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G = S3xC3xC18order 324 = 22·34

Direct product of C3xC18 and S3

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

Aliases: S3xC3xC18, C32.7C62, C6:(C3xC18), C3:(C6xC18), (C3xC6):3C18, (C3xC18):13C6, (C32xC18):1C2, C32:4(C2xC18), C33.6(C2xC6), (C32xC9):6C22, (S3xC32).3C6, (S3xC6).1C32, C32.22(S3xC6), (C32xC6).17C6, C6.11(S3xC32), C3.4(S3xC3xC6), (S3xC3xC6).2C3, (C3xC9):15(C2xC6), (C3xS3).1(C3xC6), (C3xC6).44(C3xS3), (C3xC6).18(C3xC6), SmallGroup(324,137)

Series: Derived Chief Lower central Upper central

C1C3 — S3xC3xC18
C1C3C32C3xC9C32xC9S3xC3xC9 — S3xC3xC18
C3 — S3xC3xC18
C1C3xC18

Generators and relations for S3xC3xC18
 G = < a,b,c,d | a3=b18=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 220 in 130 conjugacy classes, 70 normal (20 characteristic)
C1, C2, C2, C3, C3, C3, C22, S3, C6, C6, C6, C9, C9, C32, C32, C32, D6, C2xC6, C18, C18, C3xS3, C3xC6, C3xC6, C3xC6, C3xC9, C3xC9, C3xC9, C33, C2xC18, S3xC6, S3xC6, C62, S3xC9, C3xC18, C3xC18, C3xC18, S3xC32, C32xC6, C32xC9, S3xC18, C6xC18, S3xC3xC6, S3xC3xC9, C32xC18, S3xC3xC18
Quotients: C1, C2, C3, C22, S3, C6, C9, C32, D6, C2xC6, C18, C3xS3, C3xC6, C3xC9, C2xC18, S3xC6, C62, S3xC9, C3xC18, S3xC32, S3xC18, C6xC18, S3xC3xC6, S3xC3xC9, S3xC3xC18

Smallest permutation representation of S3xC3xC18
On 108 points
Generators in S108
(1 55 36)(2 56 19)(3 57 20)(4 58 21)(5 59 22)(6 60 23)(7 61 24)(8 62 25)(9 63 26)(10 64 27)(11 65 28)(12 66 29)(13 67 30)(14 68 31)(15 69 32)(16 70 33)(17 71 34)(18 72 35)(37 79 98)(38 80 99)(39 81 100)(40 82 101)(41 83 102)(42 84 103)(43 85 104)(44 86 105)(45 87 106)(46 88 107)(47 89 108)(48 90 91)(49 73 92)(50 74 93)(51 75 94)(52 76 95)(53 77 96)(54 78 97)
(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 67 24)(2 68 25)(3 69 26)(4 70 27)(5 71 28)(6 72 29)(7 55 30)(8 56 31)(9 57 32)(10 58 33)(11 59 34)(12 60 35)(13 61 36)(14 62 19)(15 63 20)(16 64 21)(17 65 22)(18 66 23)(37 104 73)(38 105 74)(39 106 75)(40 107 76)(41 108 77)(42 91 78)(43 92 79)(44 93 80)(45 94 81)(46 95 82)(47 96 83)(48 97 84)(49 98 85)(50 99 86)(51 100 87)(52 101 88)(53 102 89)(54 103 90)
(1 51)(2 52)(3 53)(4 54)(5 37)(6 38)(7 39)(8 40)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 95)(20 96)(21 97)(22 98)(23 99)(24 100)(25 101)(26 102)(27 103)(28 104)(29 105)(30 106)(31 107)(32 108)(33 91)(34 92)(35 93)(36 94)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(61 81)(62 82)(63 83)(64 84)(65 85)(66 86)(67 87)(68 88)(69 89)(70 90)(71 73)(72 74)

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

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

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

162 conjugacy classes

class 1 2A2B2C3A···3H3I···3Q6A···6H6I···6Q6R···6AG9A···9R9S···9AJ18A···18R18S···18AJ18AK···18BT
order12223···33···36···66···66···69···99···918···1818···1818···18
size11331···12···21···12···23···31···12···21···12···23···3

162 irreducible representations

dim11111111111122222222
type+++++
imageC1C2C2C3C3C6C6C6C6C9C18C18S3D6C3xS3C3xS3S3xC6S3xC6S3xC9S3xC18
kernelS3xC3xC18S3xC3xC9C32xC18S3xC18S3xC3xC6S3xC9C3xC18S3xC32C32xC6S3xC6C3xS3C3xC6C3xC18C3xC9C18C3xC6C9C32C6C3
# reps12162126421836181162621818

Matrix representation of S3xC3xC18 in GL3(F19) generated by

700
010
001
,
800
0140
0014
,
100
0110
0187
,
1800
0116
0188
G:=sub<GL(3,GF(19))| [7,0,0,0,1,0,0,0,1],[8,0,0,0,14,0,0,0,14],[1,0,0,0,11,18,0,0,7],[18,0,0,0,11,18,0,6,8] >;

S3xC3xC18 in GAP, Magma, Sage, TeX

S_3\times C_3\times C_{18}
% in TeX

G:=Group("S3xC3xC18");
// GroupNames label

G:=SmallGroup(324,137);
// by ID

G=gap.SmallGroup(324,137);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,93,7781]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^18=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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