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G = C3×C3⋊Dic3order 108 = 22·33

Direct product of C3 and C3⋊Dic3

direct product, metabelian, supersoluble, monomial, A-group

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

C1C32 — C3×C3⋊Dic3
C1C3C32C3×C6C32×C6 — C3×C3⋊Dic3
C32 — C3×C3⋊Dic3
C1C6

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 [×4], C3 [×4], C4, C6, C6 [×4], C6 [×4], C32, C32 [×4], C32 [×4], Dic3 [×4], C12, C3×C6, C3×C6 [×4], C3×C6 [×4], C33, C3×Dic3 [×4], C3⋊Dic3, C32×C6, C3×C3⋊Dic3
Quotients: C1, C2, C3, C4, S3 [×4], C6, Dic3 [×4], C12, C3×S3 [×4], C3⋊S3, C3×Dic3 [×4], 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 17 31)(8 18 32)(9 13 33)(10 14 34)(11 15 35)(12 16 36)
(1 20 29)(2 21 30)(3 22 25)(4 23 26)(5 24 27)(6 19 28)(7 13 35)(8 14 36)(9 15 31)(10 16 32)(11 17 33)(12 18 34)
(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 36 4 33)(2 35 5 32)(3 34 6 31)(7 27 10 30)(8 26 11 29)(9 25 12 28)(13 24 16 21)(14 23 17 20)(15 22 18 19)

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

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

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

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 3A3B3C···3N4A4B6A6B6C···6N12A12B12C12D
order12333···344666···612121212
size11112···299112···29999

36 irreducible representations

dim1111112222
type+++-
imageC1C2C3C4C6C12S3Dic3C3×S3C3×Dic3
kernelC3×C3⋊Dic3C32×C6C3⋊Dic3C33C3×C6C32C3×C6C32C6C3
# reps1122244488

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

9000
0900
0030
0003
,
1000
0100
0090
0003
,
10000
2400
0010
0001
,
8200
0500
0001
0010
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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