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G = C27.A4order 324 = 22·34

The central extension by C27 of A4

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

C1C22 — C27.A4
C1C22C2×C6C2×C18C2×C54 — C27.A4
C22 — C27.A4
C1C27

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 >

3C2
3C6
3C18
3C54
4C81

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 3A3B6A6B9A···9F18A···18F27A···27R54A···54R81A···81BB
order1233669···918···1827···2754···5481···81
size1311331···13···31···13···34···4

108 irreducible representations

dim111113333
type++
imageC1C3C9C27C81A4C3.A4C9.A4C27.A4
kernelC27.A4C2×C54C2×C18C2×C6C22C27C9C3C1
# reps126185412618

Matrix representation of C27.A4 in GL4(𝔽163) generated by

36000
05800
00580
00058
,
1000
0100
0381620
01400162
,
1000
016200
001620
02301
,
84000
0381610
001251
023230
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

Subgroup lattice of C27.A4 in TeX

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