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G = C3×C3.A4order 108 = 22·33

Direct product of C3 and C3.A4

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

Aliases: C3×C3.A4, C32.2A4, C62.1C3, (C2×C6)⋊C9, C3.2(C3×A4), C222(C3×C9), (C2×C6).3C32, SmallGroup(108,20)

Series: Derived Chief Lower central Upper central

C1C22 — C3×C3.A4
C1C22C2×C6C3.A4 — C3×C3.A4
C22 — C3×C3.A4
C1C32

Generators and relations for C3×C3.A4
 G = < a,b,c,d,e | a3=b3=c2=d2=1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

3C2
3C6
3C6
3C6
3C6
4C9
4C9
4C9
3C3×C6
4C3×C9

Smallest permutation representation of C3×C3.A4
On 54 points
Generators in S54
(1 28 38)(2 29 39)(3 30 40)(4 31 41)(5 32 42)(6 33 43)(7 34 44)(8 35 45)(9 36 37)(10 24 47)(11 25 48)(12 26 49)(13 27 50)(14 19 51)(15 20 52)(16 21 53)(17 22 54)(18 23 46)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)
(2 53)(3 54)(5 47)(6 48)(8 50)(9 51)(10 32)(11 33)(13 35)(14 36)(16 29)(17 30)(19 37)(21 39)(22 40)(24 42)(25 43)(27 45)
(1 52)(3 54)(4 46)(6 48)(7 49)(9 51)(11 33)(12 34)(14 36)(15 28)(17 30)(18 31)(19 37)(20 38)(22 40)(23 41)(25 43)(26 44)
(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 49 50 51 52 53 54)

G:=sub<Sym(54)| (1,28,38)(2,29,39)(3,30,40)(4,31,41)(5,32,42)(6,33,43)(7,34,44)(8,35,45)(9,36,37)(10,24,47)(11,25,48)(12,26,49)(13,27,50)(14,19,51)(15,20,52)(16,21,53)(17,22,54)(18,23,46), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (2,53)(3,54)(5,47)(6,48)(8,50)(9,51)(10,32)(11,33)(13,35)(14,36)(16,29)(17,30)(19,37)(21,39)(22,40)(24,42)(25,43)(27,45), (1,52)(3,54)(4,46)(6,48)(7,49)(9,51)(11,33)(12,34)(14,36)(15,28)(17,30)(18,31)(19,37)(20,38)(22,40)(23,41)(25,43)(26,44), (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)(46,47,48,49,50,51,52,53,54)>;

G:=Group( (1,28,38)(2,29,39)(3,30,40)(4,31,41)(5,32,42)(6,33,43)(7,34,44)(8,35,45)(9,36,37)(10,24,47)(11,25,48)(12,26,49)(13,27,50)(14,19,51)(15,20,52)(16,21,53)(17,22,54)(18,23,46), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (2,53)(3,54)(5,47)(6,48)(8,50)(9,51)(10,32)(11,33)(13,35)(14,36)(16,29)(17,30)(19,37)(21,39)(22,40)(24,42)(25,43)(27,45), (1,52)(3,54)(4,46)(6,48)(7,49)(9,51)(11,33)(12,34)(14,36)(15,28)(17,30)(18,31)(19,37)(20,38)(22,40)(23,41)(25,43)(26,44), (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)(46,47,48,49,50,51,52,53,54) );

G=PermutationGroup([(1,28,38),(2,29,39),(3,30,40),(4,31,41),(5,32,42),(6,33,43),(7,34,44),(8,35,45),(9,36,37),(10,24,47),(11,25,48),(12,26,49),(13,27,50),(14,19,51),(15,20,52),(16,21,53),(17,22,54),(18,23,46)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54)], [(2,53),(3,54),(5,47),(6,48),(8,50),(9,51),(10,32),(11,33),(13,35),(14,36),(16,29),(17,30),(19,37),(21,39),(22,40),(24,42),(25,43),(27,45)], [(1,52),(3,54),(4,46),(6,48),(7,49),(9,51),(11,33),(12,34),(14,36),(15,28),(17,30),(18,31),(19,37),(20,38),(22,40),(23,41),(25,43),(26,44)], [(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),(46,47,48,49,50,51,52,53,54)])

C3×C3.A4 is a maximal subgroup of
C32.3S4  C62.11C32  C62.12C32  C62.16C32  He3.A4  He3⋊A4  C62.C32  3- 1+2⋊A4  C62⋊C9  A4×C3×C9  He3.2A4  C62.9C32
C3×C3.A4 is a maximal quotient of
C62.C9  C62.12C32  C62⋊C9

36 conjugacy classes

class 1  2 3A···3H6A···6H9A···9R
order123···36···69···9
size131···13···34···4

36 irreducible representations

dim1111333
type++
imageC1C3C3C9A4C3.A4C3×A4
kernelC3×C3.A4C3.A4C62C2×C6C32C3C3
# reps16218162

Matrix representation of C3×C3.A4 in GL4(𝔽19) generated by

7000
0700
0070
0007
,
1000
0700
0070
0007
,
1000
0100
06180
017018
,
1000
01800
00180
0201
,
1000
06170
00131
0020
G:=sub<GL(4,GF(19))| [7,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[1,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[1,0,0,0,0,1,6,17,0,0,18,0,0,0,0,18],[1,0,0,0,0,18,0,2,0,0,18,0,0,0,0,1],[1,0,0,0,0,6,0,0,0,17,13,2,0,0,1,0] >;

C3×C3.A4 in GAP, Magma, Sage, TeX

C_3\times C_3.A_4
% in TeX

G:=Group("C3xC3.A4");
// GroupNames label

G:=SmallGroup(108,20);
// by ID

G=gap.SmallGroup(108,20);
# by ID

G:=PCGroup([5,-3,-3,-3,-2,2,45,1083,2029]);
// Polycyclic

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

Export

Subgroup lattice of C3×C3.A4 in TeX

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