metabelian, soluble, monomial, A-group
Aliases: C32⋊C4, C3⋊S3.C2, SmallGroup(36,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C32⋊C4 |
Generators and relations for C32⋊C4
G = < a,b,c | a3=b3=c4=1, cbc-1=ab=ba, cac-1=a-1b >
Character table of C32⋊C4
| class | 1 | 2 | 3A | 3B | 4A | 4B | |
| size | 1 | 9 | 4 | 4 | 9 | 9 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | 1 | i | -i | linear of order 4 |
| ρ4 | 1 | -1 | 1 | 1 | -i | i | linear of order 4 |
| ρ5 | 4 | 0 | 1 | -2 | 0 | 0 | orthogonal faithful |
| ρ6 | 4 | 0 | -2 | 1 | 0 | 0 | orthogonal faithful |
►On 6 points - transitive group 6T10
Generators in S6
(1 6 4)(2 3 5)
(1 4 6)
(1 2)(3 4 5 6)
G:=sub<Sym(6)| (1,6,4)(2,3,5), (1,4,6), (1,2)(3,4,5,6)>;
G:=Group( (1,6,4)(2,3,5), (1,4,6), (1,2)(3,4,5,6) );
G=PermutationGroup([[(1,6,4),(2,3,5)], [(1,4,6)], [(1,2),(3,4,5,6)]])
G:=TransitiveGroup(6,10);
►On 9 points: primitive - transitive group 9T9
Generators in S9
(1 8 6)(2 9 5)(3 7 4)
(1 4 2)(3 9 8)(5 6 7)
(2 3 4 5)(6 7 8 9)
G:=sub<Sym(9)| (1,8,6)(2,9,5)(3,7,4), (1,4,2)(3,9,8)(5,6,7), (2,3,4,5)(6,7,8,9)>;
G:=Group( (1,8,6)(2,9,5)(3,7,4), (1,4,2)(3,9,8)(5,6,7), (2,3,4,5)(6,7,8,9) );
G=PermutationGroup([[(1,8,6),(2,9,5),(3,7,4)], [(1,4,2),(3,9,8),(5,6,7)], [(2,3,4,5),(6,7,8,9)]])
G:=TransitiveGroup(9,9);
►On 12 points - transitive group 12T17
Generators in S12
(1 9 8)(2 5 10)(3 6 11)(4 12 7)
(2 10 5)(4 7 12)
(1 2 3 4)(5 6 7 8)(9 10 11 12)
G:=sub<Sym(12)| (1,9,8)(2,5,10)(3,6,11)(4,12,7), (2,10,5)(4,7,12), (1,2,3,4)(5,6,7,8)(9,10,11,12)>;
G:=Group( (1,9,8)(2,5,10)(3,6,11)(4,12,7), (2,10,5)(4,7,12), (1,2,3,4)(5,6,7,8)(9,10,11,12) );
G=PermutationGroup([[(1,9,8),(2,5,10),(3,6,11),(4,12,7)], [(2,10,5),(4,7,12)], [(1,2,3,4),(5,6,7,8),(9,10,11,12)]])
G:=TransitiveGroup(12,17);
►On 18 points - transitive group 18T10
Generators in S18
(1 14 12)(2 8 10)(3 16 7)(4 15 11)(5 9 18)(6 13 17)
(1 3 5)(2 17 15)(4 10 13)(6 11 8)(7 18 12)(9 14 16)
(1 2)(3 4 5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)
G:=sub<Sym(18)| (1,14,12)(2,8,10)(3,16,7)(4,15,11)(5,9,18)(6,13,17), (1,3,5)(2,17,15)(4,10,13)(6,11,8)(7,18,12)(9,14,16), (1,2)(3,4,5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18)>;
G:=Group( (1,14,12)(2,8,10)(3,16,7)(4,15,11)(5,9,18)(6,13,17), (1,3,5)(2,17,15)(4,10,13)(6,11,8)(7,18,12)(9,14,16), (1,2)(3,4,5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18) );
G=PermutationGroup([[(1,14,12),(2,8,10),(3,16,7),(4,15,11),(5,9,18),(6,13,17)], [(1,3,5),(2,17,15),(4,10,13),(6,11,8),(7,18,12),(9,14,16)], [(1,2),(3,4,5,6),(7,8,9,10),(11,12,13,14),(15,16,17,18)]])
G:=TransitiveGroup(18,10);
C32⋊C4 is a maximal subgroup of
F9 S3≀C2 PSU3(𝔽2) C32⋊F5 C92⋊C4 C34⋊4C4 A6 (C3×C39)⋊C4
C32⋊Dicp: C33⋊C4 C32⋊Dic5 C32⋊Dic7 C32⋊Dic11 C32⋊Dic13 ...
C32⋊C4 is a maximal quotient of
C32⋊2C8 He3⋊C4 C32⋊F5 C92⋊C4 C34⋊4C4 (C3×C39)⋊C4
C32⋊Dicp: C33⋊C4 C32⋊Dic5 C32⋊Dic7 C32⋊Dic11 C32⋊Dic13 ...
| action | f(x) | Disc(f) |
|---|---|---|
| 6T10 | x6-21x4+21x3+99x2-198x+99 | 312·54·112 |
| 9T9 | x9-36x7-45x6+297x5+459x4-858x3-1404x2+819x+1339 | 326·136·292·432 |
| 12T17 | x12-48x10-72x9+636x8+1848x7-484x6-6528x5-5625x4+3456x3+6744x2+2880x+376 | 253·314·74·172·472·2392·7875292 |
Matrix representation of C32⋊C4 ►in GL4(ℤ) generated by
| 0 | 1 | 0 | 0 |
| -1 | -1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | -1 | -1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | -1 | -1 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| -1 | -1 | 0 | 0 |
G:=sub<GL(4,Integers())| [0,-1,0,0,1,-1,0,0,0,0,0,-1,0,0,1,-1],[1,0,0,0,0,1,0,0,0,0,-1,1,0,0,-1,0],[0,0,1,-1,0,0,0,-1,1,0,0,0,0,1,0,0] >;
C32⋊C4 in GAP, Magma, Sage, TeX
C_3^2\rtimes C_4
% in TeX
G:=Group("C3^2:C4"); // GroupNames label
G:=SmallGroup(36,9);
// by ID
G=gap.SmallGroup(36,9);
# by ID
G:=PCGroup([4,-2,-2,-3,3,8,338,54,515,199]);
// Polycyclic
G:=Group<a,b,c|a^3=b^3=c^4=1,c*b*c^-1=a*b=b*a,c*a*c^-1=a^-1*b>;
// generators/relations
Export
Subgroup lattice of C32⋊C4 in TeX
Character table of C32⋊C4 in TeX