metabelian, soluble, monomial, A-group
Aliases: C27.A4, C22⋊C81, (C2×C6).C27, (C2×C54).C3, C3.(C9.A4), (C2×C18).1C9, C9.2(C3.A4), SmallGroup(324,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — C27.A4 |
Generators and relations for C27.A4
G = < a,b,c,d | a27=b2=c2=1, d3=a, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Smallest permutation representation of C27.A4
►On 162 points
Generators in S162
(1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79)(2 5 8 11 14 17 20 23 26 29 32 35 38 41 44 47 50 53 56 59 62 65 68 71 74 77 80)(3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81)(82 85 88 91 94 97 100 103 106 109 112 115 118 121 124 127 130 133 136 139 142 145 148 151 154 157 160)(83 86 89 92 95 98 101 104 107 110 113 116 119 122 125 128 131 134 137 140 143 146 149 152 155 158 161)(84 87 90 93 96 99 102 105 108 111 114 117 120 123 126 129 132 135 138 141 144 147 150 153 156 159 162)
(2 154)(3 155)(5 157)(6 158)(8 160)(9 161)(11 82)(12 83)(14 85)(15 86)(17 88)(18 89)(20 91)(21 92)(23 94)(24 95)(26 97)(27 98)(29 100)(30 101)(32 103)(33 104)(35 106)(36 107)(38 109)(39 110)(41 112)(42 113)(44 115)(45 116)(47 118)(48 119)(50 121)(51 122)(53 124)(54 125)(56 127)(57 128)(59 130)(60 131)(62 133)(63 134)(65 136)(66 137)(68 139)(69 140)(71 142)(72 143)(74 145)(75 146)(77 148)(78 149)(80 151)(81 152)
(1 153)(3 155)(4 156)(6 158)(7 159)(9 161)(10 162)(12 83)(13 84)(15 86)(16 87)(18 89)(19 90)(21 92)(22 93)(24 95)(25 96)(27 98)(28 99)(30 101)(31 102)(33 104)(34 105)(36 107)(37 108)(39 110)(40 111)(42 113)(43 114)(45 116)(46 117)(48 119)(49 120)(51 122)(52 123)(54 125)(55 126)(57 128)(58 129)(60 131)(61 132)(63 134)(64 135)(66 137)(67 138)(69 140)(70 141)(72 143)(73 144)(75 146)(76 147)(78 149)(79 150)(81 152)
(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)
G:=sub<Sym(162)| (1,4,7,10,13,16,19,22,25,28,31,34,37,40,43,46,49,52,55,58,61,64,67,70,73,76,79)(2,5,8,11,14,17,20,23,26,29,32,35,38,41,44,47,50,53,56,59,62,65,68,71,74,77,80)(3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,72,75,78,81)(82,85,88,91,94,97,100,103,106,109,112,115,118,121,124,127,130,133,136,139,142,145,148,151,154,157,160)(83,86,89,92,95,98,101,104,107,110,113,116,119,122,125,128,131,134,137,140,143,146,149,152,155,158,161)(84,87,90,93,96,99,102,105,108,111,114,117,120,123,126,129,132,135,138,141,144,147,150,153,156,159,162), (2,154)(3,155)(5,157)(6,158)(8,160)(9,161)(11,82)(12,83)(14,85)(15,86)(17,88)(18,89)(20,91)(21,92)(23,94)(24,95)(26,97)(27,98)(29,100)(30,101)(32,103)(33,104)(35,106)(36,107)(38,109)(39,110)(41,112)(42,113)(44,115)(45,116)(47,118)(48,119)(50,121)(51,122)(53,124)(54,125)(56,127)(57,128)(59,130)(60,131)(62,133)(63,134)(65,136)(66,137)(68,139)(69,140)(71,142)(72,143)(74,145)(75,146)(77,148)(78,149)(80,151)(81,152), (1,153)(3,155)(4,156)(6,158)(7,159)(9,161)(10,162)(12,83)(13,84)(15,86)(16,87)(18,89)(19,90)(21,92)(22,93)(24,95)(25,96)(27,98)(28,99)(30,101)(31,102)(33,104)(34,105)(36,107)(37,108)(39,110)(40,111)(42,113)(43,114)(45,116)(46,117)(48,119)(49,120)(51,122)(52,123)(54,125)(55,126)(57,128)(58,129)(60,131)(61,132)(63,134)(64,135)(66,137)(67,138)(69,140)(70,141)(72,143)(73,144)(75,146)(76,147)(78,149)(79,150)(81,152), (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)>;
G:=Group( (1,4,7,10,13,16,19,22,25,28,31,34,37,40,43,46,49,52,55,58,61,64,67,70,73,76,79)(2,5,8,11,14,17,20,23,26,29,32,35,38,41,44,47,50,53,56,59,62,65,68,71,74,77,80)(3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,72,75,78,81)(82,85,88,91,94,97,100,103,106,109,112,115,118,121,124,127,130,133,136,139,142,145,148,151,154,157,160)(83,86,89,92,95,98,101,104,107,110,113,116,119,122,125,128,131,134,137,140,143,146,149,152,155,158,161)(84,87,90,93,96,99,102,105,108,111,114,117,120,123,126,129,132,135,138,141,144,147,150,153,156,159,162), (2,154)(3,155)(5,157)(6,158)(8,160)(9,161)(11,82)(12,83)(14,85)(15,86)(17,88)(18,89)(20,91)(21,92)(23,94)(24,95)(26,97)(27,98)(29,100)(30,101)(32,103)(33,104)(35,106)(36,107)(38,109)(39,110)(41,112)(42,113)(44,115)(45,116)(47,118)(48,119)(50,121)(51,122)(53,124)(54,125)(56,127)(57,128)(59,130)(60,131)(62,133)(63,134)(65,136)(66,137)(68,139)(69,140)(71,142)(72,143)(74,145)(75,146)(77,148)(78,149)(80,151)(81,152), (1,153)(3,155)(4,156)(6,158)(7,159)(9,161)(10,162)(12,83)(13,84)(15,86)(16,87)(18,89)(19,90)(21,92)(22,93)(24,95)(25,96)(27,98)(28,99)(30,101)(31,102)(33,104)(34,105)(36,107)(37,108)(39,110)(40,111)(42,113)(43,114)(45,116)(46,117)(48,119)(49,120)(51,122)(52,123)(54,125)(55,126)(57,128)(58,129)(60,131)(61,132)(63,134)(64,135)(66,137)(67,138)(69,140)(70,141)(72,143)(73,144)(75,146)(76,147)(78,149)(79,150)(81,152), (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) );
G=PermutationGroup([[(1,4,7,10,13,16,19,22,25,28,31,34,37,40,43,46,49,52,55,58,61,64,67,70,73,76,79),(2,5,8,11,14,17,20,23,26,29,32,35,38,41,44,47,50,53,56,59,62,65,68,71,74,77,80),(3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,72,75,78,81),(82,85,88,91,94,97,100,103,106,109,112,115,118,121,124,127,130,133,136,139,142,145,148,151,154,157,160),(83,86,89,92,95,98,101,104,107,110,113,116,119,122,125,128,131,134,137,140,143,146,149,152,155,158,161),(84,87,90,93,96,99,102,105,108,111,114,117,120,123,126,129,132,135,138,141,144,147,150,153,156,159,162)], [(2,154),(3,155),(5,157),(6,158),(8,160),(9,161),(11,82),(12,83),(14,85),(15,86),(17,88),(18,89),(20,91),(21,92),(23,94),(24,95),(26,97),(27,98),(29,100),(30,101),(32,103),(33,104),(35,106),(36,107),(38,109),(39,110),(41,112),(42,113),(44,115),(45,116),(47,118),(48,119),(50,121),(51,122),(53,124),(54,125),(56,127),(57,128),(59,130),(60,131),(62,133),(63,134),(65,136),(66,137),(68,139),(69,140),(71,142),(72,143),(74,145),(75,146),(77,148),(78,149),(80,151),(81,152)], [(1,153),(3,155),(4,156),(6,158),(7,159),(9,161),(10,162),(12,83),(13,84),(15,86),(16,87),(18,89),(19,90),(21,92),(22,93),(24,95),(25,96),(27,98),(28,99),(30,101),(31,102),(33,104),(34,105),(36,107),(37,108),(39,110),(40,111),(42,113),(43,114),(45,116),(46,117),(48,119),(49,120),(51,122),(52,123),(54,125),(55,126),(57,128),(58,129),(60,131),(61,132),(63,134),(64,135),(66,137),(67,138),(69,140),(70,141),(72,143),(73,144),(75,146),(76,147),(78,149),(79,150),(81,152)], [(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)]])
108 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 9A | ··· | 9F | 18A | ··· | 18F | 27A | ··· | 27R | 54A | ··· | 54R | 81A | ··· | 81BB |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 18 | ··· | 18 | 27 | ··· | 27 | 54 | ··· | 54 | 81 | ··· | 81 |
| size | 1 | 3 | 1 | 1 | 3 | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | |||||||
| image | C1 | C3 | C9 | C27 | C81 | A4 | C3.A4 | C9.A4 | C27.A4 |
| kernel | C27.A4 | C2×C54 | C2×C18 | C2×C6 | C22 | C27 | C9 | C3 | C1 |
| # reps | 1 | 2 | 6 | 18 | 54 | 1 | 2 | 6 | 18 |
Matrix representation of C27.A4 ►in GL4(𝔽163) generated by
| 36 | 0 | 0 | 0 |
| 0 | 58 | 0 | 0 |
| 0 | 0 | 58 | 0 |
| 0 | 0 | 0 | 58 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 38 | 162 | 0 |
| 0 | 140 | 0 | 162 |
| 1 | 0 | 0 | 0 |
| 0 | 162 | 0 | 0 |
| 0 | 0 | 162 | 0 |
| 0 | 23 | 0 | 1 |
| 84 | 0 | 0 | 0 |
| 0 | 38 | 161 | 0 |
| 0 | 0 | 125 | 1 |
| 0 | 23 | 23 | 0 |
G:=sub<GL(4,GF(163))| [36,0,0,0,0,58,0,0,0,0,58,0,0,0,0,58],[1,0,0,0,0,1,38,140,0,0,162,0,0,0,0,162],[1,0,0,0,0,162,0,23,0,0,162,0,0,0,0,1],[84,0,0,0,0,38,0,23,0,161,125,23,0,0,1,0] >;
C27.A4 in GAP, Magma, Sage, TeX
C_{27}.A_4 % in TeX
G:=Group("C27.A4"); // GroupNames label
G:=SmallGroup(324,3);
// by ID
G=gap.SmallGroup(324,3);
# by ID
G:=PCGroup([6,-3,-3,-3,-3,-2,2,18,43,68,4864,8753]);
// Polycyclic
G:=Group<a,b,c,d|a^27=b^2=c^2=1,d^3=a,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
Export