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G = A4×C18order 216 = 23·33

Direct product of C18 and A4

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

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

C1C22 — A4×C18
C1C22C2×C6C2×C18C9×A4 — A4×C18
C22 — A4×C18
C1C18

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 >

3C2
3C2
4C3
4C3
4C3
3C22
3C22
3C6
3C6
4C6
4C6
4C6
4C9
4C9
4C32
3C2×C6
3C2×C6
3C18
3C18
4C18
4C3×C6
4C18
4C3×C9
3C2×C18
3C2×C18
4C3×C18

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

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

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

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

A4×C18 is a maximal subgroup of   A4⋊Dic9

72 conjugacy classes

class 1 2A2B2C3A3B3C···3H6A6B6C6D6E6F6G···6L9A···9F9G···9R18A···18F18G···18R18S···18AD
order1222333···36666666···69···99···918···1818···1818···18
size1133114···41133334···41···14···41···13···34···4

72 irreducible representations

dim1111111111333333
type++++
imageC1C2C3C3C3C6C6C6C9C18A4C2×A4C3×A4C6×A4C9×A4A4×C18
kernelA4×C18C9×A4C2×C3.A4C22×C18C6×A4C3.A4C2×C18C3×A4C2×A4A4C18C9C6C3C2C1
# reps114224221818112266

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

3000
0800
0080
0008
,
1000
0100
00180
014018
,
1000
01800
00180
05111
,
7000
0010
014817
08111
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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Subgroup lattice of A4×C18 in TeX

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