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## G = A4×C3×C9order 324 = 22·34

### Direct product of C3×C9 and A4

Aliases: A4×C3×C9, C62.23C32, (C6×C18)⋊7C3, (C2×C18)⋊2C32, C3.A44C32, (C2×C6).1C33, C3.1(C32×A4), C221(C32×C9), (C32×A4).4C3, (C3×A4).5C32, C32.20(C3×A4), (C2×C6)⋊1(C3×C9), (C3×C3.A4)⋊10C3, SmallGroup(324,126)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C3×C9
 Chief series C1 — C22 — C2×C6 — C3×A4 — C32×A4 — A4×C3×C9
 Lower central C22 — A4×C3×C9
 Upper central C1 — C3×C9

Generators and relations for A4×C3×C9
G = < a,b,c,d,e | a3=b9=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Subgroups: 250 in 110 conjugacy classes, 60 normal (10 characteristic)
C1, C2, C3, C3, C3, C22, C6, C9, C9, C32, C32, A4, C2×C6, C2×C6, C18, C3×C6, C3×C9, C3×C9, C33, C3.A4, C2×C18, C3×A4, C62, C3×C18, C32×C9, C9×A4, C3×C3.A4, C6×C18, C32×A4, A4×C3×C9
Quotients: C1, C3, C9, C32, A4, C3×C9, C33, C3×A4, C32×C9, C9×A4, C32×A4, A4×C3×C9

Smallest permutation representation of A4×C3×C9
On 108 points
Generators in S108
(1 105 69)(2 106 70)(3 107 71)(4 108 72)(5 100 64)(6 101 65)(7 102 66)(8 103 67)(9 104 68)(10 73 37)(11 74 38)(12 75 39)(13 76 40)(14 77 41)(15 78 42)(16 79 43)(17 80 44)(18 81 45)(19 82 46)(20 83 47)(21 84 48)(22 85 49)(23 86 50)(24 87 51)(25 88 52)(26 89 53)(27 90 54)(28 91 55)(29 92 56)(30 93 57)(31 94 58)(32 95 59)(33 96 60)(34 97 61)(35 98 62)(36 99 63)
(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 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 105)(11 106)(12 107)(13 108)(14 100)(15 101)(16 102)(17 103)(18 104)(19 35)(20 36)(21 28)(22 29)(23 30)(24 31)(25 32)(26 33)(27 34)(46 62)(47 63)(48 55)(49 56)(50 57)(51 58)(52 59)(53 60)(54 61)(64 77)(65 78)(66 79)(67 80)(68 81)(69 73)(70 74)(71 75)(72 76)(82 98)(83 99)(84 91)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)
(1 52)(2 53)(3 54)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 32)(11 33)(12 34)(13 35)(14 36)(15 28)(16 29)(17 30)(18 31)(19 108)(20 100)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(37 59)(38 60)(39 61)(40 62)(41 63)(42 55)(43 56)(44 57)(45 58)(64 83)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 82)(73 95)(74 96)(75 97)(76 98)(77 99)(78 91)(79 92)(80 93)(81 94)
(10 25 32)(11 26 33)(12 27 34)(13 19 35)(14 20 36)(15 21 28)(16 22 29)(17 23 30)(18 24 31)(37 52 59)(38 53 60)(39 54 61)(40 46 62)(41 47 63)(42 48 55)(43 49 56)(44 50 57)(45 51 58)(73 88 95)(74 89 96)(75 90 97)(76 82 98)(77 83 99)(78 84 91)(79 85 92)(80 86 93)(81 87 94)

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

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

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

108 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Z 6A ··· 6H 9A ··· 9R 9S ··· 9BB 18A ··· 18R order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 3 1 ··· 1 4 ··· 4 3 ··· 3 1 ··· 1 4 ··· 4 3 ··· 3

108 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 type + + image C1 C3 C3 C3 C3 C9 A4 C3×A4 C3×A4 C9×A4 kernel A4×C3×C9 C9×A4 C3×C3.A4 C6×C18 C32×A4 C3×A4 C3×C9 C9 C32 C3 # reps 1 18 4 2 2 54 1 6 2 18

Matrix representation of A4×C3×C9 in GL4(𝔽19) generated by

 7 0 0 0 0 7 0 0 0 0 7 0 0 0 0 7
,
 7 0 0 0 0 6 0 0 0 0 6 0 0 0 0 6
,
 1 0 0 0 0 0 18 1 0 0 18 0 0 1 18 0
,
 1 0 0 0 0 18 0 0 0 18 0 1 0 18 1 0
,
 11 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0
G:=sub<GL(4,GF(19))| [7,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[7,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[1,0,0,0,0,0,0,1,0,18,18,18,0,1,0,0],[1,0,0,0,0,18,18,18,0,0,0,1,0,0,1,0],[11,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;

A4×C3×C9 in GAP, Magma, Sage, TeX

A_4\times C_3\times C_9
% in TeX

G:=Group("A4xC3xC9");
// GroupNames label

G:=SmallGroup(324,126);
// by ID

G=gap.SmallGroup(324,126);
# by ID

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

G:=Group<a,b,c,d,e|a^3=b^9=c^2=d^2=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations

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