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## G = C9×A4order 108 = 22·33

### Direct product of C9 and A4

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

 Derived series C1 — C22 — C9×A4
 Chief series C1 — C22 — C2×C6 — C3×A4 — C9×A4
 Lower central C22 — C9×A4
 Upper central C1 — C9

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 >

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

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

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

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

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 3A 3B 3C ··· 3H 6A 6B 9A ··· 9F 9G ··· 9R 18A ··· 18F order 1 2 3 3 3 ··· 3 6 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 3 1 1 4 ··· 4 3 3 1 ··· 1 4 ··· 4 3 ··· 3

36 irreducible representations

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

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

 9 0 0 0 9 0 0 0 9
,
 0 18 1 0 18 0 1 18 0
,
 18 0 0 18 0 1 18 1 0
,
 0 11 0 0 0 11 11 0 0
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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