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## G = C9×C32⋊C4order 324 = 22·34

### Direct product of C9 and C32⋊C4

Aliases: C9×C32⋊C4, C322C36, C33.3C12, C3⋊S3.C18, (C32×C9)⋊1C4, (C3×C32⋊C4).C3, (C9×C3⋊S3).1C2, (C3×C3⋊S3).2C6, C3.1(C3×C32⋊C4), SmallGroup(324,109)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C9×C32⋊C4
 Chief series C1 — C32 — C33 — C3×C3⋊S3 — C9×C3⋊S3 — C9×C32⋊C4
 Lower central C32 — C9×C32⋊C4
 Upper central C1 — C9

Generators and relations for C9×C32⋊C4
G = < a,b,c,d | a9=b3=c3=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >

Smallest permutation representation of C9×C32⋊C4
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 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 31 34)(29 32 35)(30 33 36)
(10 13 16)(11 14 17)(12 15 18)(28 34 31)(29 35 32)(30 36 33)
(1 10 27 31)(2 11 19 32)(3 12 20 33)(4 13 21 34)(5 14 22 35)(6 15 23 36)(7 16 24 28)(8 17 25 29)(9 18 26 30)

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

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

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

54 conjugacy classes

 class 1 2 3A 3B 3C ··· 3H 4A 4B 6A 6B 9A ··· 9F 9G ··· 9R 12A 12B 12C 12D 18A ··· 18F 36A ··· 36L order 1 2 3 3 3 ··· 3 4 4 6 6 9 ··· 9 9 ··· 9 12 12 12 12 18 ··· 18 36 ··· 36 size 1 9 1 1 4 ··· 4 9 9 9 9 1 ··· 1 4 ··· 4 9 9 9 9 9 ··· 9 9 ··· 9

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 4 4 4 type + + + image C1 C2 C3 C4 C6 C9 C12 C18 C36 C32⋊C4 C3×C32⋊C4 C9×C32⋊C4 kernel C9×C32⋊C4 C9×C3⋊S3 C3×C32⋊C4 C32×C9 C3×C3⋊S3 C32⋊C4 C33 C3⋊S3 C32 C9 C3 C1 # reps 1 1 2 2 2 6 4 6 12 2 4 12

Matrix representation of C9×C32⋊C4 in GL4(𝔽37) generated by

 33 0 0 0 0 33 0 0 0 0 33 0 0 0 0 33
,
 10 0 0 0 0 1 0 0 0 0 1 0 0 0 0 26
,
 10 0 0 0 0 10 0 0 0 0 26 0 0 0 0 26
,
 0 36 0 0 0 0 0 36 36 0 0 0 0 0 36 0
G:=sub<GL(4,GF(37))| [33,0,0,0,0,33,0,0,0,0,33,0,0,0,0,33],[10,0,0,0,0,1,0,0,0,0,1,0,0,0,0,26],[10,0,0,0,0,10,0,0,0,0,26,0,0,0,0,26],[0,0,36,0,36,0,0,0,0,0,0,36,0,36,0,0] >;

C9×C32⋊C4 in GAP, Magma, Sage, TeX

C_9\times C_3^2\rtimes C_4
% in TeX

G:=Group("C9xC3^2:C4");
// GroupNames label

G:=SmallGroup(324,109);
// by ID

G=gap.SmallGroup(324,109);
# by ID

G:=PCGroup([6,-2,-3,-2,-3,-3,3,36,79,7564,376,10373,1313]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^3=c^3=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;
// generators/relations

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