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G = S3xC2xC18order 216 = 23·33

Direct product of C2xC18 and S3

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

Aliases: S3xC2xC18, C62.12C6, C6:(C2xC18), (C2xC6):5C18, (C6xC18):3C2, C3:(C22xC18), (C3xC9):2C23, (S3xC6).5C6, C6.34(S3xC6), (C3xC18):2C22, C32.2(C22xC6), C3.4(S3xC2xC6), (C3xS3).(C2xC6), (S3xC2xC6).2C3, (C3xC6).23(C2xC6), (C2xC6).21(C3xS3), SmallGroup(216,109)

Series: Derived Chief Lower central Upper central

C1C3 — S3xC2xC18
C1C3C32C3xC9S3xC9S3xC18 — S3xC2xC18
C3 — S3xC2xC18
C1C2xC18

Generators and relations for S3xC2xC18
 G = < a,b,c,d | a2=b18=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 182 in 106 conjugacy classes, 63 normal (15 characteristic)
C1, C2, C2, C3, C3, C22, C22, S3, C6, C6, C23, C9, C9, C32, D6, C2xC6, C2xC6, C18, C18, C3xS3, C3xC6, C22xS3, C22xC6, C3xC9, C2xC18, C2xC18, S3xC6, C62, S3xC9, C3xC18, C22xC18, S3xC2xC6, S3xC18, C6xC18, S3xC2xC18
Quotients: C1, C2, C3, C22, S3, C6, C23, C9, D6, C2xC6, C18, C3xS3, C22xS3, C22xC6, C2xC18, S3xC6, S3xC9, C22xC18, S3xC2xC6, S3xC18, S3xC2xC18

Smallest permutation representation of S3xC2xC18
On 72 points
Generators in S72
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)
(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)
(1 7 13)(2 8 14)(3 9 15)(4 10 16)(5 11 17)(6 12 18)(19 25 31)(20 26 32)(21 27 33)(22 28 34)(23 29 35)(24 30 36)(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)
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 49)(9 50)(10 51)(11 52)(12 53)(13 54)(14 37)(15 38)(16 39)(17 40)(18 41)(19 70)(20 71)(21 72)(22 55)(23 56)(24 57)(25 58)(26 59)(27 60)(28 61)(29 62)(30 63)(31 64)(32 65)(33 66)(34 67)(35 68)(36 69)

G:=sub<Sym(72)| (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72), (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), (1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36)(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), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,49)(9,50)(10,51)(11,52)(12,53)(13,54)(14,37)(15,38)(16,39)(17,40)(18,41)(19,70)(20,71)(21,72)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)>;

G:=Group( (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72), (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), (1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36)(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), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,49)(9,50)(10,51)(11,52)(12,53)(13,54)(14,37)(15,38)(16,39)(17,40)(18,41)(19,70)(20,71)(21,72)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69) );

G=PermutationGroup([[(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,33),(8,34),(9,35),(10,36),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24),(17,25),(18,26),(37,55),(38,56),(39,57),(40,58),(41,59),(42,60),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72)], [(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)], [(1,7,13),(2,8,14),(3,9,15),(4,10,16),(5,11,17),(6,12,18),(19,25,31),(20,26,32),(21,27,33),(22,28,34),(23,29,35),(24,30,36),(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)], [(1,42),(2,43),(3,44),(4,45),(5,46),(6,47),(7,48),(8,49),(9,50),(10,51),(11,52),(12,53),(13,54),(14,37),(15,38),(16,39),(17,40),(18,41),(19,70),(20,71),(21,72),(22,55),(23,56),(24,57),(25,58),(26,59),(27,60),(28,61),(29,62),(30,63),(31,64),(32,65),(33,66),(34,67),(35,68),(36,69)]])

S3xC2xC18 is a maximal subgroup of   D6:Dic9

108 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E6A···6F6G···6O6P···6W9A···9F9G···9L18A···18R18S···18AJ18AK···18BH
order12222222333336···66···66···69···99···918···1818···1818···18
size11113333112221···12···23···31···12···21···12···23···3

108 irreducible representations

dim111111111222222
type+++++
imageC1C2C2C3C6C6C9C18C18S3D6C3xS3S3xC6S3xC9S3xC18
kernelS3xC2xC18S3xC18C6xC18S3xC2xC6S3xC6C62C22xS3D6C2xC6C2xC18C18C2xC6C6C22C2
# reps161212263661326618

Matrix representation of S3xC2xC18 in GL3(F19) generated by

1800
0180
0018
,
300
0170
0017
,
100
077
0011
,
1800
010
0618
G:=sub<GL(3,GF(19))| [18,0,0,0,18,0,0,0,18],[3,0,0,0,17,0,0,0,17],[1,0,0,0,7,0,0,7,11],[18,0,0,0,1,6,0,0,18] >;

S3xC2xC18 in GAP, Magma, Sage, TeX

S_3\times C_2\times C_{18}
% in TeX

G:=Group("S3xC2xC18");
// GroupNames label

G:=SmallGroup(216,109);
// by ID

G=gap.SmallGroup(216,109);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,93,5189]);
// Polycyclic

G:=Group<a,b,c,d|a^2=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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