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## G = A4×C33order 324 = 22·34

### Direct product of C33 and A4

Aliases: A4×C33, C22⋊C34, C624C32, (C2×C6)⋊C33, (C3×C62)⋊5C3, SmallGroup(324,171)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C33
 Chief series C1 — C22 — A4 — C3×A4 — C32×A4 — A4×C33
 Lower central C22 — A4×C33
 Upper central C1 — C33

Generators and relations for A4×C33
G = < a,b,c,d,e,f | a3=b3=c3=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 1060 in 452 conjugacy classes, 240 normal (5 characteristic)
C1, C2, C3, C3, C22, C6, C32, C32, A4, C2×C6, C3×C6, C33, C33, C3×A4, C62, C32×C6, C34, C32×A4, C3×C62, A4×C33
Quotients: C1, C3, C32, A4, C33, C3×A4, C34, C32×A4, A4×C33

Smallest permutation representation of A4×C33
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)
(1 104 78)(2 105 76)(3 103 77)(4 14 27)(5 15 25)(6 13 26)(7 17 30)(8 18 28)(9 16 29)(10 89 36)(11 90 34)(12 88 35)(19 32 55)(20 33 56)(21 31 57)(22 83 96)(23 84 94)(24 82 95)(37 53 63)(38 54 61)(39 52 62)(40 102 66)(41 100 64)(42 101 65)(43 97 69)(44 98 67)(45 99 68)(46 50 72)(47 51 70)(48 49 71)(58 73 86)(59 74 87)(60 75 85)(79 92 108)(80 93 106)(81 91 107)
(1 67 56)(2 68 57)(3 69 55)(4 82 61)(5 83 62)(6 84 63)(7 85 64)(8 86 65)(9 87 66)(10 91 70)(11 92 71)(12 93 72)(13 94 37)(14 95 38)(15 96 39)(16 59 40)(17 60 41)(18 58 42)(19 103 43)(20 104 44)(21 105 45)(22 52 25)(23 53 26)(24 54 27)(28 73 101)(29 74 102)(30 75 100)(31 76 99)(32 77 97)(33 78 98)(34 79 49)(35 80 50)(36 81 51)(46 88 106)(47 89 107)(48 90 108)
(1 24)(2 22)(3 23)(4 20)(5 21)(6 19)(7 81)(8 79)(9 80)(10 41)(11 42)(12 40)(13 32)(14 33)(15 31)(16 93)(17 91)(18 92)(25 57)(26 55)(27 56)(28 108)(29 106)(30 107)(34 65)(35 66)(36 64)(37 97)(38 98)(39 99)(43 63)(44 61)(45 62)(46 74)(47 75)(48 73)(49 86)(50 87)(51 85)(52 68)(53 69)(54 67)(58 71)(59 72)(60 70)(76 96)(77 94)(78 95)(82 104)(83 105)(84 103)(88 102)(89 100)(90 101)
(1 107)(2 108)(3 106)(4 64)(5 65)(6 66)(7 82)(8 83)(9 84)(10 33)(11 31)(12 32)(13 40)(14 41)(15 42)(16 94)(17 95)(18 96)(19 35)(20 36)(21 34)(22 28)(23 29)(24 30)(25 101)(26 102)(27 100)(37 59)(38 60)(39 58)(43 50)(44 51)(45 49)(46 69)(47 67)(48 68)(52 73)(53 74)(54 75)(55 88)(56 89)(57 90)(61 85)(62 86)(63 87)(70 98)(71 99)(72 97)(76 92)(77 93)(78 91)(79 105)(80 103)(81 104)
(1 33 44)(2 31 45)(3 32 43)(4 47 17)(5 48 18)(6 46 16)(7 27 70)(8 25 71)(9 26 72)(10 85 24)(11 86 22)(12 87 23)(13 50 29)(14 51 30)(15 49 28)(19 69 77)(20 67 78)(21 68 76)(34 73 96)(35 74 94)(36 75 95)(37 80 102)(38 81 100)(39 79 101)(40 63 106)(41 61 107)(42 62 108)(52 92 65)(53 93 66)(54 91 64)(55 97 103)(56 98 104)(57 99 105)(58 83 90)(59 84 88)(60 82 89)

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

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

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

108 conjugacy classes

 class 1 2 3A ··· 3Z 3AA ··· 3CB 6A ··· 6Z order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 size 1 3 1 ··· 1 4 ··· 4 3 ··· 3

108 irreducible representations

 dim 1 1 1 3 3 type + + image C1 C3 C3 A4 C3×A4 kernel A4×C33 C32×A4 C3×C62 C33 C32 # reps 1 78 2 1 26

Matrix representation of A4×C33 in GL5(𝔽7)

 4 0 0 0 0 0 4 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 4 0 0 0 0 0 1 0 0 0 0 0 2 0 0 0 0 0 2 0 0 0 0 0 2
,
 1 0 0 0 0 0 4 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 6 1 0 0 0 6 0 0 0 1 6 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 6 0 0 0 0 6 0 1 0 0 6 1 0
,
 1 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 4 0 0 4 0 0

G:=sub<GL(5,GF(7))| [4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,2],[1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,6,6,6,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,6,6,6,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,4,0,0,0,0,0,4,0] >;

A4×C33 in GAP, Magma, Sage, TeX

A_4\times C_3^3
% in TeX

G:=Group("A4xC3^3");
// GroupNames label

G:=SmallGroup(324,171);
// by ID

G=gap.SmallGroup(324,171);
# by ID

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

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

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