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## G = C6×F9order 432 = 24·33

### Direct product of C6 and F9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C6×F9
 Chief series C1 — C32 — C3⋊S3 — C32⋊C4 — C3×C32⋊C4 — C3×F9 — C6×F9
 Lower central C32 — C6×F9
 Upper central C1 — C6

Generators and relations for C6×F9
G = < a,b,c,d | a6=b3=c3=d8=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Subgroups: 308 in 58 conjugacy classes, 26 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, C2×C4, C32, C32, C12, D6, C2×C6, C2×C8, C3×S3, C3⋊S3, C3×C6, C3×C6, C24, C2×C12, C33, C32⋊C4, S3×C6, C2×C3⋊S3, C2×C24, C3×C3⋊S3, C32×C6, F9, C2×C32⋊C4, C3×C32⋊C4, C6×C3⋊S3, C2×F9, C3×F9, C6×C32⋊C4, C6×F9
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C12, C2×C6, C2×C8, C24, C2×C12, C2×C24, F9, C2×F9, C3×F9, C6×F9

Smallest permutation representation of C6×F9
On 48 points
Generators in S48
(1 30 9 24 45 35)(2 31 10 17 46 36)(3 32 11 18 47 37)(4 25 12 19 48 38)(5 26 13 20 41 39)(6 27 14 21 42 40)(7 28 15 22 43 33)(8 29 16 23 44 34)
(2 10 46)(3 11 47)(4 48 12)(6 42 14)(7 43 15)(8 16 44)(17 36 31)(18 37 32)(19 25 38)(21 27 40)(22 28 33)(23 34 29)
(1 9 45)(3 11 47)(4 12 48)(5 41 13)(7 43 15)(8 44 16)(18 37 32)(19 38 25)(20 26 39)(22 28 33)(23 29 34)(24 35 30)
(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)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I 8A ··· 8H 12A ··· 12H 24A ··· 24P order 1 2 2 2 3 3 3 3 3 4 4 4 4 6 6 6 6 6 6 6 6 6 8 ··· 8 12 ··· 12 24 ··· 24 size 1 1 9 9 1 1 8 8 8 9 9 9 9 1 1 8 8 8 9 9 9 9 9 ··· 9 9 ··· 9 9 ··· 9

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8 8 8 8 type + + + + + image C1 C2 C2 C3 C4 C4 C6 C6 C8 C8 C12 C12 C24 C24 F9 C2×F9 C3×F9 C6×F9 kernel C6×F9 C3×F9 C6×C32⋊C4 C2×F9 C3×C32⋊C4 C6×C3⋊S3 F9 C2×C32⋊C4 C3×C3⋊S3 C32×C6 C32⋊C4 C2×C3⋊S3 C3⋊S3 C3×C6 C6 C3 C2 C1 # reps 1 2 1 2 2 2 4 2 4 4 4 4 8 8 1 1 2 2

Matrix representation of C6×F9 in GL9(𝔽73)

 9 0 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 0 64
,
 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 64 0 0 0 0 0 0 70 3 64 8 0 0 0 0 0 40 72 29 0 8 0 0 0 0 5 54 0 0 0 64 0 0 0 24 62 0 0 0 0 64 0 0 57 58 53 0 0 0 0 8
,
 1 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 49 64 8 0 0 0 0 0 28 0 0 0 64 0 0 0 0 0 25 29 0 0 8 0 0 0 46 11 14 0 0 0 1 0 0 16 62 14 0 0 0 0 1
,
 72 0 0 0 0 0 0 0 0 0 68 64 13 0 63 0 0 0 0 5 54 13 0 0 63 0 0 0 24 62 59 0 0 0 63 0 0 0 0 0 0 27 46 72 1 0 0 0 0 0 5 9 60 0 0 0 0 0 1 68 19 60 0 0 0 0 0 0 49 11 14 0 0 0 0 0 0 71 62 14 0

G:=sub<GL(9,GF(73))| [9,0,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,64],[1,0,0,0,0,0,0,0,0,0,1,0,0,70,40,5,24,57,0,0,1,0,3,72,54,62,58,0,0,0,64,64,29,0,0,53,0,0,0,0,8,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,0,0,0,8,0,0,0,28,0,46,16,0,0,64,0,49,0,25,11,62,0,0,0,64,64,0,29,14,14,0,0,0,0,8,0,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,0,0,0,68,5,24,0,0,0,0,0,0,64,54,62,0,0,0,0,0,0,13,13,59,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,63,0,0,27,5,68,49,71,0,0,63,0,46,9,19,11,62,0,0,0,63,72,60,60,14,14,0,0,0,0,1,0,0,0,0] >;

C6×F9 in GAP, Magma, Sage, TeX

C_6\times F_9
% in TeX

G:=Group("C6xF9");
// GroupNames label

G:=SmallGroup(432,751);
// by ID

G=gap.SmallGroup(432,751);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,84,80,6053,1202,201,16470,1595,622]);
// Polycyclic

G:=Group<a,b,c,d|a^6=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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