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## G = Dic3×C33order 324 = 22·34

### Direct product of C33 and Dic3

Aliases: Dic3×C33, C345C4, C339C12, C3⋊(C32×C12), C6.(C32×C6), C2.(S3×C33), C324(C3×C12), (C33×C6).1C2, C6.14(S3×C32), (C32×C6).24S3, (C32×C6).19C6, (C3×C6).21(C3×C6), (C3×C6).45(C3×S3), SmallGroup(324,155)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — Dic3×C33
 Chief series C1 — C3 — C6 — C3×C6 — C32×C6 — C33×C6 — Dic3×C33
 Lower central C3 — Dic3×C33
 Upper central C1 — C32×C6

Generators and relations for Dic3×C33
G = < a,b,c,d,e | a3=b3=c3=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 536 in 324 conjugacy classes, 140 normal (10 characteristic)
C1, C2, C3, C3, C3, C4, C6, C6, C6, C32, C32, Dic3, C12, C3×C6, C3×C6, C33, C33, C33, C3×Dic3, C3×C12, C32×C6, C32×C6, C32×C6, C34, C32×Dic3, C32×C12, C33×C6, Dic3×C33
Quotients: C1, C2, C3, C4, S3, C6, C32, Dic3, C12, C3×S3, C3×C6, C33, C3×Dic3, C3×C12, S3×C32, C32×C6, C32×Dic3, C32×C12, S3×C33, Dic3×C33

Smallest permutation representation of Dic3×C33
On 108 points
Generators in S108
(1 47 35)(2 48 36)(3 43 31)(4 44 32)(5 45 33)(6 46 34)(7 49 19)(8 50 20)(9 51 21)(10 52 22)(11 53 23)(12 54 24)(13 37 25)(14 38 26)(15 39 27)(16 40 28)(17 41 29)(18 42 30)(55 97 85)(56 98 86)(57 99 87)(58 100 88)(59 101 89)(60 102 90)(61 103 73)(62 104 74)(63 105 75)(64 106 76)(65 107 77)(66 108 78)(67 91 79)(68 92 80)(69 93 81)(70 94 82)(71 95 83)(72 96 84)
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)(49 51 53)(50 52 54)(55 59 57)(56 60 58)(61 65 63)(62 66 64)(67 71 69)(68 72 70)(73 77 75)(74 78 76)(79 83 81)(80 84 82)(85 89 87)(86 90 88)(91 95 93)(92 96 94)(97 101 99)(98 102 100)(103 107 105)(104 108 106)
(1 17 11)(2 18 12)(3 13 7)(4 14 8)(5 15 9)(6 16 10)(19 31 25)(20 32 26)(21 33 27)(22 34 28)(23 35 29)(24 36 30)(37 49 43)(38 50 44)(39 51 45)(40 52 46)(41 53 47)(42 54 48)(55 67 61)(56 68 62)(57 69 63)(58 70 64)(59 71 65)(60 72 66)(73 85 79)(74 86 80)(75 87 81)(76 88 82)(77 89 83)(78 90 84)(91 103 97)(92 104 98)(93 105 99)(94 106 100)(95 107 101)(96 108 102)
(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 58 4 55)(2 57 5 60)(3 56 6 59)(7 62 10 65)(8 61 11 64)(9 66 12 63)(13 68 16 71)(14 67 17 70)(15 72 18 69)(19 74 22 77)(20 73 23 76)(21 78 24 75)(25 80 28 83)(26 79 29 82)(27 84 30 81)(31 86 34 89)(32 85 35 88)(33 90 36 87)(37 92 40 95)(38 91 41 94)(39 96 42 93)(43 98 46 101)(44 97 47 100)(45 102 48 99)(49 104 52 107)(50 103 53 106)(51 108 54 105)

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

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

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

162 conjugacy classes

 class 1 2 3A ··· 3Z 3AA ··· 3BA 4A 4B 6A ··· 6Z 6AA ··· 6BA 12A ··· 12AZ order 1 2 3 ··· 3 3 ··· 3 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 ··· 1 2 ··· 2 3 3 1 ··· 1 2 ··· 2 3 ··· 3

162 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C3 C4 C6 C12 S3 Dic3 C3×S3 C3×Dic3 kernel Dic3×C33 C33×C6 C32×Dic3 C34 C32×C6 C33 C32×C6 C33 C3×C6 C32 # reps 1 1 26 2 26 52 1 1 26 26

Matrix representation of Dic3×C33 in GL4(𝔽13) generated by

 9 0 0 0 0 1 0 0 0 0 9 0 0 0 0 9
,
 9 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3
,
 1 0 0 0 0 1 0 0 0 0 3 0 0 0 0 3
,
 1 0 0 0 0 1 0 0 0 0 4 0 0 0 3 10
,
 1 0 0 0 0 1 0 0 0 0 4 8 0 0 6 9
G:=sub<GL(4,GF(13))| [9,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[9,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,4,3,0,0,0,10],[1,0,0,0,0,1,0,0,0,0,4,6,0,0,8,9] >;

Dic3×C33 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_3^3
% in TeX

G:=Group("Dic3xC3^3");
// GroupNames label

G:=SmallGroup(324,155);
// by ID

G=gap.SmallGroup(324,155);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-2,-3,324,7781]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^6=1,e^2=d^3,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,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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