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G = Dic3xC33order 324 = 22·34

Direct product of C33 and Dic3

direct product, metabelian, supersoluble, monomial, A-group

Aliases: Dic3xC33, C34:5C4, C33:9C12, C3:(C32xC12), C6.(C32xC6), C2.(S3xC33), C32:4(C3xC12), (C33xC6).1C2, C6.14(S3xC32), (C32xC6).24S3, (C32xC6).19C6, (C3xC6).21(C3xC6), (C3xC6).45(C3xS3), SmallGroup(324,155)

Series: Derived Chief Lower central Upper central

C1C3 — Dic3xC33
C1C3C6C3xC6C32xC6C33xC6 — Dic3xC33
C3 — Dic3xC33
C1C32xC6

Generators and relations for Dic3xC33
 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, C3xC6, C3xC6, C33, C33, C33, C3xDic3, C3xC12, C32xC6, C32xC6, C32xC6, C34, C32xDic3, C32xC12, C33xC6, Dic3xC33
Quotients: C1, C2, C3, C4, S3, C6, C32, Dic3, C12, C3xS3, C3xC6, C33, C3xDic3, C3xC12, S3xC32, C32xC6, C32xDic3, C32xC12, S3xC33, Dic3xC33

Smallest permutation representation of Dic3xC33
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···3Z3AA···3BA4A4B6A···6Z6AA···6BA12A···12AZ
order123···33···3446···66···612···12
size111···12···2331···12···23···3

162 irreducible representations

dim1111112222
type+++-
imageC1C2C3C4C6C12S3Dic3C3xS3C3xDic3
kernelDic3xC33C33xC6C32xDic3C34C32xC6C33C32xC6C33C3xC6C32
# reps112622652112626

Matrix representation of Dic3xC33 in GL4(F13) generated by

9000
0100
0090
0009
,
9000
0300
0030
0003
,
1000
0100
0030
0003
,
1000
0100
0040
00310
,
1000
0100
0048
0069
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] >;

Dic3xC33 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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