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## G = C27.A4order 324 = 22·34

### The central extension by C27 of A4

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

 Derived series C1 — C22 — C27.A4
 Chief series C1 — C22 — C2×C6 — C2×C18 — C2×C54 — C27.A4
 Lower central C22 — C27.A4
 Upper central C1 — C27

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 94)(3 95)(5 97)(6 98)(8 100)(9 101)(11 103)(12 104)(14 106)(15 107)(17 109)(18 110)(20 112)(21 113)(23 115)(24 116)(26 118)(27 119)(29 121)(30 122)(32 124)(33 125)(35 127)(36 128)(38 130)(39 131)(41 133)(42 134)(44 136)(45 137)(47 139)(48 140)(50 142)(51 143)(53 145)(54 146)(56 148)(57 149)(59 151)(60 152)(62 154)(63 155)(65 157)(66 158)(68 160)(69 161)(71 82)(72 83)(74 85)(75 86)(77 88)(78 89)(80 91)(81 92)
(1 93)(3 95)(4 96)(6 98)(7 99)(9 101)(10 102)(12 104)(13 105)(15 107)(16 108)(18 110)(19 111)(21 113)(22 114)(24 116)(25 117)(27 119)(28 120)(30 122)(31 123)(33 125)(34 126)(36 128)(37 129)(39 131)(40 132)(42 134)(43 135)(45 137)(46 138)(48 140)(49 141)(51 143)(52 144)(54 146)(55 147)(57 149)(58 150)(60 152)(61 153)(63 155)(64 156)(66 158)(67 159)(69 161)(70 162)(72 83)(73 84)(75 86)(76 87)(78 89)(79 90)(81 92)
(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), (2,94)(3,95)(5,97)(6,98)(8,100)(9,101)(11,103)(12,104)(14,106)(15,107)(17,109)(18,110)(20,112)(21,113)(23,115)(24,116)(26,118)(27,119)(29,121)(30,122)(32,124)(33,125)(35,127)(36,128)(38,130)(39,131)(41,133)(42,134)(44,136)(45,137)(47,139)(48,140)(50,142)(51,143)(53,145)(54,146)(56,148)(57,149)(59,151)(60,152)(62,154)(63,155)(65,157)(66,158)(68,160)(69,161)(71,82)(72,83)(74,85)(75,86)(77,88)(78,89)(80,91)(81,92), (1,93)(3,95)(4,96)(6,98)(7,99)(9,101)(10,102)(12,104)(13,105)(15,107)(16,108)(18,110)(19,111)(21,113)(22,114)(24,116)(25,117)(27,119)(28,120)(30,122)(31,123)(33,125)(34,126)(36,128)(37,129)(39,131)(40,132)(42,134)(43,135)(45,137)(46,138)(48,140)(49,141)(51,143)(52,144)(54,146)(55,147)(57,149)(58,150)(60,152)(61,153)(63,155)(64,156)(66,158)(67,159)(69,161)(70,162)(72,83)(73,84)(75,86)(76,87)(78,89)(79,90)(81,92), (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), (2,94)(3,95)(5,97)(6,98)(8,100)(9,101)(11,103)(12,104)(14,106)(15,107)(17,109)(18,110)(20,112)(21,113)(23,115)(24,116)(26,118)(27,119)(29,121)(30,122)(32,124)(33,125)(35,127)(36,128)(38,130)(39,131)(41,133)(42,134)(44,136)(45,137)(47,139)(48,140)(50,142)(51,143)(53,145)(54,146)(56,148)(57,149)(59,151)(60,152)(62,154)(63,155)(65,157)(66,158)(68,160)(69,161)(71,82)(72,83)(74,85)(75,86)(77,88)(78,89)(80,91)(81,92), (1,93)(3,95)(4,96)(6,98)(7,99)(9,101)(10,102)(12,104)(13,105)(15,107)(16,108)(18,110)(19,111)(21,113)(22,114)(24,116)(25,117)(27,119)(28,120)(30,122)(31,123)(33,125)(34,126)(36,128)(37,129)(39,131)(40,132)(42,134)(43,135)(45,137)(46,138)(48,140)(49,141)(51,143)(52,144)(54,146)(55,147)(57,149)(58,150)(60,152)(61,153)(63,155)(64,156)(66,158)(67,159)(69,161)(70,162)(72,83)(73,84)(75,86)(76,87)(78,89)(79,90)(81,92), (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)], [(2,94),(3,95),(5,97),(6,98),(8,100),(9,101),(11,103),(12,104),(14,106),(15,107),(17,109),(18,110),(20,112),(21,113),(23,115),(24,116),(26,118),(27,119),(29,121),(30,122),(32,124),(33,125),(35,127),(36,128),(38,130),(39,131),(41,133),(42,134),(44,136),(45,137),(47,139),(48,140),(50,142),(51,143),(53,145),(54,146),(56,148),(57,149),(59,151),(60,152),(62,154),(63,155),(65,157),(66,158),(68,160),(69,161),(71,82),(72,83),(74,85),(75,86),(77,88),(78,89),(80,91),(81,92)], [(1,93),(3,95),(4,96),(6,98),(7,99),(9,101),(10,102),(12,104),(13,105),(15,107),(16,108),(18,110),(19,111),(21,113),(22,114),(24,116),(25,117),(27,119),(28,120),(30,122),(31,123),(33,125),(34,126),(36,128),(37,129),(39,131),(40,132),(42,134),(43,135),(45,137),(46,138),(48,140),(49,141),(51,143),(52,144),(54,146),(55,147),(57,149),(58,150),(60,152),(61,153),(63,155),(64,156),(66,158),(67,159),(69,161),(70,162),(72,83),(73,84),(75,86),(76,87),(78,89),(79,90),(81,92)], [(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`

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