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G = C18×S4  order 432 = 24·33

Direct product of C18 and S4

direct product, non-abelian, soluble, monomial

Aliases: C18×S4, (C2×A4)⋊C18, A4⋊(C2×C18), (C6×S4).C3, (C3×S4).C6, C23⋊(S3×C9), (C2×C18)⋊2D6, C3.4(C6×S4), C22⋊(S3×C18), (A4×C18)⋊1C2, C6.19(C3×S4), (C6×A4).6C6, (C9×A4)⋊2C22, (C22×C18)⋊1S3, (C2×C6).1(S3×C6), (C3×A4).2(C2×C6), (C22×C6).2(C3×S3), SmallGroup(432,532)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — A4 — C18×S4
Chief series
C1 — C22 — A4 — C3×A4 — C9×A4 — C9×S4 — C18×S4
Lower central
A4 — C18×S4
Upper central
C1 — C18

Generators and relations for C18×S4
 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, C2×C4, D4, C23, C23, C9, C9, C32, C12, A4, A4, D6, C2×C6, C2×C6, C2×D4, C18, C18, C3×S3, C3×C6, C2×C12, C3×D4, S4, C2×A4, C2×A4, C22×C6, C22×C6, C3×C9, C36, C3.A4, C2×C18, C2×C18, C3×A4, S3×C6, C6×D4, C2×S4, S3×C9, C3×C18, C2×C36, D4×C9, C2×C3.A4, C22×C18, C22×C18, C3×S4, C6×A4, C9×A4, S3×C18, D4×C18, C6×S4, C9×S4, A4×C18, C18×S4
Quotients: C1, C2, C3, C22, S3, C6, C9, D6, C2×C6, C18, C3×S3, S4, C2×C18, S3×C6, C2×S4, S3×C9, C3×S4, S3×C18, C6×S4, C9×S4, C18×S4

Smallest permutation representation of C18×S4
►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+++++++
imageC1C2C2C3C6C6C9C18C18S3D6C3×S3S3×C6S3×C9S3×C18S4C2×S4C3×S4C6×S4C9×S4C18×S4
kernelC18×S4C9×S4A4×C18C6×S4C3×S4C6×A4C2×S4S4C2×A4C22×C18C2×C18C22×C6C2×C6C23C22C18C9C6C3C2C1
# reps121242612611226622441212

Matrix representation of C18×S4 ►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] >;
 

C18×S4 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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