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G = C32⋊C4order 36 = 22·32

The semidirect product of C32 and C4 acting faithfully

metabelian, soluble, monomial, A-group

Aliases: C32⋊C4, C3⋊S3.C2, SmallGroup(36,9)

Series: Derived Chief Lower central Upper central

C1C32 — C32⋊C4
C1C32C3⋊S3 — C32⋊C4
C32 — C32⋊C4
C1

Generators and relations for C32⋊C4
 G = < a,b,c | a3=b3=c4=1, cbc-1=ab=ba, cac-1=a-1b >

9C2
2C3
2C3
9C4
6S3
6S3

Character table of C32⋊C4

 class 123A3B4A4B
 size 194499
ρ1111111    trivial
ρ21111-1-1    linear of order 2
ρ31-111i-i    linear of order 4
ρ41-111-ii    linear of order 4
ρ5401-200    orthogonal faithful
ρ640-2100    orthogonal faithful

Permutation representations of C32⋊C4
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 3 5)(2 15 17)(4 11 10)(6 8 13)(7 18 14)(9 12 16)
(1 14 12)(2 8 10)(3 7 16)(4 15 13)(5 18 9)(6 11 17)
(1 2)(3 4 5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)

G:=sub<Sym(18)| (1,3,5)(2,15,17)(4,11,10)(6,8,13)(7,18,14)(9,12,16), (1,14,12)(2,8,10)(3,7,16)(4,15,13)(5,18,9)(6,11,17), (1,2)(3,4,5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18)>;

G:=Group( (1,3,5)(2,15,17)(4,11,10)(6,8,13)(7,18,14)(9,12,16), (1,14,12)(2,8,10)(3,7,16)(4,15,13)(5,18,9)(6,11,17), (1,2)(3,4,5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18) );

G=PermutationGroup([(1,3,5),(2,15,17),(4,11,10),(6,8,13),(7,18,14),(9,12,16)], [(1,14,12),(2,8,10),(3,7,16),(4,15,13),(5,18,9),(6,11,17)], [(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  C344C4  A6  (C3×C39)⋊C4
 C32⋊Dicp: C33⋊C4  C32⋊Dic5  C32⋊Dic7  C32⋊Dic11  C32⋊Dic13 ...
C32⋊C4 is a maximal quotient of
C322C8  He3⋊C4  C32⋊F5  C92⋊C4  C344C4  (C3×C39)⋊C4
 C32⋊Dicp: C33⋊C4  C32⋊Dic5  C32⋊Dic7  C32⋊Dic11  C32⋊Dic13 ...

Polynomial with Galois group C32⋊C4 over ℚ
actionf(x)Disc(f)
6T10x6-21x4+21x3+99x2-198x+99312·54·112
9T9x9-36x7-45x6+297x5+459x4-858x3-1404x2+819x+1339326·136·292·432
12T17x12-48x10-72x9+636x8+1848x7-484x6-6528x5-5625x4+3456x3+6744x2+2880x+376253·314·74·172·472·2392·7875292

Matrix representation of C32⋊C4 in GL4(ℤ) generated by

0100
-1-100
0001
00-1-1
,
1000
0100
00-1-1
0010
,
0010
0001
1000
-1-100
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

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