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## G = A4×C18order 216 = 23·33

### Direct product of C18 and A4

Aliases: A4×C18, (C2×C18)⋊6C6, C3.A45C6, C6.3(C3×A4), C3.1(C6×A4), C231(C3×C9), (C6×A4).2C3, (C3×A4).3C6, (C22×C18)⋊1C3, C221(C3×C18), (C22×C6).1C32, (C2×C3.A4)⋊3C3, (C2×C6).1(C3×C6), SmallGroup(216,103)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C18
 Chief series C1 — C22 — C2×C6 — C2×C18 — C9×A4 — A4×C18
 Lower central C22 — A4×C18
 Upper central C1 — C18

Generators and relations for A4×C18
G = < a,b,c,d | a18=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Smallest permutation representation of A4×C18
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)
(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 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)
(1 37 27)(2 38 28)(3 39 29)(4 40 30)(5 41 31)(6 42 32)(7 43 33)(8 44 34)(9 45 35)(10 46 36)(11 47 19)(12 48 20)(13 49 21)(14 50 22)(15 51 23)(16 52 24)(17 53 25)(18 54 26)

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

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

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

A4×C18 is a maximal subgroup of   A4⋊Dic9

72 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 6A 6B 6C 6D 6E 6F 6G ··· 6L 9A ··· 9F 9G ··· 9R 18A ··· 18F 18G ··· 18R 18S ··· 18AD order 1 2 2 2 3 3 3 ··· 3 6 6 6 6 6 6 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 18 ··· 18 18 ··· 18 size 1 1 3 3 1 1 4 ··· 4 1 1 3 3 3 3 4 ··· 4 1 ··· 1 4 ··· 4 1 ··· 1 3 ··· 3 4 ··· 4

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 type + + + + image C1 C2 C3 C3 C3 C6 C6 C6 C9 C18 A4 C2×A4 C3×A4 C6×A4 C9×A4 A4×C18 kernel A4×C18 C9×A4 C2×C3.A4 C22×C18 C6×A4 C3.A4 C2×C18 C3×A4 C2×A4 A4 C18 C9 C6 C3 C2 C1 # reps 1 1 4 2 2 4 2 2 18 18 1 1 2 2 6 6

Matrix representation of A4×C18 in GL4(𝔽19) generated by

 3 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8
,
 1 0 0 0 0 1 0 0 0 0 18 0 0 14 0 18
,
 1 0 0 0 0 18 0 0 0 0 18 0 0 5 11 1
,
 7 0 0 0 0 0 1 0 0 14 8 17 0 8 1 11
G:=sub<GL(4,GF(19))| [3,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,1,0,14,0,0,18,0,0,0,0,18],[1,0,0,0,0,18,0,5,0,0,18,11,0,0,0,1],[7,0,0,0,0,0,14,8,0,1,8,1,0,0,17,11] >;

A4×C18 in GAP, Magma, Sage, TeX

A_4\times C_{18}
% in TeX

G:=Group("A4xC18");
// GroupNames label

G:=SmallGroup(216,103);
// by ID

G=gap.SmallGroup(216,103);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-2,2,68,1630,2927]);
// Polycyclic

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

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