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Properties

Label 18T25
Degree $18$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_6\times A_4$

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Show commands: Magma

magma: G := TransitiveGroup(18, 25);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $25$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_6\times A_4$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $6$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,9,15,2,10,16)(3,11,17,4,12,18)(5,7,14,6,8,13), (1,3,6,2,4,5)(7,10,11)(8,9,12)(13,16,18,14,15,17)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$ x 4
$6$:  $C_6$ x 4
$9$:  $C_3^2$
$12$:  $A_4$
$18$:  $C_6 \times C_3$
$24$:  $A_4\times C_2$
$36$:  $C_3\times A_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$ x 4

Degree 6: $A_4\times C_2$

Degree 9: $C_3^2$

Low degree siblings

24T71 x 3, 36T18, 36T31

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $(13,14)(15,16)(17,18)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$
$ 6, 6, 3, 3 $ $3$ $6$ $( 1, 3, 6, 2, 4, 5)( 7, 9,11, 8,10,12)(13,15,18)(14,16,17)$
$ 6, 6, 6 $ $1$ $6$ $( 1, 3, 6, 2, 4, 5)( 7, 9,11, 8,10,12)(13,16,18,14,15,17)$
$ 6, 3, 3, 3, 3 $ $3$ $6$ $( 1, 3, 6, 2, 4, 5)( 7,10,11)( 8, 9,12)(13,15,18)(14,16,17)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 4, 6)( 2, 3, 5)( 7,10,11)( 8, 9,12)(13,15,18)(14,16,17)$
$ 6, 6, 3, 3 $ $3$ $6$ $( 1, 5, 4, 2, 6, 3)( 7,11,10)( 8,12, 9)(13,17,15,14,18,16)$
$ 6, 3, 3, 3, 3 $ $3$ $6$ $( 1, 5, 4, 2, 6, 3)( 7,11,10)( 8,12, 9)(13,18,15)(14,17,16)$
$ 6, 6, 6 $ $1$ $6$ $( 1, 5, 4, 2, 6, 3)( 7,12,10, 8,11, 9)(13,17,15,14,18,16)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 6, 4)( 2, 5, 3)( 7,11,10)( 8,12, 9)(13,18,15)(14,17,16)$
$ 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 7,17)( 2, 8,18)( 3, 9,13)( 4,10,14)( 5,12,15)( 6,11,16)$
$ 6, 6, 6 $ $4$ $6$ $( 1, 7,17, 2, 8,18)( 3, 9,13, 4,10,14)( 5,12,15, 6,11,16)$
$ 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 9,16)( 2,10,15)( 3,11,18)( 4,12,17)( 5, 7,13)( 6, 8,14)$
$ 6, 6, 6 $ $4$ $6$ $( 1, 9,16, 2,10,15)( 3,11,18, 4,12,17)( 5, 7,13, 6, 8,14)$
$ 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1,11,13)( 2,12,14)( 3, 8,16)( 4, 7,15)( 5, 9,17)( 6,10,18)$
$ 6, 6, 6 $ $4$ $6$ $( 1,11,13, 2,12,14)( 3, 8,16, 4, 7,15)( 5, 9,17, 6,10,18)$
$ 6, 6, 6 $ $4$ $6$ $( 1,13,11, 2,14,12)( 3,16, 8, 4,15, 7)( 5,17, 9, 6,18,10)$
$ 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1,13,12)( 2,14,11)( 3,16, 7)( 4,15, 8)( 5,17,10)( 6,18, 9)$
$ 6, 6, 6 $ $4$ $6$ $( 1,15,10, 2,16, 9)( 3,17,12, 4,18,11)( 5,14, 8, 6,13, 7)$
$ 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1,15, 9)( 2,16,10)( 3,17,11)( 4,18,12)( 5,14, 7)( 6,13, 8)$
$ 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1,17, 7)( 2,18, 8)( 3,13, 9)( 4,14,10)( 5,15,12)( 6,16,11)$
$ 6, 6, 6 $ $4$ $6$ $( 1,17, 8, 2,18, 7)( 3,13,10, 4,14, 9)( 5,15,11, 6,16,12)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  72.47
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A1 3A-1 3B1 3B-1 3C1 3C-1 3D1 3D-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 6D1 6D-1 6E1 6E-1 6F1 6F-1
Size 1 1 3 3 1 1 4 4 4 4 4 4 1 1 3 3 3 3 4 4 4 4 4 4
2 P 1A 1A 1A 1A 3A-1 3A1 3B-1 3B1 3C-1 3D1 3C1 3D-1 3A1 3A-1 3A-1 3A1 3A1 3A-1 3D-1 3B1 3B-1 3C1 3D1 3C-1
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2B 2B 2C 2C 2A 2A 2A 2A 2A 2A
Type
72.47.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.47.1b R 1 −1 −1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1 −1 −1 −1 −1
72.47.1c1 C 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1 1
72.47.1c2 C 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1 1
72.47.1d1 C 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3 ζ3−1
72.47.1d2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3−1 ζ3
72.47.1e1 C 1 1 1 1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3−1 ζ3
72.47.1e2 C 1 1 1 1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3 ζ3−1
72.47.1f1 C 1 1 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 1 1 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1
72.47.1f2 C 1 1 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3
72.47.1g1 C 1 −1 −1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 1 1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −1 −1
72.47.1g2 C 1 −1 −1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −1 −1
72.47.1h1 C 1 −1 −1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 −1 −1 −ζ3 −ζ3−1
72.47.1h2 C 1 −1 −1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 −1 −1 −ζ3−1 −ζ3
72.47.1i1 C 1 −1 −1 1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3 −1 −1 −ζ3 −ζ3−1 −ζ3−1 −ζ3
72.47.1i2 C 1 −1 −1 1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1 −1 −1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
72.47.1j1 C 1 −1 −1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 −1 −1 −1 −1 1 1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1
72.47.1j2 C 1 −1 −1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 −1 −1 −1 −1 1 1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3
72.47.3a R 3 3 −1 −1 3 3 0 0 0 0 0 0 3 3 −1 −1 −1 −1 0 0 0 0 0 0
72.47.3b R 3 −3 1 −1 3 3 0 0 0 0 0 0 −3 −3 1 1 −1 −1 0 0 0 0 0 0
72.47.3c1 C 3 3 −1 −1 3ζ3−1 3ζ3 0 0 0 0 0 0 3ζ3 3ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 0 0 0 0 0 0
72.47.3c2 C 3 3 −1 −1 3ζ3 3ζ3−1 0 0 0 0 0 0 3ζ3−1 3ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 0 0 0 0 0 0
72.47.3d1 C 3 −3 1 −1 3ζ3−1 3ζ3 0 0 0 0 0 0 −3ζ3 −3ζ3−1 ζ3 ζ3−1 −ζ3−1 −ζ3 0 0 0 0 0 0
72.47.3d2 C 3 −3 1 −1 3ζ3 3ζ3−1 0 0 0 0 0 0 −3ζ3−1 −3ζ3 ζ3−1 ζ3 −ζ3 −ζ3−1 0 0 0 0 0 0

magma: CharacterTable(G);