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## G = C3×C3⋊Dic3order 108 = 22·33

### Direct product of C3 and C3⋊Dic3

Aliases: C3×C3⋊Dic3, C334C4, C325C12, C324Dic3, C3⋊(C3×Dic3), C6.7(C3×S3), (C3×C6).9S3, (C3×C6).8C6, C6.6(C3⋊S3), (C32×C6).2C2, C2.(C3×C3⋊S3), SmallGroup(108,33)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C3⋊Dic3
G = < a,b,c,d | a3=b3=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 100 in 52 conjugacy classes, 26 normal (10 characteristic)
C1, C2, C3, C3, C3, C4, C6, C6, C6, C32, C32, C32, Dic3, C12, C3×C6, C3×C6, C3×C6, C33, C3×Dic3, C3⋊Dic3, C32×C6, C3×C3⋊Dic3
Quotients: C1, C2, C3, C4, S3, C6, Dic3, C12, C3×S3, C3⋊S3, C3×Dic3, C3⋊Dic3, C3×C3⋊S3, C3×C3⋊Dic3

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

C3×C3⋊Dic3 is a maximal subgroup of
C334C8  C3×S3×Dic3  C338(C2×C4)  C337D4  C334Q8  C339(C2×C4)  C339D4  C335Q8  C12×C3⋊S3  C32⋊C36  C32⋊Dic9  C322Dic9  C33⋊Dic3  C334C12  C33.Dic3  He36Dic3  C3⋊Dic3.2A4
C3×C3⋊Dic3 is a maximal quotient of
C334C12  C33.Dic3  He3.4Dic3  He3.5C12

36 conjugacy classes

 class 1 2 3A 3B 3C ··· 3N 4A 4B 6A 6B 6C ··· 6N 12A 12B 12C 12D order 1 2 3 3 3 ··· 3 4 4 6 6 6 ··· 6 12 12 12 12 size 1 1 1 1 2 ··· 2 9 9 1 1 2 ··· 2 9 9 9 9

36 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 C3×C3⋊Dic3 C32×C6 C3⋊Dic3 C33 C3×C6 C32 C3×C6 C32 C6 C3 # reps 1 1 2 2 2 4 4 4 8 8

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

 9 0 0 0 0 9 0 0 0 0 3 0 0 0 0 3
,
 1 0 0 0 0 1 0 0 0 0 9 0 0 0 0 3
,
 10 0 0 0 2 4 0 0 0 0 1 0 0 0 0 1
,
 8 2 0 0 0 5 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,3],[10,2,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[8,0,0,0,2,5,0,0,0,0,0,1,0,0,1,0] >;

C3×C3⋊Dic3 in GAP, Magma, Sage, TeX

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

G:=Group("C3xC3:Dic3");
// GroupNames label

G:=SmallGroup(108,33);
// by ID

G=gap.SmallGroup(108,33);
# by ID

G:=PCGroup([5,-2,-3,-2,-3,-3,30,483,1804]);
// Polycyclic

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

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