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## G = C27⋊A4order 324 = 22·34

### The semidirect product of C27 and A4 acting via A4/C22=C3

Aliases: C27⋊A4, (C3×A4).C9, (C9×A4).C3, C3.A4.C9, (C2×C54)⋊2C3, C9.A41C3, C9.3(C3×A4), C3.3(C9×A4), C221(C27⋊C3), (C2×C18).3C32, (C2×C6).2(C3×C9), SmallGroup(324,43)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C27⋊A4
G = < a,b,c,d | a27=b2=c2=d3=1, ab=ba, ac=ca, dad-1=a19, dbd-1=bc=cb, dcd-1=b >

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

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

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

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

60 conjugacy classes

 class 1 2 3A 3B 3C 3D 6A 6B 9A ··· 9F 9G 9H 9I 9J 18A ··· 18F 27A ··· 27F 27G ··· 27R 54A ··· 54R order 1 2 3 3 3 3 6 6 9 ··· 9 9 9 9 9 18 ··· 18 27 ··· 27 27 ··· 27 54 ··· 54 size 1 3 1 1 12 12 3 3 1 ··· 1 12 12 12 12 3 ··· 3 3 ··· 3 12 ··· 12 3 ··· 3

60 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 3 type + + image C1 C3 C3 C3 C9 C9 A4 C3×A4 C27⋊C3 C9×A4 C27⋊A4 kernel C27⋊A4 C9.A4 C2×C54 C9×A4 C3.A4 C3×A4 C27 C9 C22 C3 C1 # reps 1 4 2 2 12 6 1 2 6 6 18

Matrix representation of C27⋊A4 in GL3(𝔽109) generated by

 50 89 61 28 98 48 81 20 70
,
 0 0 1 108 108 108 1 0 0
,
 0 1 0 1 0 0 108 108 108
,
 1 0 0 0 0 1 108 108 108
G:=sub<GL(3,GF(109))| [50,28,81,89,98,20,61,48,70],[0,108,1,0,108,0,1,108,0],[0,1,108,1,0,108,0,0,108],[1,0,108,0,0,108,0,1,108] >;

C27⋊A4 in GAP, Magma, Sage, TeX

C_{27}\rtimes A_4
% in TeX

G:=Group("C27:A4");
// GroupNames label

G:=SmallGroup(324,43);
// by ID

G=gap.SmallGroup(324,43);
# by ID

G:=PCGroup([6,-3,-3,-3,-3,-2,2,361,43,68,4864,8753]);
// Polycyclic

G:=Group<a,b,c,d|a^27=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^19,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations

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