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## G = A4×C27order 324 = 22·34

### Direct product of C27 and A4

Aliases: A4×C27, (C2×C54)⋊1C3, C9.A43C3, C3.1(C9×A4), C9.2(C3×A4), (C3×A4).2C9, (C9×A4).2C3, C221(C3×C27), C3.A4.2C9, (C2×C18).2C32, (C2×C6).1(C3×C9), SmallGroup(324,42)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C27
 Chief series C1 — C22 — C2×C6 — C2×C18 — C9×A4 — A4×C27
 Lower central C22 — A4×C27
 Upper central C1 — C27

Generators and relations for A4×C27
G = < a,b,c,d | a27=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Smallest permutation representation of A4×C27
On 108 points
Generators in S108
(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)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 28)(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)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 61)(17 62)(18 63)(19 64)(20 65)(21 66)(22 67)(23 68)(24 69)(25 70)(26 71)(27 72)(28 104)(29 105)(30 106)(31 107)(32 108)(33 82)(34 83)(35 84)(36 85)(37 86)(38 87)(39 88)(40 89)(41 90)(42 91)(43 92)(44 93)(45 94)(46 95)(47 96)(48 97)(49 98)(50 99)(51 100)(52 101)(53 102)(54 103)
(1 10 19)(2 11 20)(3 12 21)(4 13 22)(5 14 23)(6 15 24)(7 16 25)(8 17 26)(9 18 27)(28 81 95)(29 55 96)(30 56 97)(31 57 98)(32 58 99)(33 59 100)(34 60 101)(35 61 102)(36 62 103)(37 63 104)(38 64 105)(39 65 106)(40 66 107)(41 67 108)(42 68 82)(43 69 83)(44 70 84)(45 71 85)(46 72 86)(47 73 87)(48 74 88)(49 75 89)(50 76 90)(51 77 91)(52 78 92)(53 79 93)(54 80 94)

G:=sub<Sym(108)| (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), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,28)(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), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,61)(17,62)(18,63)(19,64)(20,65)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,104)(29,105)(30,106)(31,107)(32,108)(33,82)(34,83)(35,84)(36,85)(37,86)(38,87)(39,88)(40,89)(41,90)(42,91)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(50,99)(51,100)(52,101)(53,102)(54,103), (1,10,19)(2,11,20)(3,12,21)(4,13,22)(5,14,23)(6,15,24)(7,16,25)(8,17,26)(9,18,27)(28,81,95)(29,55,96)(30,56,97)(31,57,98)(32,58,99)(33,59,100)(34,60,101)(35,61,102)(36,62,103)(37,63,104)(38,64,105)(39,65,106)(40,66,107)(41,67,108)(42,68,82)(43,69,83)(44,70,84)(45,71,85)(46,72,86)(47,73,87)(48,74,88)(49,75,89)(50,76,90)(51,77,91)(52,78,92)(53,79,93)(54,80,94)>;

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), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,28)(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), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,61)(17,62)(18,63)(19,64)(20,65)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,104)(29,105)(30,106)(31,107)(32,108)(33,82)(34,83)(35,84)(36,85)(37,86)(38,87)(39,88)(40,89)(41,90)(42,91)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(50,99)(51,100)(52,101)(53,102)(54,103), (1,10,19)(2,11,20)(3,12,21)(4,13,22)(5,14,23)(6,15,24)(7,16,25)(8,17,26)(9,18,27)(28,81,95)(29,55,96)(30,56,97)(31,57,98)(32,58,99)(33,59,100)(34,60,101)(35,61,102)(36,62,103)(37,63,104)(38,64,105)(39,65,106)(40,66,107)(41,67,108)(42,68,82)(43,69,83)(44,70,84)(45,71,85)(46,72,86)(47,73,87)(48,74,88)(49,75,89)(50,76,90)(51,77,91)(52,78,92)(53,79,93)(54,80,94) );

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)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,28),(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)], [(1,73),(2,74),(3,75),(4,76),(5,77),(6,78),(7,79),(8,80),(9,81),(10,55),(11,56),(12,57),(13,58),(14,59),(15,60),(16,61),(17,62),(18,63),(19,64),(20,65),(21,66),(22,67),(23,68),(24,69),(25,70),(26,71),(27,72),(28,104),(29,105),(30,106),(31,107),(32,108),(33,82),(34,83),(35,84),(36,85),(37,86),(38,87),(39,88),(40,89),(41,90),(42,91),(43,92),(44,93),(45,94),(46,95),(47,96),(48,97),(49,98),(50,99),(51,100),(52,101),(53,102),(54,103)], [(1,10,19),(2,11,20),(3,12,21),(4,13,22),(5,14,23),(6,15,24),(7,16,25),(8,17,26),(9,18,27),(28,81,95),(29,55,96),(30,56,97),(31,57,98),(32,58,99),(33,59,100),(34,60,101),(35,61,102),(36,62,103),(37,63,104),(38,64,105),(39,65,106),(40,66,107),(41,67,108),(42,68,82),(43,69,83),(44,70,84),(45,71,85),(46,72,86),(47,73,87),(48,74,88),(49,75,89),(50,76,90),(51,77,91),(52,78,92),(53,79,93),(54,80,94)]])

108 conjugacy classes

 class 1 2 3A 3B 3C ··· 3H 6A 6B 9A ··· 9F 9G ··· 9R 18A ··· 18F 27A ··· 27R 27S ··· 27BB 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 3 1 1 4 ··· 4 3 3 1 ··· 1 4 ··· 4 3 ··· 3 1 ··· 1 4 ··· 4 3 ··· 3

108 irreducible representations

 dim 1 1 1 1 1 1 1 3 3 3 3 type + + image C1 C3 C3 C3 C9 C9 C27 A4 C3×A4 C9×A4 A4×C27 kernel A4×C27 C9.A4 C2×C54 C9×A4 C3.A4 C3×A4 A4 C27 C9 C3 C1 # reps 1 4 2 2 12 6 54 1 2 6 18

Matrix representation of A4×C27 in GL4(𝔽109) generated by

 22 0 0 0 0 63 0 0 0 0 63 0 0 0 0 63
,
 1 0 0 0 0 0 108 1 0 0 108 0 0 1 108 0
,
 1 0 0 0 0 108 0 0 0 108 0 1 0 108 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0
G:=sub<GL(4,GF(109))| [22,0,0,0,0,63,0,0,0,0,63,0,0,0,0,63],[1,0,0,0,0,0,0,1,0,108,108,108,0,1,0,0],[1,0,0,0,0,108,108,108,0,0,0,1,0,0,1,0],[1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;

A4×C27 in GAP, Magma, Sage, TeX

A_4\times C_{27}
% in TeX

G:=Group("A4xC27");
// GroupNames label

G:=SmallGroup(324,42);
// by ID

G=gap.SmallGroup(324,42);
# by ID

G:=PCGroup([6,-3,-3,-3,-3,-2,2,43,68,4864,8753]);
// Polycyclic

G:=Group<a,b,c,d|a^27=b^2=c^2=d^3=1,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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