p-group, metabelian, nilpotent (class 3), monomial
Aliases: C3≀C3, AΣL1(𝔽27), He3⋊1C3, C33⋊1C3, C3.2He3, C32.1C32, 3- 1+2⋊1C3, 3-Sylow(S9), SmallGroup(81,7)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
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 >
Character table of C3≀C3
| class | 1 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 3I | 3J | 3K | 3L | 9A | 9B | 9C | 9D | |
| size | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 9 | 9 | 9 | 9 | 9 | 9 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | 1 | linear of order 3 |
| ρ3 | 1 | 1 | 1 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | 1 | ζ32 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | 1 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ8 | 1 | 1 | 1 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | 1 | ζ3 | linear of order 3 |
| ρ9 | 1 | 1 | 1 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ10 | 3 | -3+3√-3/2 | -3-3√-3/2 | -3-√-3/2 | 0 | 3+√-3/2 | 3-√-3/2 | -3+√-3/2 | -√-3 | √-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ11 | 3 | -3-3√-3/2 | -3+3√-3/2 | -√-3 | 0 | -3-√-3/2 | -3+√-3/2 | √-3 | 3-√-3/2 | 3+√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ12 | 3 | 3 | 3 | 0 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3 |
| ρ13 | 3 | -3+3√-3/2 | -3-3√-3/2 | 3-√-3/2 | 0 | -√-3 | √-3 | 3+√-3/2 | -3+√-3/2 | -3-√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ14 | 3 | -3-3√-3/2 | -3+3√-3/2 | -3+√-3/2 | 0 | 3-√-3/2 | 3+√-3/2 | -3-√-3/2 | √-3 | -√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ15 | 3 | -3-3√-3/2 | -3+3√-3/2 | 3+√-3/2 | 0 | √-3 | -√-3 | 3-√-3/2 | -3-√-3/2 | -3+√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ16 | 3 | -3+3√-3/2 | -3-3√-3/2 | √-3 | 0 | -3+√-3/2 | -3-√-3/2 | -√-3 | 3+√-3/2 | 3-√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ17 | 3 | 3 | 3 | 0 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3 |
►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 He3⋊2A4 C62.6C32 C33⋊2A4 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 He3⋊2A4 C62.6C32 C33⋊2A4
| action | f(x) | Disc(f) |
|---|---|---|
| 9T17 | x9-4x8-2x7+22x6-14x5-22x4+20x3+2x2-5x+1 | 76·132·432 |
Matrix representation of C3≀C3 ►in GL3(𝔽7) generated by
| 2 | 0 | 5 |
| 5 | 0 | 5 |
| 1 | 1 | 5 |
| 4 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 1 | 0 | 0 |
| 4 | 4 | 0 |
| 6 | 0 | 2 |
| 0 | 3 | 6 |
| 1 | 5 | 1 |
| 0 | 0 | 1 |
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