direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3⋊S3×C27, C32⋊4C54, C33.7C18, C3⋊(S3×C27), (C3×C27)⋊9S3, (C32×C27)⋊2C2, C32.20(S3×C9), (C32×C9).28C6, C9.6(C3×C3⋊S3), C3.5(C9×C3⋊S3), (C3×C3⋊S3).2C9, (C9×C3⋊S3).2C3, (C3×C9).51(C3×S3), SmallGroup(486,161)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C3⋊S3×C27 |
Generators and relations for C3⋊S3×C27
G = < a,b,c,d | a27=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 160 in 72 conjugacy classes, 28 normal (12 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, C18, C3×S3, C3⋊S3, C27, C27, C3×C9, C3×C9, C33, C54, S3×C9, C3×C3⋊S3, C3×C27, C3×C27, C32×C9, S3×C27, C9×C3⋊S3, C32×C27, C3⋊S3×C27
Quotients: C1, C2, C3, S3, C6, C9, C18, C3×S3, C3⋊S3, C27, C54, S3×C9, C3×C3⋊S3, S3×C27, C9×C3⋊S3, C3⋊S3×C27
►On 162 points
Generators in S162
(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)
(1 97 34)(2 98 35)(3 99 36)(4 100 37)(5 101 38)(6 102 39)(7 103 40)(8 104 41)(9 105 42)(10 106 43)(11 107 44)(12 108 45)(13 82 46)(14 83 47)(15 84 48)(16 85 49)(17 86 50)(18 87 51)(19 88 52)(20 89 53)(21 90 54)(22 91 28)(23 92 29)(24 93 30)(25 94 31)(26 95 32)(27 96 33)(55 115 161)(56 116 162)(57 117 136)(58 118 137)(59 119 138)(60 120 139)(61 121 140)(62 122 141)(63 123 142)(64 124 143)(65 125 144)(66 126 145)(67 127 146)(68 128 147)(69 129 148)(70 130 149)(71 131 150)(72 132 151)(73 133 152)(74 134 153)(75 135 154)(76 109 155)(77 110 156)(78 111 157)(79 112 158)(80 113 159)(81 114 160)
(1 43 88)(2 44 89)(3 45 90)(4 46 91)(5 47 92)(6 48 93)(7 49 94)(8 50 95)(9 51 96)(10 52 97)(11 53 98)(12 54 99)(13 28 100)(14 29 101)(15 30 102)(16 31 103)(17 32 104)(18 33 105)(19 34 106)(20 35 107)(21 36 108)(22 37 82)(23 38 83)(24 39 84)(25 40 85)(26 41 86)(27 42 87)(55 152 124)(56 153 125)(57 154 126)(58 155 127)(59 156 128)(60 157 129)(61 158 130)(62 159 131)(63 160 132)(64 161 133)(65 162 134)(66 136 135)(67 137 109)(68 138 110)(69 139 111)(70 140 112)(71 141 113)(72 142 114)(73 143 115)(74 144 116)(75 145 117)(76 146 118)(77 147 119)(78 148 120)(79 149 121)(80 150 122)(81 151 123)
(1 125)(2 126)(3 127)(4 128)(5 129)(6 130)(7 131)(8 132)(9 133)(10 134)(11 135)(12 109)(13 110)(14 111)(15 112)(16 113)(17 114)(18 115)(19 116)(20 117)(21 118)(22 119)(23 120)(24 121)(25 122)(26 123)(27 124)(28 138)(29 139)(30 140)(31 141)(32 142)(33 143)(34 144)(35 145)(36 146)(37 147)(38 148)(39 149)(40 150)(41 151)(42 152)(43 153)(44 154)(45 155)(46 156)(47 157)(48 158)(49 159)(50 160)(51 161)(52 162)(53 136)(54 137)(55 87)(56 88)(57 89)(58 90)(59 91)(60 92)(61 93)(62 94)(63 95)(64 96)(65 97)(66 98)(67 99)(68 100)(69 101)(70 102)(71 103)(72 104)(73 105)(74 106)(75 107)(76 108)(77 82)(78 83)(79 84)(80 85)(81 86)
G:=sub<Sym(162)| (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), (1,97,34)(2,98,35)(3,99,36)(4,100,37)(5,101,38)(6,102,39)(7,103,40)(8,104,41)(9,105,42)(10,106,43)(11,107,44)(12,108,45)(13,82,46)(14,83,47)(15,84,48)(16,85,49)(17,86,50)(18,87,51)(19,88,52)(20,89,53)(21,90,54)(22,91,28)(23,92,29)(24,93,30)(25,94,31)(26,95,32)(27,96,33)(55,115,161)(56,116,162)(57,117,136)(58,118,137)(59,119,138)(60,120,139)(61,121,140)(62,122,141)(63,123,142)(64,124,143)(65,125,144)(66,126,145)(67,127,146)(68,128,147)(69,129,148)(70,130,149)(71,131,150)(72,132,151)(73,133,152)(74,134,153)(75,135,154)(76,109,155)(77,110,156)(78,111,157)(79,112,158)(80,113,159)(81,114,160), (1,43,88)(2,44,89)(3,45,90)(4,46,91)(5,47,92)(6,48,93)(7,49,94)(8,50,95)(9,51,96)(10,52,97)(11,53,98)(12,54,99)(13,28,100)(14,29,101)(15,30,102)(16,31,103)(17,32,104)(18,33,105)(19,34,106)(20,35,107)(21,36,108)(22,37,82)(23,38,83)(24,39,84)(25,40,85)(26,41,86)(27,42,87)(55,152,124)(56,153,125)(57,154,126)(58,155,127)(59,156,128)(60,157,129)(61,158,130)(62,159,131)(63,160,132)(64,161,133)(65,162,134)(66,136,135)(67,137,109)(68,138,110)(69,139,111)(70,140,112)(71,141,113)(72,142,114)(73,143,115)(74,144,116)(75,145,117)(76,146,118)(77,147,119)(78,148,120)(79,149,121)(80,150,122)(81,151,123), (1,125)(2,126)(3,127)(4,128)(5,129)(6,130)(7,131)(8,132)(9,133)(10,134)(11,135)(12,109)(13,110)(14,111)(15,112)(16,113)(17,114)(18,115)(19,116)(20,117)(21,118)(22,119)(23,120)(24,121)(25,122)(26,123)(27,124)(28,138)(29,139)(30,140)(31,141)(32,142)(33,143)(34,144)(35,145)(36,146)(37,147)(38,148)(39,149)(40,150)(41,151)(42,152)(43,153)(44,154)(45,155)(46,156)(47,157)(48,158)(49,159)(50,160)(51,161)(52,162)(53,136)(54,137)(55,87)(56,88)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,99)(68,100)(69,101)(70,102)(71,103)(72,104)(73,105)(74,106)(75,107)(76,108)(77,82)(78,83)(79,84)(80,85)(81,86)>;
G:=Group( (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), (1,97,34)(2,98,35)(3,99,36)(4,100,37)(5,101,38)(6,102,39)(7,103,40)(8,104,41)(9,105,42)(10,106,43)(11,107,44)(12,108,45)(13,82,46)(14,83,47)(15,84,48)(16,85,49)(17,86,50)(18,87,51)(19,88,52)(20,89,53)(21,90,54)(22,91,28)(23,92,29)(24,93,30)(25,94,31)(26,95,32)(27,96,33)(55,115,161)(56,116,162)(57,117,136)(58,118,137)(59,119,138)(60,120,139)(61,121,140)(62,122,141)(63,123,142)(64,124,143)(65,125,144)(66,126,145)(67,127,146)(68,128,147)(69,129,148)(70,130,149)(71,131,150)(72,132,151)(73,133,152)(74,134,153)(75,135,154)(76,109,155)(77,110,156)(78,111,157)(79,112,158)(80,113,159)(81,114,160), (1,43,88)(2,44,89)(3,45,90)(4,46,91)(5,47,92)(6,48,93)(7,49,94)(8,50,95)(9,51,96)(10,52,97)(11,53,98)(12,54,99)(13,28,100)(14,29,101)(15,30,102)(16,31,103)(17,32,104)(18,33,105)(19,34,106)(20,35,107)(21,36,108)(22,37,82)(23,38,83)(24,39,84)(25,40,85)(26,41,86)(27,42,87)(55,152,124)(56,153,125)(57,154,126)(58,155,127)(59,156,128)(60,157,129)(61,158,130)(62,159,131)(63,160,132)(64,161,133)(65,162,134)(66,136,135)(67,137,109)(68,138,110)(69,139,111)(70,140,112)(71,141,113)(72,142,114)(73,143,115)(74,144,116)(75,145,117)(76,146,118)(77,147,119)(78,148,120)(79,149,121)(80,150,122)(81,151,123), (1,125)(2,126)(3,127)(4,128)(5,129)(6,130)(7,131)(8,132)(9,133)(10,134)(11,135)(12,109)(13,110)(14,111)(15,112)(16,113)(17,114)(18,115)(19,116)(20,117)(21,118)(22,119)(23,120)(24,121)(25,122)(26,123)(27,124)(28,138)(29,139)(30,140)(31,141)(32,142)(33,143)(34,144)(35,145)(36,146)(37,147)(38,148)(39,149)(40,150)(41,151)(42,152)(43,153)(44,154)(45,155)(46,156)(47,157)(48,158)(49,159)(50,160)(51,161)(52,162)(53,136)(54,137)(55,87)(56,88)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,99)(68,100)(69,101)(70,102)(71,103)(72,104)(73,105)(74,106)(75,107)(76,108)(77,82)(78,83)(79,84)(80,85)(81,86) );
G=PermutationGroup([[(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)], [(1,97,34),(2,98,35),(3,99,36),(4,100,37),(5,101,38),(6,102,39),(7,103,40),(8,104,41),(9,105,42),(10,106,43),(11,107,44),(12,108,45),(13,82,46),(14,83,47),(15,84,48),(16,85,49),(17,86,50),(18,87,51),(19,88,52),(20,89,53),(21,90,54),(22,91,28),(23,92,29),(24,93,30),(25,94,31),(26,95,32),(27,96,33),(55,115,161),(56,116,162),(57,117,136),(58,118,137),(59,119,138),(60,120,139),(61,121,140),(62,122,141),(63,123,142),(64,124,143),(65,125,144),(66,126,145),(67,127,146),(68,128,147),(69,129,148),(70,130,149),(71,131,150),(72,132,151),(73,133,152),(74,134,153),(75,135,154),(76,109,155),(77,110,156),(78,111,157),(79,112,158),(80,113,159),(81,114,160)], [(1,43,88),(2,44,89),(3,45,90),(4,46,91),(5,47,92),(6,48,93),(7,49,94),(8,50,95),(9,51,96),(10,52,97),(11,53,98),(12,54,99),(13,28,100),(14,29,101),(15,30,102),(16,31,103),(17,32,104),(18,33,105),(19,34,106),(20,35,107),(21,36,108),(22,37,82),(23,38,83),(24,39,84),(25,40,85),(26,41,86),(27,42,87),(55,152,124),(56,153,125),(57,154,126),(58,155,127),(59,156,128),(60,157,129),(61,158,130),(62,159,131),(63,160,132),(64,161,133),(65,162,134),(66,136,135),(67,137,109),(68,138,110),(69,139,111),(70,140,112),(71,141,113),(72,142,114),(73,143,115),(74,144,116),(75,145,117),(76,146,118),(77,147,119),(78,148,120),(79,149,121),(80,150,122),(81,151,123)], [(1,125),(2,126),(3,127),(4,128),(5,129),(6,130),(7,131),(8,132),(9,133),(10,134),(11,135),(12,109),(13,110),(14,111),(15,112),(16,113),(17,114),(18,115),(19,116),(20,117),(21,118),(22,119),(23,120),(24,121),(25,122),(26,123),(27,124),(28,138),(29,139),(30,140),(31,141),(32,142),(33,143),(34,144),(35,145),(36,146),(37,147),(38,148),(39,149),(40,150),(41,151),(42,152),(43,153),(44,154),(45,155),(46,156),(47,157),(48,158),(49,159),(50,160),(51,161),(52,162),(53,136),(54,137),(55,87),(56,88),(57,89),(58,90),(59,91),(60,92),(61,93),(62,94),(63,95),(64,96),(65,97),(66,98),(67,99),(68,100),(69,101),(70,102),(71,103),(72,104),(73,105),(74,106),(75,107),(76,108),(77,82),(78,83),(79,84),(80,85),(81,86)]])
162 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3N | 6A | 6B | 9A | ··· | 9F | 9G | ··· | 9AD | 18A | ··· | 18F | 27A | ··· | 27R | 27S | ··· | 27CL | 54A | ··· | 54R |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 | 27 | ··· | 27 | 27 | ··· | 27 | 54 | ··· | 54 |
| size | 1 | 9 | 1 | 1 | 2 | ··· | 2 | 9 | 9 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 |
162 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | C27 | C54 | S3 | C3×S3 | S3×C9 | S3×C27 |
| kernel | C3⋊S3×C27 | C32×C27 | C9×C3⋊S3 | C32×C9 | C3×C3⋊S3 | C33 | C3⋊S3 | C32 | C3×C27 | C3×C9 | C32 | C3 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 18 | 18 | 4 | 8 | 24 | 72 |
Matrix representation of C3⋊S3×C27 ►in GL4(𝔽109) generated by
| 7 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 |
| 0 | 0 | 49 | 0 |
| 0 | 0 | 0 | 49 |
| 45 | 0 | 0 | 0 |
| 0 | 63 | 0 | 0 |
| 0 | 0 | 45 | 0 |
| 0 | 0 | 0 | 63 |
| 45 | 0 | 0 | 0 |
| 0 | 63 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(109))| [7,0,0,0,0,7,0,0,0,0,49,0,0,0,0,49],[45,0,0,0,0,63,0,0,0,0,45,0,0,0,0,63],[45,0,0,0,0,63,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C3⋊S3×C27 in GAP, Magma, Sage, TeX
C_3\rtimes S_3\times C_{27} % in TeX
G:=Group("C3:S3xC27"); // GroupNames label
G:=SmallGroup(486,161);
// by ID
G=gap.SmallGroup(486,161);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,68,3244,11669]);
// Polycyclic
G:=Group<a,b,c,d|a^27=b^3=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations