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G = C9×A4order 108 = 22·33

Direct product of C9 and A4

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

Aliases: C9×A4, (C2×C18)⋊1C3, C9(C3.A4), C3.A43C3, C3.1(C3×A4), C221(C3×C9), (C3×A4).2C3, (C2×C6).1C32, SmallGroup(108,18)

Series: Derived Chief Lower central Upper central

C1C22 — C9×A4
C1C22C2×C6C3×A4 — C9×A4
C22 — C9×A4
C1C9

Generators and relations for C9×A4
 G = < a,b,c,d | a9=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

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

Smallest permutation representation of C9×A4
On 36 points
Generators in S36
(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)
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(19 35)(20 36)(21 28)(22 29)(23 30)(24 31)(25 32)(26 33)(27 34)
(1 25)(2 26)(3 27)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 32)(11 33)(12 34)(13 35)(14 36)(15 28)(16 29)(17 30)(18 31)
(1 7 4)(2 8 5)(3 9 6)(10 22 35)(11 23 36)(12 24 28)(13 25 29)(14 26 30)(15 27 31)(16 19 32)(17 20 33)(18 21 34)

G:=sub<Sym(36)| (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), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(19,35)(20,36)(21,28)(22,29)(23,30)(24,31)(25,32)(26,33)(27,34), (1,25)(2,26)(3,27)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,32)(11,33)(12,34)(13,35)(14,36)(15,28)(16,29)(17,30)(18,31), (1,7,4)(2,8,5)(3,9,6)(10,22,35)(11,23,36)(12,24,28)(13,25,29)(14,26,30)(15,27,31)(16,19,32)(17,20,33)(18,21,34)>;

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), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(19,35)(20,36)(21,28)(22,29)(23,30)(24,31)(25,32)(26,33)(27,34), (1,25)(2,26)(3,27)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,32)(11,33)(12,34)(13,35)(14,36)(15,28)(16,29)(17,30)(18,31), (1,7,4)(2,8,5)(3,9,6)(10,22,35)(11,23,36)(12,24,28)(13,25,29)(14,26,30)(15,27,31)(16,19,32)(17,20,33)(18,21,34) );

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)], [(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(19,35),(20,36),(21,28),(22,29),(23,30),(24,31),(25,32),(26,33),(27,34)], [(1,25),(2,26),(3,27),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(10,32),(11,33),(12,34),(13,35),(14,36),(15,28),(16,29),(17,30),(18,31)], [(1,7,4),(2,8,5),(3,9,6),(10,22,35),(11,23,36),(12,24,28),(13,25,29),(14,26,30),(15,27,31),(16,19,32),(17,20,33),(18,21,34)])

C9×A4 is a maximal subgroup of   C9⋊S4  C27⋊A4  C62.25C32  C62.9C32  C3.A42
C9×A4 is a maximal quotient of   C27⋊A4  C62.11C32  C62.16C32  C3.A42

36 conjugacy classes

class 1  2 3A3B3C···3H6A6B9A···9F9G···9R18A···18F
order12333···3669···99···918···18
size13114···4331···14···43···3

36 irreducible representations

dim11111333
type++
imageC1C3C3C3C9A4C3×A4C9×A4
kernelC9×A4C3.A4C2×C18C3×A4A4C9C3C1
# reps142218126

Matrix representation of C9×A4 in GL3(𝔽19) generated by

900
090
009
,
0181
0180
1180
,
1800
1801
1810
,
0110
0011
1100
G:=sub<GL(3,GF(19))| [9,0,0,0,9,0,0,0,9],[0,0,1,18,18,18,1,0,0],[18,18,18,0,0,1,0,1,0],[0,0,11,11,0,0,0,11,0] >;

C9×A4 in GAP, Magma, Sage, TeX

C_9\times A_4
% in TeX

G:=Group("C9xA4");
// GroupNames label

G:=SmallGroup(108,18);
// by ID

G=gap.SmallGroup(108,18);
# by ID

G:=PCGroup([5,-3,-3,-3,-2,2,36,1083,2029]);
// Polycyclic

G:=Group<a,b,c,d|a^9=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 C9×A4 in TeX

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