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## G = C18×S4order 432 = 24·33

### Direct product of C18 and S4

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 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 9A ··· 9F 9G ··· 9L 12A 12B 12C 12D 18A ··· 18F 18G ··· 18R 18S ··· 18AD 18AE ··· 18AJ 36A ··· 36L order 1 2 2 2 2 2 3 3 3 3 3 4 4 6 6 6 6 6 6 6 6 6 6 6 6 6 9 ··· 9 9 ··· 9 12 12 12 12 18 ··· 18 18 ··· 18 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 3 3 6 6 1 1 8 8 8 6 6 1 1 3 3 3 3 6 6 6 6 8 8 8 1 ··· 1 8 ··· 8 6 6 6 6 1 ··· 1 3 ··· 3 6 ··· 6 8 ··· 8 6 ··· 6

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 type + + + + + + + image C1 C2 C2 C3 C6 C6 C9 C18 C18 S3 D6 C3×S3 S3×C6 S3×C9 S3×C18 S4 C2×S4 C3×S4 C6×S4 C9×S4 C18×S4 kernel C18×S4 C9×S4 A4×C18 C6×S4 C3×S4 C6×A4 C2×S4 S4 C2×A4 C22×C18 C2×C18 C22×C6 C2×C6 C23 C22 C18 C9 C6 C3 C2 C1 # reps 1 2 1 2 4 2 6 12 6 1 1 2 2 6 6 2 2 4 4 12 12

Matrix representation of C18×S4 in GL3(𝔽37) generated by

 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 0 1 1 0 0 0 1 0
,
 36 0 0 0 0 36 0 36 0
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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