# Properties

 Label 12T18 Degree $12$ Order $36$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_6\times S_3$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(12, 18);

## Group action invariants

 Degree $n$: $12$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $18$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_6\times S_3$ CHM label: $[3^{2}]E(4)$ 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: (1,10)(2,5)(3,12)(4,7)(6,9)(8,11), (2,6,10)(4,8,12), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $S_3$, $C_6$ x 3
$12$:  $D_{6}$, $C_6\times C_2$
$18$:  $S_3\times C_3$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 3: None

Degree 4: $C_2^2$

Degree 6: $S_3\times C_3$

## Low degree siblings

18T6 x 2, 36T6

Siblings are shown with degree $\leq 47$

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

## Conjugacy classes

 Label Cycle Type Size Order Representative $1^{12}$ $1$ $1$ $()$ $3^{2},1^{6}$ $2$ $3$ $( 2, 6,10)( 4, 8,12)$ $3^{2},1^{6}$ $2$ $3$ $( 2,10, 6)( 4,12, 8)$ $2^{6}$ $3$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$ $6^{2}$ $3$ $6$ $( 1, 2, 5, 6, 9,10)( 3, 4, 7, 8,11,12)$ $6^{2}$ $3$ $6$ $( 1, 2, 9,10, 5, 6)( 3, 4,11,12, 7, 8)$ $6^{2}$ $1$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$ $6,2^{3}$ $2$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 8)( 4,10)( 6,12)$ $6^{2}$ $2$ $6$ $( 1, 3, 5, 7, 9,11)( 2,12,10, 8, 6, 4)$ $6^{2}$ $3$ $6$ $( 1, 4, 5, 8, 9,12)( 2, 3, 6, 7,10,11)$ $6^{2}$ $3$ $6$ $( 1, 4, 9,12, 5, 8)( 2, 7,10, 3, 6,11)$ $2^{6}$ $3$ $2$ $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$ $3^{4}$ $1$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$ $3^{4}$ $2$ $3$ $( 1, 5, 9)( 2,10, 6)( 3, 7,11)( 4,12, 8)$ $2^{6}$ $1$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$ $6,2^{3}$ $2$ $6$ $( 1, 7)( 2,12,10, 8, 6, 4)( 3, 9)( 5,11)$ $3^{4}$ $1$ $3$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$ $6^{2}$ $1$ $6$ $( 1,11, 9, 7, 5, 3)( 2,12,10, 8, 6, 4)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $36=2^{2} \cdot 3^{2}$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: not nilpotent Label: 36.12 magma: IdentifyGroup(G); Character table:

 1A 2A 2B 2C 3A1 3A-1 3B 3C1 3C-1 6A1 6A-1 6B 6C1 6C-1 6D1 6D-1 6E1 6E-1 Size 1 1 3 3 1 1 2 2 2 1 1 2 2 2 3 3 3 3 2 P 1A 1A 1A 1A 3A-1 3A1 3B 3C-1 3C1 3A-1 3A1 3C-1 3B 3C1 3A1 3A-1 3A1 3A-1 3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2B 2B 2C 2C Type 36.12.1a R $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 36.12.1b R $1$ $−1$ $−1$ $1$ $1$ $1$ $1$ $1$ $1$ $−1$ $−1$ $−1$ $−1$ $−1$ $−1$ $−1$ $1$ $1$ 36.12.1c R $1$ $−1$ $1$ $−1$ $1$ $1$ $1$ $1$ $1$ $−1$ $−1$ $−1$ $−1$ $−1$ $1$ $1$ $−1$ $−1$ 36.12.1d R $1$ $1$ $−1$ $−1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $−1$ $−1$ $−1$ $−1$ 36.12.1e1 C $1$ $1$ $1$ $1$ $ζ3−1$ $ζ3$ $1$ $ζ3−1$ $ζ3$ $ζ3−1$ $ζ3$ $1$ $ζ3$ $ζ3−1$ $ζ3$ $ζ3−1$ $ζ3$ $ζ3−1$ 36.12.1e2 C $1$ $1$ $1$ $1$ $ζ3$ $ζ3−1$ $1$ $ζ3$ $ζ3−1$ $ζ3$ $ζ3−1$ $1$ $ζ3−1$ $ζ3$ $ζ3−1$ $ζ3$ $ζ3−1$ $ζ3$ 36.12.1f1 C $1$ $−1$ $−1$ $1$ $ζ3−1$ $ζ3$ $1$ $ζ3−1$ $ζ3$ $−ζ3−1$ $−ζ3$ $−1$ $−ζ3$ $−ζ3−1$ $−ζ3$ $−ζ3−1$ $ζ3$ $ζ3−1$ 36.12.1f2 C $1$ $−1$ $−1$ $1$ $ζ3$ $ζ3−1$ $1$ $ζ3$ $ζ3−1$ $−ζ3$ $−ζ3−1$ $−1$ $−ζ3−1$ $−ζ3$ $−ζ3−1$ $−ζ3$ $ζ3−1$ $ζ3$ 36.12.1g1 C $1$ $−1$ $1$ $−1$ $ζ3−1$ $ζ3$ $1$ $ζ3−1$ $ζ3$ $−ζ3−1$ $−ζ3$ $−1$ $−ζ3$ $−ζ3−1$ $ζ3$ $ζ3−1$ $−ζ3$ $−ζ3−1$ 36.12.1g2 C $1$ $−1$ $1$ $−1$ $ζ3$ $ζ3−1$ $1$ $ζ3$ $ζ3−1$ $−ζ3$ $−ζ3−1$ $−1$ $−ζ3−1$ $−ζ3$ $ζ3−1$ $ζ3$ $−ζ3−1$ $−ζ3$ 36.12.1h1 C $1$ $1$ $−1$ $−1$ $ζ3−1$ $ζ3$ $1$ $ζ3−1$ $ζ3$ $ζ3−1$ $ζ3$ $1$ $ζ3$ $ζ3−1$ $−ζ3$ $−ζ3−1$ $−ζ3$ $−ζ3−1$ 36.12.1h2 C $1$ $1$ $−1$ $−1$ $ζ3$ $ζ3−1$ $1$ $ζ3$ $ζ3−1$ $ζ3$ $ζ3−1$ $1$ $ζ3−1$ $ζ3$ $−ζ3−1$ $−ζ3$ $−ζ3−1$ $−ζ3$ 36.12.2a R $2$ $2$ $0$ $0$ $2$ $2$ $−1$ $−1$ $−1$ $2$ $2$ $−1$ $−1$ $−1$ $0$ $0$ $0$ $0$ 36.12.2b R $2$ $−2$ $0$ $0$ $2$ $2$ $−1$ $−1$ $−1$ $−2$ $−2$ $1$ $1$ $1$ $0$ $0$ $0$ $0$ 36.12.2c1 C $2$ $2$ $0$ $0$ $2ζ3−1$ $2ζ3$ $−1$ $−ζ3−1$ $−ζ3$ $2ζ3−1$ $2ζ3$ $−1$ $−ζ3$ $−ζ3−1$ $0$ $0$ $0$ $0$ 36.12.2c2 C $2$ $2$ $0$ $0$ $2ζ3$ $2ζ3−1$ $−1$ $−ζ3$ $−ζ3−1$ $2ζ3$ $2ζ3−1$ $−1$ $−ζ3−1$ $−ζ3$ $0$ $0$ $0$ $0$ 36.12.2d1 C $2$ $−2$ $0$ $0$ $2ζ3−1$ $2ζ3$ $−1$ $−ζ3−1$ $−ζ3$ $−2ζ3−1$ $−2ζ3$ $1$ $ζ3$ $ζ3−1$ $0$ $0$ $0$ $0$ 36.12.2d2 C $2$ $−2$ $0$ $0$ $2ζ3$ $2ζ3−1$ $−1$ $−ζ3$ $−ζ3−1$ $−2ζ3$ $−2ζ3−1$ $1$ $ζ3−1$ $ζ3$ $0$ $0$ $0$ $0$

magma: CharacterTable(G);