(17,218,38,239),(18,217,39,238),(19,216,40,237),(20,215,41,236),(21,214,42,235),(43,145,64,166),(44,144,65,165),(45,143,66,164),(46,142,67,163),(47,141,68,162),(48,140,69,161),(49,139,70,160),(50,138,71,159),(51,137,72,158),(52,136,73,157),(53,135,74,156),(54,134,75,155),(55,133,76,154),(56,132,77,153),(57,131,78,152),(58,130,79,151),(59,129,80,150),(60,128,81,149),(61,127,82,148),(62,168,83,147),(63,167,84,146),(85,187,106,208),(86,186,107,207),(87,185,108,206),(88,184,109,205),(89,183,110,204),(90,182,111,203),(91,181,112,202),(92,180,113,201),(93,179,114,200),(94,178,115,199),(95,177,116,198),(96,176,117,197),(97,175,118,196),(98,174,119,195),(99,173,120,194),(100,172,121,193),(101,171,122,192),(102,170,123,191),(103,169,124,190),(104,210,125,189),(105,209,126,188)]])
66 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 7A | 7B | 7C | 14A | 14B | 14C | 21A | ··· | 21X | 42A | ··· | 42X |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 14 | 14 | 14 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 63 | 63 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | - | |
| image | C1 | C2 | C4 | S3 | Dic3 | D7 | Dic7 | D21 | Dic21 |
| kernel | C3⋊Dic21 | C3×C42 | C3×C21 | C42 | C21 | C3×C6 | C32 | C6 | C3 |
| # reps | 1 | 1 | 2 | 4 | 4 | 3 | 3 | 24 | 24 |
Matrix representation of C3⋊Dic21 ►in GL4(𝔽337) generated by
| 236 | 230 | 0 | 0 |
| 107 | 100 | 0 | 0 |
| 0 | 0 | 100 | 107 |
| 0 | 0 | 230 | 236 |
| 107 | 101 | 0 | 0 |
| 236 | 59 | 0 | 0 |
| 0 | 0 | 0 | 336 |
| 0 | 0 | 1 | 194 |
| 82 | 175 | 0 | 0 |
| 106 | 255 | 0 | 0 |
| 0 | 0 | 255 | 162 |
| 0 | 0 | 231 | 82 |
G:=sub<GL(4,GF(337))| [236,107,0,0,230,100,0,0,0,0,100,230,0,0,107,236],[107,236,0,0,101,59,0,0,0,0,0,1,0,0,336,194],[82,106,0,0,175,255,0,0,0,0,255,231,0,0,162,82] >;
C3⋊Dic21 in GAP, Magma, Sage, TeX
C_3\rtimes {\rm Dic}_{21} % in TeX
G:=Group("C3:Dic21"); // GroupNames label
G:=SmallGroup(252,24);
// by ID
G=gap.SmallGroup(252,24);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-7,10,122,483,5404]);
// Polycyclic
G:=Group<a,b,c|a^3=b^42=1,c^2=b^21,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations
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