metabelian, supersoluble, monomial, A-group
Aliases: C3⋊D63, C9⋊D21, C63⋊1S3, C21⋊1D9, C32.3D21, C7⋊(C9⋊S3), (C3×C9)⋊3D7, (C3×C63)⋊3C2, C3.(C3⋊D21), (C3×C21).3S3, C21.1(C3⋊S3), SmallGroup(378,42)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C63 — C3⋊D63 |
Generators and relations for C3⋊D63
G = < a,b,c | a3=b63=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Smallest permutation representation of C3⋊D63
►On 189 points
Generators in S189
(1 145 121)(2 146 122)(3 147 123)(4 148 124)(5 149 125)(6 150 126)(7 151 64)(8 152 65)(9 153 66)(10 154 67)(11 155 68)(12 156 69)(13 157 70)(14 158 71)(15 159 72)(16 160 73)(17 161 74)(18 162 75)(19 163 76)(20 164 77)(21 165 78)(22 166 79)(23 167 80)(24 168 81)(25 169 82)(26 170 83)(27 171 84)(28 172 85)(29 173 86)(30 174 87)(31 175 88)(32 176 89)(33 177 90)(34 178 91)(35 179 92)(36 180 93)(37 181 94)(38 182 95)(39 183 96)(40 184 97)(41 185 98)(42 186 99)(43 187 100)(44 188 101)(45 189 102)(46 127 103)(47 128 104)(48 129 105)(49 130 106)(50 131 107)(51 132 108)(52 133 109)(53 134 110)(54 135 111)(55 136 112)(56 137 113)(57 138 114)(58 139 115)(59 140 116)(60 141 117)(61 142 118)(62 143 119)(63 144 120)
(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)
(2 63)(3 62)(4 61)(5 60)(6 59)(7 58)(8 57)(9 56)(10 55)(11 54)(12 53)(13 52)(14 51)(15 50)(16 49)(17 48)(18 47)(19 46)(20 45)(21 44)(22 43)(23 42)(24 41)(25 40)(26 39)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(64 139)(65 138)(66 137)(67 136)(68 135)(69 134)(70 133)(71 132)(72 131)(73 130)(74 129)(75 128)(76 127)(77 189)(78 188)(79 187)(80 186)(81 185)(82 184)(83 183)(84 182)(85 181)(86 180)(87 179)(88 178)(89 177)(90 176)(91 175)(92 174)(93 173)(94 172)(95 171)(96 170)(97 169)(98 168)(99 167)(100 166)(101 165)(102 164)(103 163)(104 162)(105 161)(106 160)(107 159)(108 158)(109 157)(110 156)(111 155)(112 154)(113 153)(114 152)(115 151)(116 150)(117 149)(118 148)(119 147)(120 146)(121 145)(122 144)(123 143)(124 142)(125 141)(126 140)
G:=sub<Sym(189)| (1,145,121)(2,146,122)(3,147,123)(4,148,124)(5,149,125)(6,150,126)(7,151,64)(8,152,65)(9,153,66)(10,154,67)(11,155,68)(12,156,69)(13,157,70)(14,158,71)(15,159,72)(16,160,73)(17,161,74)(18,162,75)(19,163,76)(20,164,77)(21,165,78)(22,166,79)(23,167,80)(24,168,81)(25,169,82)(26,170,83)(27,171,84)(28,172,85)(29,173,86)(30,174,87)(31,175,88)(32,176,89)(33,177,90)(34,178,91)(35,179,92)(36,180,93)(37,181,94)(38,182,95)(39,183,96)(40,184,97)(41,185,98)(42,186,99)(43,187,100)(44,188,101)(45,189,102)(46,127,103)(47,128,104)(48,129,105)(49,130,106)(50,131,107)(51,132,108)(52,133,109)(53,134,110)(54,135,111)(55,136,112)(56,137,113)(57,138,114)(58,139,115)(59,140,116)(60,141,117)(61,142,118)(62,143,119)(63,144,120), (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), (2,63)(3,62)(4,61)(5,60)(6,59)(7,58)(8,57)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(64,139)(65,138)(66,137)(67,136)(68,135)(69,134)(70,133)(71,132)(72,131)(73,130)(74,129)(75,128)(76,127)(77,189)(78,188)(79,187)(80,186)(81,185)(82,184)(83,183)(84,182)(85,181)(86,180)(87,179)(88,178)(89,177)(90,176)(91,175)(92,174)(93,173)(94,172)(95,171)(96,170)(97,169)(98,168)(99,167)(100,166)(101,165)(102,164)(103,163)(104,162)(105,161)(106,160)(107,159)(108,158)(109,157)(110,156)(111,155)(112,154)(113,153)(114,152)(115,151)(116,150)(117,149)(118,148)(119,147)(120,146)(121,145)(122,144)(123,143)(124,142)(125,141)(126,140)>;
G:=Group( (1,145,121)(2,146,122)(3,147,123)(4,148,124)(5,149,125)(6,150,126)(7,151,64)(8,152,65)(9,153,66)(10,154,67)(11,155,68)(12,156,69)(13,157,70)(14,158,71)(15,159,72)(16,160,73)(17,161,74)(18,162,75)(19,163,76)(20,164,77)(21,165,78)(22,166,79)(23,167,80)(24,168,81)(25,169,82)(26,170,83)(27,171,84)(28,172,85)(29,173,86)(30,174,87)(31,175,88)(32,176,89)(33,177,90)(34,178,91)(35,179,92)(36,180,93)(37,181,94)(38,182,95)(39,183,96)(40,184,97)(41,185,98)(42,186,99)(43,187,100)(44,188,101)(45,189,102)(46,127,103)(47,128,104)(48,129,105)(49,130,106)(50,131,107)(51,132,108)(52,133,109)(53,134,110)(54,135,111)(55,136,112)(56,137,113)(57,138,114)(58,139,115)(59,140,116)(60,141,117)(61,142,118)(62,143,119)(63,144,120), (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), (2,63)(3,62)(4,61)(5,60)(6,59)(7,58)(8,57)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(64,139)(65,138)(66,137)(67,136)(68,135)(69,134)(70,133)(71,132)(72,131)(73,130)(74,129)(75,128)(76,127)(77,189)(78,188)(79,187)(80,186)(81,185)(82,184)(83,183)(84,182)(85,181)(86,180)(87,179)(88,178)(89,177)(90,176)(91,175)(92,174)(93,173)(94,172)(95,171)(96,170)(97,169)(98,168)(99,167)(100,166)(101,165)(102,164)(103,163)(104,162)(105,161)(106,160)(107,159)(108,158)(109,157)(110,156)(111,155)(112,154)(113,153)(114,152)(115,151)(116,150)(117,149)(118,148)(119,147)(120,146)(121,145)(122,144)(123,143)(124,142)(125,141)(126,140) );
G=PermutationGroup([[(1,145,121),(2,146,122),(3,147,123),(4,148,124),(5,149,125),(6,150,126),(7,151,64),(8,152,65),(9,153,66),(10,154,67),(11,155,68),(12,156,69),(13,157,70),(14,158,71),(15,159,72),(16,160,73),(17,161,74),(18,162,75),(19,163,76),(20,164,77),(21,165,78),(22,166,79),(23,167,80),(24,168,81),(25,169,82),(26,170,83),(27,171,84),(28,172,85),(29,173,86),(30,174,87),(31,175,88),(32,176,89),(33,177,90),(34,178,91),(35,179,92),(36,180,93),(37,181,94),(38,182,95),(39,183,96),(40,184,97),(41,185,98),(42,186,99),(43,187,100),(44,188,101),(45,189,102),(46,127,103),(47,128,104),(48,129,105),(49,130,106),(50,131,107),(51,132,108),(52,133,109),(53,134,110),(54,135,111),(55,136,112),(56,137,113),(57,138,114),(58,139,115),(59,140,116),(60,141,117),(61,142,118),(62,143,119),(63,144,120)], [(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)], [(2,63),(3,62),(4,61),(5,60),(6,59),(7,58),(8,57),(9,56),(10,55),(11,54),(12,53),(13,52),(14,51),(15,50),(16,49),(17,48),(18,47),(19,46),(20,45),(21,44),(22,43),(23,42),(24,41),(25,40),(26,39),(27,38),(28,37),(29,36),(30,35),(31,34),(32,33),(64,139),(65,138),(66,137),(67,136),(68,135),(69,134),(70,133),(71,132),(72,131),(73,130),(74,129),(75,128),(76,127),(77,189),(78,188),(79,187),(80,186),(81,185),(82,184),(83,183),(84,182),(85,181),(86,180),(87,179),(88,178),(89,177),(90,176),(91,175),(92,174),(93,173),(94,172),(95,171),(96,170),(97,169),(98,168),(99,167),(100,166),(101,165),(102,164),(103,163),(104,162),(105,161),(106,160),(107,159),(108,158),(109,157),(110,156),(111,155),(112,154),(113,153),(114,152),(115,151),(116,150),(117,149),(118,148),(119,147),(120,146),(121,145),(122,144),(123,143),(124,142),(125,141),(126,140)]])
96 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 7A | 7B | 7C | 9A | ··· | 9I | 21A | ··· | 21X | 63A | ··· | 63BB |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 7 | 7 | 7 | 9 | ··· | 9 | 21 | ··· | 21 | 63 | ··· | 63 |
| size | 1 | 189 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | S3 | S3 | D7 | D9 | D21 | D21 | D63 |
| kernel | C3⋊D63 | C3×C63 | C63 | C3×C21 | C3×C9 | C21 | C9 | C32 | C3 |
| # reps | 1 | 1 | 3 | 1 | 3 | 9 | 18 | 6 | 54 |
Matrix representation of C3⋊D63 ►in GL4(𝔽127) generated by
| 126 | 126 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 118 | 96 | 0 | 0 |
| 31 | 22 | 0 | 0 |
| 0 | 0 | 98 | 124 |
| 0 | 0 | 3 | 101 |
| 1 | 0 | 0 | 0 |
| 126 | 126 | 0 | 0 |
| 0 | 0 | 40 | 104 |
| 0 | 0 | 64 | 87 |
G:=sub<GL(4,GF(127))| [126,1,0,0,126,0,0,0,0,0,1,0,0,0,0,1],[118,31,0,0,96,22,0,0,0,0,98,3,0,0,124,101],[1,126,0,0,0,126,0,0,0,0,40,64,0,0,104,87] >;
C3⋊D63 in GAP, Magma, Sage, TeX
C_3\rtimes D_{63} % in TeX
G:=Group("C3:D63"); // GroupNames label
G:=SmallGroup(378,42);
// by ID
G=gap.SmallGroup(378,42);
# by ID
G:=PCGroup([5,-2,-3,-3,-7,-3,1301,1266,182,2163,6304]);
// Polycyclic
G:=Group<a,b,c|a^3=b^63=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export