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G = C18xS4  order 432 = 24·33

Direct product of C18 and S4

direct product, non-abelian, soluble, monomial

Aliases: C18xS4, (C2xA4):C18, A4:(C2xC18), (C6xS4).C3, (C3xS4).C6, C23:(S3xC9), (C2xC18):2D6, C3.4(C6xS4), C22:(S3xC18), (A4xC18):1C2, C6.19(C3xS4), (C6xA4).6C6, (C9xA4):2C22, (C22xC18):1S3, (C2xC6).1(S3xC6), (C3xA4).2(C2xC6), (C22xC6).2(C3xS3), SmallGroup(432,532)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — A4 — C18xS4
Chief series
C1 — C22 — A4 — C3xA4 — C9xA4 — C9xS4 — C18xS4
Lower central
A4 — C18xS4
Upper central
C1 — C18

Generators and relations for C18xS4
 G = < a,b,c,d,e | a18=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >

Subgroups: 334 in 107 conjugacy classes, 27 normal (21 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, D4, C23, C23, C9, C9, C32, C12, A4, A4, D6, C2xC6, C2xC6, C2xD4, C18, C18, C3xS3, C3xC6, C2xC12, C3xD4, S4, C2xA4, C2xA4, C22xC6, C22xC6, C3xC9, C36, C3.A4, C2xC18, C2xC18, C3xA4, S3xC6, C6xD4, C2xS4, S3xC9, C3xC18, C2xC36, D4xC9, C2xC3.A4, C22xC18, C22xC18, C3xS4, C6xA4, C9xA4, S3xC18, D4xC18, C6xS4, C9xS4, A4xC18, C18xS4
Quotients: C1, C2, C3, C22, S3, C6, C9, D6, C2xC6, C18, C3xS3, S4, C2xC18, S3xC6, C2xS4, S3xC9, C3xS4, S3xC18, C6xS4, C9xS4, C18xS4

Smallest permutation representation of C18xS4
►On 54 points
Generators in S54
(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)

(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)

(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)

(1 50 28)(2 51 29)(3 52 30)(4 53 31)(5 54 32)(6 37 33)(7 38 34)(8 39 35)(9 40 36)(10 41 19)(11 42 20)(12 43 21)(13 44 22)(14 45 23)(15 46 24)(16 47 25)(17 48 26)(18 49 27)

(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(19 50)(20 51)(21 52)(22 53)(23 54)(24 37)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)
 
G:=sub<Sym(54)| (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), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54), (19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54), (1,50,28)(2,51,29)(3,52,30)(4,53,31)(5,54,32)(6,37,33)(7,38,34)(8,39,35)(9,40,36)(10,41,19)(11,42,20)(12,43,21)(13,44,22)(14,45,23)(15,46,24)(16,47,25)(17,48,26)(18,49,27), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(19,50)(20,51)(21,52)(22,53)(23,54)(24,37)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)>;
 
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), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54), (19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54), (1,50,28)(2,51,29)(3,52,30)(4,53,31)(5,54,32)(6,37,33)(7,38,34)(8,39,35)(9,40,36)(10,41,19)(11,42,20)(12,43,21)(13,44,22)(14,45,23)(15,46,24)(16,47,25)(17,48,26)(18,49,27), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(19,50)(20,51)(21,52)(22,53)(23,54)(24,37)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49) );
 
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)], [(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(37,46),(38,47),(39,48),(40,49),(41,50),(42,51),(43,52),(44,53),(45,54)], [(19,28),(20,29),(21,30),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36),(37,46),(38,47),(39,48),(40,49),(41,50),(42,51),(43,52),(44,53),(45,54)], [(1,50,28),(2,51,29),(3,52,30),(4,53,31),(5,54,32),(6,37,33),(7,38,34),(8,39,35),(9,40,36),(10,41,19),(11,42,20),(12,43,21),(13,44,22),(14,45,23),(15,46,24),(16,47,25),(17,48,26),(18,49,27)], [(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(19,50),(20,51),(21,52),(22,53),(23,54),(24,37),(25,38),(26,39),(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,49)]])
 

90 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D3E4A4B6A6B6C6D6E6F6G6H6I6J6K6L6M9A···9F9G···9L12A12B12C12D18A···18F18G···18R18S···18AD18AE···18AJ36A···36L
order122222333334466666666666669···99···91212121218···1818···1818···1818···1836···36
size113366118886611333366668881···18···866661···13···36···68···86···6

90 irreducible representations

dim111111111222222333333
type+++++++
imageC1C2C2C3C6C6C9C18C18S3D6C3xS3S3xC6S3xC9S3xC18S4C2xS4C3xS4C6xS4C9xS4C18xS4
kernelC18xS4C9xS4A4xC18C6xS4C3xS4C6xA4C2xS4S4C2xA4C22xC18C2xC18C22xC6C2xC6C23C22C18C9C6C3C2C1
# reps121242612611226622441212

Matrix representation of C18xS4 ►in GL3(F37) generated by

2800
0280
0028
,
3600
0360
001
,
100
0360
0036
,
001
100
010
,
3600
0036
0360
G:=sub<GL(3,GF(37))| [28,0,0,0,28,0,0,0,28],[36,0,0,0,36,0,0,0,1],[1,0,0,0,36,0,0,0,36],[0,1,0,0,0,1,1,0,0],[36,0,0,0,0,36,0,36,0] >;
 

C18xS4 in GAP, Magma, Sage, TeX 

C_{18}\times S_4
 
% in TeX
 
G:=Group("C18xS4");
 
// GroupNames label
 
G:=SmallGroup(432,532);
 
// by ID
 
G=gap.SmallGroup(432,532);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,79,2524,9077,285,5298,475]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^18=b^2=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

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