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G = C3≀C3order 81 = 34

Wreath product of C3 by C3

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C3C3, AΣL1(𝔽27), He31C3, C331C3, C3.2He3, C32.1C32, 3- 1+21C3, 3-Sylow(S9), SmallGroup(81,7)

Series: Derived Chief Lower central Upper central Jennings

C1C32 — C3≀C3
C1C3C32C33 — C3≀C3
C1C3C32 — C3≀C3
C1C3C32 — C3≀C3
C1C3C32 — C3≀C3

Generators and relations for C3≀C3
 G = < a,b,c,d | a3=b3=c3=d3=1, ab=ba, cac-1=ab-1, ad=da, bc=cb, bd=db, dcd-1=ab-1c >

3C3
3C3
3C3
3C3
9C3
3C9
3C32
3C32
3C9
3C32
3C32
3C32

Character table of C3≀C3

 class 13A3B3C3D3E3F3G3H3I3J3K3L9A9B9C9D
 size 11133333333999999
ρ111111111111111111    trivial
ρ2111ζ31ζ32ζ3ζ32ζ32ζ31ζ3ζ32ζ321ζ31    linear of order 3
ρ3111ζ321ζ3ζ32ζ3ζ3ζ321ζ3ζ321ζ31ζ32    linear of order 3
ρ4111ζ321ζ3ζ32ζ3ζ3ζ321ζ32ζ3ζ31ζ321    linear of order 3
ρ511111111111ζ32ζ3ζ32ζ3ζ3ζ32    linear of order 3
ρ6111ζ31ζ32ζ3ζ32ζ32ζ3111ζ3ζ3ζ32ζ32    linear of order 3
ρ711111111111ζ3ζ32ζ3ζ32ζ32ζ3    linear of order 3
ρ8111ζ31ζ32ζ3ζ32ζ32ζ31ζ32ζ31ζ321ζ3    linear of order 3
ρ9111ζ321ζ3ζ32ζ3ζ3ζ32111ζ32ζ32ζ3ζ3    linear of order 3
ρ103-3+3-3/2-3-3-3/2-3--3/203+-3/23--3/2-3+-3/2--3-30000000    complex faithful
ρ113-3-3-3/2-3+3-3/2--30-3--3/2-3+-3/2-33--3/23+-3/20000000    complex faithful
ρ123330-3+3-3/200000-3-3-3/2000000    complex lifted from He3
ρ133-3+3-3/2-3-3-3/23--3/20--3-33+-3/2-3+-3/2-3--3/20000000    complex faithful
ρ143-3-3-3/2-3+3-3/2-3+-3/203--3/23+-3/2-3--3/2-3--30000000    complex faithful
ρ153-3-3-3/2-3+3-3/23+-3/20-3--33--3/2-3--3/2-3+-3/20000000    complex faithful
ρ163-3+3-3/2-3-3-3/2-30-3+-3/2-3--3/2--33+-3/23--3/20000000    complex faithful
ρ173330-3-3-3/200000-3+3-3/2000000    complex lifted from He3

Permutation representations of C3≀C3
On 9 points - transitive group 9T17
Generators in S9
(4 5 6)(7 8 9)
(1 3 2)(4 5 6)(7 9 8)
(1 7 4)(2 8 6)(3 9 5)
(1 3 2)(4 5 6)(7 8 9)

G:=sub<Sym(9)| (4,5,6)(7,8,9), (1,3,2)(4,5,6)(7,9,8), (1,7,4)(2,8,6)(3,9,5), (1,3,2)(4,5,6)(7,8,9)>;

G:=Group( (4,5,6)(7,8,9), (1,3,2)(4,5,6)(7,9,8), (1,7,4)(2,8,6)(3,9,5), (1,3,2)(4,5,6)(7,8,9) );

G=PermutationGroup([(4,5,6),(7,8,9)], [(1,3,2),(4,5,6),(7,9,8)], [(1,7,4),(2,8,6),(3,9,5)], [(1,3,2),(4,5,6),(7,8,9)])

G:=TransitiveGroup(9,17);

On 27 points - transitive group 27T19
Generators in S27
(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)
(1 3 2)(4 6 5)(7 9 8)(10 11 12)(13 14 15)(16 17 18)(19 21 20)(22 24 23)(25 27 26)
(1 19 10)(2 20 12)(3 21 11)(4 22 14)(5 23 13)(6 24 15)(7 25 17)(8 26 16)(9 27 18)
(1 7 4)(2 8 5)(3 9 6)(10 17 14)(11 18 15)(12 16 13)(19 27 23)(20 25 24)(21 26 22)

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

G:=Group( (10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27), (1,3,2)(4,6,5)(7,9,8)(10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,24,23)(25,27,26), (1,19,10)(2,20,12)(3,21,11)(4,22,14)(5,23,13)(6,24,15)(7,25,17)(8,26,16)(9,27,18), (1,7,4)(2,8,5)(3,9,6)(10,17,14)(11,18,15)(12,16,13)(19,27,23)(20,25,24)(21,26,22) );

G=PermutationGroup([(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24),(25,26,27)], [(1,3,2),(4,6,5),(7,9,8),(10,11,12),(13,14,15),(16,17,18),(19,21,20),(22,24,23),(25,27,26)], [(1,19,10),(2,20,12),(3,21,11),(4,22,14),(5,23,13),(6,24,15),(7,25,17),(8,26,16),(9,27,18)], [(1,7,4),(2,8,5),(3,9,6),(10,17,14),(11,18,15),(12,16,13),(19,27,23),(20,25,24),(21,26,22)])

G:=TransitiveGroup(27,19);

On 27 points - transitive group 27T21
Generators in S27
(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)
(1 14 10)(2 15 11)(3 13 12)(4 27 8)(5 25 9)(6 26 7)(16 23 19)(17 24 20)(18 22 21)
(2 15 11)(3 12 13)(4 9 7)(5 6 27)(8 25 26)(16 18 20)(17 23 22)(19 21 24)
(1 25 16)(2 26 17)(3 27 18)(4 21 12)(5 19 10)(6 20 11)(7 24 15)(8 22 13)(9 23 14)

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

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), (1,14,10)(2,15,11)(3,13,12)(4,27,8)(5,25,9)(6,26,7)(16,23,19)(17,24,20)(18,22,21), (2,15,11)(3,12,13)(4,9,7)(5,6,27)(8,25,26)(16,18,20)(17,23,22)(19,21,24), (1,25,16)(2,26,17)(3,27,18)(4,21,12)(5,19,10)(6,20,11)(7,24,15)(8,22,13)(9,23,14) );

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)], [(1,14,10),(2,15,11),(3,13,12),(4,27,8),(5,25,9),(6,26,7),(16,23,19),(17,24,20),(18,22,21)], [(2,15,11),(3,12,13),(4,9,7),(5,6,27),(8,25,26),(16,18,20),(17,23,22),(19,21,24)], [(1,25,16),(2,26,17),(3,27,18),(4,21,12),(5,19,10),(6,20,11),(7,24,15),(8,22,13),(9,23,14)])

G:=TransitiveGroup(27,21);

On 27 points - transitive group 27T27
Generators in S27
(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)
(1 13 11)(2 14 12)(3 15 10)(4 26 8)(5 27 9)(6 25 7)(16 19 22)(17 20 23)(18 21 24)
(1 5 24)(2 25 19)(3 8 17)(4 20 15)(6 16 12)(7 22 14)(9 21 11)(10 26 23)(13 27 18)
(1 13 11)(2 14 12)(3 15 10)(4 25 9)(5 26 7)(6 27 8)(16 17 18)(19 20 21)(22 23 24)

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

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), (1,13,11)(2,14,12)(3,15,10)(4,26,8)(5,27,9)(6,25,7)(16,19,22)(17,20,23)(18,21,24), (1,5,24)(2,25,19)(3,8,17)(4,20,15)(6,16,12)(7,22,14)(9,21,11)(10,26,23)(13,27,18), (1,13,11)(2,14,12)(3,15,10)(4,25,9)(5,26,7)(6,27,8)(16,17,18)(19,20,21)(22,23,24) );

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)], [(1,13,11),(2,14,12),(3,15,10),(4,26,8),(5,27,9),(6,25,7),(16,19,22),(17,20,23),(18,21,24)], [(1,5,24),(2,25,19),(3,8,17),(4,20,15),(6,16,12),(7,22,14),(9,21,11),(10,26,23),(13,27,18)], [(1,13,11),(2,14,12),(3,15,10),(4,25,9),(5,26,7),(6,27,8),(16,17,18),(19,20,21),(22,23,24)])

G:=TransitiveGroup(27,27);

C3≀C3 is a maximal subgroup of
C3≀S3  C33⋊C6  C33⋊S3  C9.He3  C33⋊C32  C9.2He3  He32A4  C62.6C32  C332A4  C33⋊A4
C3≀C3 is a maximal quotient of
C32.24He3  C33.C32  C33.3C32  C32.27He3  C32.28He3  C33⋊C9  He3⋊C9  3- 1+2⋊C9  He32A4  C62.6C32  C332A4

Polynomial with Galois group C3≀C3 over ℚ
actionf(x)Disc(f)
9T17x9-4x8-2x7+22x6-14x5-22x4+20x3+2x2-5x+176·132·432

Matrix representation of C3≀C3 in GL3(𝔽7) generated by

205
505
115
,
400
040
004
,
100
440
602
,
036
151
001
G:=sub<GL(3,GF(7))| [2,5,1,0,0,1,5,5,5],[4,0,0,0,4,0,0,0,4],[1,4,6,0,4,0,0,0,2],[0,1,0,3,5,0,6,1,1] >;

C3≀C3 in GAP, Magma, Sage, TeX

C_3\wr C_3
% in TeX

G:=Group("C3wrC3");
// GroupNames label

G:=SmallGroup(81,7);
// by ID

G=gap.SmallGroup(81,7);
# by ID

G:=PCGroup([4,-3,3,-3,-3,97,434]);
// Polycyclic

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

Export

Subgroup lattice of C3≀C3 in TeX
Character table of C3≀C3 in TeX

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