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## G = C4×F9order 288 = 25·32

### Direct product of C4 and F9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C4×F9
 Chief series C1 — C32 — C3⋊S3 — C32⋊C4 — C2×C32⋊C4 — C2×F9 — C4×F9
 Lower central C32 — C4×F9
 Upper central C1 — C4

Generators and relations for C4×F9
G = < a,b,c,d | a4=b3=c3=d8=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Smallest permutation representation of C4×F9
On 36 points
Generators in S36
(1 3 2 4)(5 32 19 21)(6 33 20 22)(7 34 13 23)(8 35 14 24)(9 36 15 25)(10 29 16 26)(11 30 17 27)(12 31 18 28)
(1 6 10)(2 20 16)(3 33 29)(4 22 26)(5 12 7)(8 9 11)(13 19 18)(14 15 17)(21 28 23)(24 25 27)(30 35 36)(31 34 32)
(1 7 11)(2 13 17)(3 34 30)(4 23 27)(5 8 6)(9 10 12)(14 20 19)(15 16 18)(21 24 22)(25 26 28)(29 31 36)(32 35 33)
(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)

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C ··· 4L 6 8A ··· 8P 12A 12B order 1 2 2 2 3 4 4 4 ··· 4 6 8 ··· 8 12 12 size 1 1 9 9 8 1 1 9 ··· 9 8 9 ··· 9 8 8

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 8 8 8 type + + + + + image C1 C2 C2 C4 C4 C4 C8 C8 C8 F9 C2×F9 C4×F9 kernel C4×F9 C4×C32⋊C4 C2×F9 C4×C3⋊S3 F9 C2×C32⋊C4 C3⋊Dic3 C3×C12 C32⋊C4 C4 C2 C1 # reps 1 1 2 2 8 2 4 4 8 1 1 2

Matrix representation of C4×F9 in GL9(𝔽73)

 27 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 0 72 0 1 0 0 0 0 0 0 72 0 0 0 1 0 0 0 0 72 0 0 0 0 1 0 0 0 72 0 0 0 0 0 1 0 0 72 0 0 0 0 0 0 1 0 72 0 0 0 0 0 0 0 1 72 0 0
,
 1 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 72 0 0 1 0 0 0 0 0 72 1 0 0 0 0 0 0 0 72 0 1 0 0 0 0 0 0 72 0 0 0 0 0 1 0 0 72 0 0 0 1 0 0 0 0 72 0 0 0 0 1 0
,
 51 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

G:=sub<GL(9,GF(73))| [27,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,72,72,72,72,72,72,72,72,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,72,72,72,72,72,72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[51,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0] >;

C4×F9 in GAP, Magma, Sage, TeX

C_4\times F_9
% in TeX

G:=Group("C4xF9");
// GroupNames label

G:=SmallGroup(288,863);
// by ID

G=gap.SmallGroup(288,863);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,64,100,4037,2371,201,10982,3156,622]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^3=c^3=d^8=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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