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## G = C32×C3⋊Dic3order 324 = 22·34

### Direct product of C32 and C3⋊Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C32×C3⋊Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — C33×C6 — C32×C3⋊Dic3
 Lower central C32 — C32×C3⋊Dic3
 Upper central C1 — C3×C6

Generators and relations for C32×C3⋊Dic3
G = < a,b,c,d,e | a3=b3=c3=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 556 in 284 conjugacy classes, 78 normal (10 characteristic)
C1, C2, C3, C3, C4, C6, C6, C32, C32, C32, Dic3, C12, C3×C6, C3×C6, C3×C6, C33, C33, C3×Dic3, C3⋊Dic3, C3×C12, C32×C6, C32×C6, C34, C32×Dic3, C3×C3⋊Dic3, C33×C6, C32×C3⋊Dic3
Quotients: C1, C2, C3, C4, S3, C6, C32, Dic3, C12, C3×S3, C3⋊S3, C3×C6, C3×Dic3, C3⋊Dic3, C3×C12, S3×C32, C3×C3⋊S3, C32×Dic3, C3×C3⋊Dic3, C32×C3⋊S3, C32×C3⋊Dic3

Smallest permutation representation of C32×C3⋊Dic3
On 36 points
Generators in S36
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 33 35)(32 34 36)
(1 22 27)(2 23 28)(3 24 29)(4 19 30)(5 20 25)(6 21 26)(7 18 33)(8 13 34)(9 14 35)(10 15 36)(11 16 31)(12 17 32)
(1 20 29)(2 21 30)(3 22 25)(4 23 26)(5 24 27)(6 19 28)(7 31 14)(8 32 15)(9 33 16)(10 34 17)(11 35 18)(12 36 13)
(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 7 4 10)(2 12 5 9)(3 11 6 8)(13 24 16 21)(14 23 17 20)(15 22 18 19)(25 35 28 32)(26 34 29 31)(27 33 30 36)

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

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

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

108 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3AR 4A 4B 6A ··· 6H 6I ··· 6AR 12A ··· 12P order 1 2 3 ··· 3 3 ··· 3 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 ··· 1 2 ··· 2 9 9 1 ··· 1 2 ··· 2 9 ··· 9

108 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C3 C4 C6 C12 S3 Dic3 C3×S3 C3×Dic3 kernel C32×C3⋊Dic3 C33×C6 C3×C3⋊Dic3 C34 C32×C6 C33 C32×C6 C33 C3×C6 C32 # reps 1 1 8 2 8 16 4 4 32 32

Matrix representation of C32×C3⋊Dic3 in GL4(𝔽13) generated by

 9 0 0 0 0 9 0 0 0 0 9 0 0 0 0 9
,
 9 0 0 0 0 9 0 0 0 0 3 0 0 0 0 3
,
 3 0 0 0 0 9 0 0 0 0 1 0 0 0 0 1
,
 4 0 0 0 0 10 0 0 0 0 9 0 0 0 0 3
,
 0 1 0 0 12 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[9,0,0,0,0,9,0,0,0,0,3,0,0,0,0,3],[3,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,10,0,0,0,0,9,0,0,0,0,3],[0,12,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C32×C3⋊Dic3 in GAP, Magma, Sage, TeX

C_3^2\times C_3\rtimes {\rm Dic}_3
% in TeX

G:=Group("C3^2xC3:Dic3");
// GroupNames label

G:=SmallGroup(324,156);
// by ID

G=gap.SmallGroup(324,156);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,-3,-3,108,2164,7781]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^6=1,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations

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