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## G = C3×C9⋊A4order 324 = 22·34

### Direct product of C3 and C9⋊A4

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C9⋊A4
G = < a,b,c,d,e | a3=b9=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b7, ece-1=cd=dc, ede-1=c >

Subgroups: 250 in 86 conjugacy classes, 42 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, 3- 1+2, C33, C3.A4, C2×C18, C3×A4, C3×A4, C62, C3×C18, C3×3- 1+2, C9⋊A4, C3×C3.A4, C6×C18, C32×A4, C3×C9⋊A4
Quotients: C1, C3, C32, A4, 3- 1+2, C33, C3×A4, C3×3- 1+2, C9⋊A4, C32×A4, C3×C9⋊A4

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

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

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

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

60 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3N 6A ··· 6H 9A ··· 9F 9G ··· 9R 18A ··· 18R order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 3 1 ··· 1 12 ··· 12 3 ··· 3 3 ··· 3 12 ··· 12 3 ··· 3

60 irreducible representations

 dim 1 1 1 1 1 3 3 3 3 3 type + + image C1 C3 C3 C3 C3 A4 3- 1+2 C3×A4 C3×A4 C9⋊A4 kernel C3×C9⋊A4 C9⋊A4 C3×C3.A4 C6×C18 C32×A4 C3×C9 C2×C6 C9 C32 C3 # reps 1 18 4 2 2 1 6 6 2 18

Matrix representation of C3×C9⋊A4 in GL6(𝔽19)

 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 4 0 0 0 0 0 18 1 0 0 0 11 18 0
,
 0 0 1 0 0 0 18 18 18 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 18 18 18 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 11 0 0 0 0 0 0 0 11 0 0 0 8 8 8 0 0 0 0 0 0 7 0 0 0 0 0 8 1 0 0 0 0 1 0 11

G:=sub<GL(6,GF(19))| [11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,11,0,0,0,4,18,18,0,0,0,0,1,0],[0,18,1,0,0,0,0,18,0,0,0,0,1,18,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,18,0,0,0,1,0,18,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,0,8,0,0,0,0,0,8,0,0,0,0,11,8,0,0,0,0,0,0,7,8,1,0,0,0,0,1,0,0,0,0,0,0,11] >;

C3×C9⋊A4 in GAP, Magma, Sage, TeX

C_3\times C_9\rtimes A_4
% in TeX

G:=Group("C3xC9:A4");
// GroupNames label

G:=SmallGroup(324,127);
// by ID

G=gap.SmallGroup(324,127);
# by ID

G:=PCGroup([6,-3,-3,-3,-3,-2,2,650,68,4864,8753]);
// 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,e*b*e^-1=b^7,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations

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