metabelian, supersoluble, monomial, A-group
Aliases: C3⋊D69, C69⋊1S3, C32⋊2D23, C23⋊(C3⋊S3), (C3×C69)⋊1C2, SmallGroup(414,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C69 — C3⋊D69 |
Generators and relations for C3⋊D69
G = < a,b,c | a3=b69=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Smallest permutation representation of C3⋊D69
►On 207 points
Generators in S207
(1 140 129)(2 141 130)(3 142 131)(4 143 132)(5 144 133)(6 145 134)(7 146 135)(8 147 136)(9 148 137)(10 149 138)(11 150 70)(12 151 71)(13 152 72)(14 153 73)(15 154 74)(16 155 75)(17 156 76)(18 157 77)(19 158 78)(20 159 79)(21 160 80)(22 161 81)(23 162 82)(24 163 83)(25 164 84)(26 165 85)(27 166 86)(28 167 87)(29 168 88)(30 169 89)(31 170 90)(32 171 91)(33 172 92)(34 173 93)(35 174 94)(36 175 95)(37 176 96)(38 177 97)(39 178 98)(40 179 99)(41 180 100)(42 181 101)(43 182 102)(44 183 103)(45 184 104)(46 185 105)(47 186 106)(48 187 107)(49 188 108)(50 189 109)(51 190 110)(52 191 111)(53 192 112)(54 193 113)(55 194 114)(56 195 115)(57 196 116)(58 197 117)(59 198 118)(60 199 119)(61 200 120)(62 201 121)(63 202 122)(64 203 123)(65 204 124)(66 205 125)(67 206 126)(68 207 127)(69 139 128)
(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)
(1 69)(2 68)(3 67)(4 66)(5 65)(6 64)(7 63)(8 62)(9 61)(10 60)(11 59)(12 58)(13 57)(14 56)(15 55)(16 54)(17 53)(18 52)(19 51)(20 50)(21 49)(22 48)(23 47)(24 46)(25 45)(26 44)(27 43)(28 42)(29 41)(30 40)(31 39)(32 38)(33 37)(34 36)(70 198)(71 197)(72 196)(73 195)(74 194)(75 193)(76 192)(77 191)(78 190)(79 189)(80 188)(81 187)(82 186)(83 185)(84 184)(85 183)(86 182)(87 181)(88 180)(89 179)(90 178)(91 177)(92 176)(93 175)(94 174)(95 173)(96 172)(97 171)(98 170)(99 169)(100 168)(101 167)(102 166)(103 165)(104 164)(105 163)(106 162)(107 161)(108 160)(109 159)(110 158)(111 157)(112 156)(113 155)(114 154)(115 153)(116 152)(117 151)(118 150)(119 149)(120 148)(121 147)(122 146)(123 145)(124 144)(125 143)(126 142)(127 141)(128 140)(129 139)(130 207)(131 206)(132 205)(133 204)(134 203)(135 202)(136 201)(137 200)(138 199)
G:=sub<Sym(207)| (1,140,129)(2,141,130)(3,142,131)(4,143,132)(5,144,133)(6,145,134)(7,146,135)(8,147,136)(9,148,137)(10,149,138)(11,150,70)(12,151,71)(13,152,72)(14,153,73)(15,154,74)(16,155,75)(17,156,76)(18,157,77)(19,158,78)(20,159,79)(21,160,80)(22,161,81)(23,162,82)(24,163,83)(25,164,84)(26,165,85)(27,166,86)(28,167,87)(29,168,88)(30,169,89)(31,170,90)(32,171,91)(33,172,92)(34,173,93)(35,174,94)(36,175,95)(37,176,96)(38,177,97)(39,178,98)(40,179,99)(41,180,100)(42,181,101)(43,182,102)(44,183,103)(45,184,104)(46,185,105)(47,186,106)(48,187,107)(49,188,108)(50,189,109)(51,190,110)(52,191,111)(53,192,112)(54,193,113)(55,194,114)(56,195,115)(57,196,116)(58,197,117)(59,198,118)(60,199,119)(61,200,120)(62,201,121)(63,202,122)(64,203,123)(65,204,124)(66,205,125)(67,206,126)(68,207,127)(69,139,128), (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), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,60)(11,59)(12,58)(13,57)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,48)(23,47)(24,46)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36)(70,198)(71,197)(72,196)(73,195)(74,194)(75,193)(76,192)(77,191)(78,190)(79,189)(80,188)(81,187)(82,186)(83,185)(84,184)(85,183)(86,182)(87,181)(88,180)(89,179)(90,178)(91,177)(92,176)(93,175)(94,174)(95,173)(96,172)(97,171)(98,170)(99,169)(100,168)(101,167)(102,166)(103,165)(104,164)(105,163)(106,162)(107,161)(108,160)(109,159)(110,158)(111,157)(112,156)(113,155)(114,154)(115,153)(116,152)(117,151)(118,150)(119,149)(120,148)(121,147)(122,146)(123,145)(124,144)(125,143)(126,142)(127,141)(128,140)(129,139)(130,207)(131,206)(132,205)(133,204)(134,203)(135,202)(136,201)(137,200)(138,199)>;
G:=Group( (1,140,129)(2,141,130)(3,142,131)(4,143,132)(5,144,133)(6,145,134)(7,146,135)(8,147,136)(9,148,137)(10,149,138)(11,150,70)(12,151,71)(13,152,72)(14,153,73)(15,154,74)(16,155,75)(17,156,76)(18,157,77)(19,158,78)(20,159,79)(21,160,80)(22,161,81)(23,162,82)(24,163,83)(25,164,84)(26,165,85)(27,166,86)(28,167,87)(29,168,88)(30,169,89)(31,170,90)(32,171,91)(33,172,92)(34,173,93)(35,174,94)(36,175,95)(37,176,96)(38,177,97)(39,178,98)(40,179,99)(41,180,100)(42,181,101)(43,182,102)(44,183,103)(45,184,104)(46,185,105)(47,186,106)(48,187,107)(49,188,108)(50,189,109)(51,190,110)(52,191,111)(53,192,112)(54,193,113)(55,194,114)(56,195,115)(57,196,116)(58,197,117)(59,198,118)(60,199,119)(61,200,120)(62,201,121)(63,202,122)(64,203,123)(65,204,124)(66,205,125)(67,206,126)(68,207,127)(69,139,128), (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), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,60)(11,59)(12,58)(13,57)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,48)(23,47)(24,46)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36)(70,198)(71,197)(72,196)(73,195)(74,194)(75,193)(76,192)(77,191)(78,190)(79,189)(80,188)(81,187)(82,186)(83,185)(84,184)(85,183)(86,182)(87,181)(88,180)(89,179)(90,178)(91,177)(92,176)(93,175)(94,174)(95,173)(96,172)(97,171)(98,170)(99,169)(100,168)(101,167)(102,166)(103,165)(104,164)(105,163)(106,162)(107,161)(108,160)(109,159)(110,158)(111,157)(112,156)(113,155)(114,154)(115,153)(116,152)(117,151)(118,150)(119,149)(120,148)(121,147)(122,146)(123,145)(124,144)(125,143)(126,142)(127,141)(128,140)(129,139)(130,207)(131,206)(132,205)(133,204)(134,203)(135,202)(136,201)(137,200)(138,199) );
G=PermutationGroup([[(1,140,129),(2,141,130),(3,142,131),(4,143,132),(5,144,133),(6,145,134),(7,146,135),(8,147,136),(9,148,137),(10,149,138),(11,150,70),(12,151,71),(13,152,72),(14,153,73),(15,154,74),(16,155,75),(17,156,76),(18,157,77),(19,158,78),(20,159,79),(21,160,80),(22,161,81),(23,162,82),(24,163,83),(25,164,84),(26,165,85),(27,166,86),(28,167,87),(29,168,88),(30,169,89),(31,170,90),(32,171,91),(33,172,92),(34,173,93),(35,174,94),(36,175,95),(37,176,96),(38,177,97),(39,178,98),(40,179,99),(41,180,100),(42,181,101),(43,182,102),(44,183,103),(45,184,104),(46,185,105),(47,186,106),(48,187,107),(49,188,108),(50,189,109),(51,190,110),(52,191,111),(53,192,112),(54,193,113),(55,194,114),(56,195,115),(57,196,116),(58,197,117),(59,198,118),(60,199,119),(61,200,120),(62,201,121),(63,202,122),(64,203,123),(65,204,124),(66,205,125),(67,206,126),(68,207,127),(69,139,128)], [(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)], [(1,69),(2,68),(3,67),(4,66),(5,65),(6,64),(7,63),(8,62),(9,61),(10,60),(11,59),(12,58),(13,57),(14,56),(15,55),(16,54),(17,53),(18,52),(19,51),(20,50),(21,49),(22,48),(23,47),(24,46),(25,45),(26,44),(27,43),(28,42),(29,41),(30,40),(31,39),(32,38),(33,37),(34,36),(70,198),(71,197),(72,196),(73,195),(74,194),(75,193),(76,192),(77,191),(78,190),(79,189),(80,188),(81,187),(82,186),(83,185),(84,184),(85,183),(86,182),(87,181),(88,180),(89,179),(90,178),(91,177),(92,176),(93,175),(94,174),(95,173),(96,172),(97,171),(98,170),(99,169),(100,168),(101,167),(102,166),(103,165),(104,164),(105,163),(106,162),(107,161),(108,160),(109,159),(110,158),(111,157),(112,156),(113,155),(114,154),(115,153),(116,152),(117,151),(118,150),(119,149),(120,148),(121,147),(122,146),(123,145),(124,144),(125,143),(126,142),(127,141),(128,140),(129,139),(130,207),(131,206),(132,205),(133,204),(134,203),(135,202),(136,201),(137,200),(138,199)]])
105 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 23A | ··· | 23K | 69A | ··· | 69CJ |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 23 | ··· | 23 | 69 | ··· | 69 |
| size | 1 | 207 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
105 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | S3 | D23 | D69 |
| kernel | C3⋊D69 | C3×C69 | C69 | C32 | C3 |
| # reps | 1 | 1 | 4 | 11 | 88 |
Matrix representation of C3⋊D69 ►in GL4(𝔽139) generated by
| 23 | 56 | 0 | 0 |
| 77 | 115 | 0 | 0 |
| 0 | 0 | 96 | 126 |
| 0 | 0 | 0 | 42 |
| 54 | 133 | 0 | 0 |
| 96 | 64 | 0 | 0 |
| 0 | 0 | 91 | 84 |
| 0 | 0 | 0 | 55 |
| 117 | 111 | 0 | 0 |
| 52 | 22 | 0 | 0 |
| 0 | 0 | 63 | 44 |
| 0 | 0 | 112 | 76 |
G:=sub<GL(4,GF(139))| [23,77,0,0,56,115,0,0,0,0,96,0,0,0,126,42],[54,96,0,0,133,64,0,0,0,0,91,0,0,0,84,55],[117,52,0,0,111,22,0,0,0,0,63,112,0,0,44,76] >;
C3⋊D69 in GAP, Magma, Sage, TeX
C_3\rtimes D_{69} % in TeX
G:=Group("C3:D69"); // GroupNames label
G:=SmallGroup(414,9);
// by ID
G=gap.SmallGroup(414,9);
# by ID
G:=PCGroup([4,-2,-3,-3,-23,33,146,6339]);
// Polycyclic
G:=Group<a,b,c|a^3=b^69=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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