# Properties

 Label 12T170 Order $$648$$ n $$12$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No

# Related objects

## Group action invariants

 Degree $n$ : $12$ Transitive number $t$ : $170$ CHM label : $[3^{4}:2]4$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,5)(2,10)(4,8)(7,11), (4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12) $|\Aut(F/K)|$: $1$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
6:  $S_3$ x 2
8:  $C_4\times C_2$
12:  $D_{6}$ x 2
24:  $S_3 \times C_4$ x 2
36:  $S_3^2$, $C_3^2:C_4$
72:  12T39, 12T40
216:  12T119 x 2

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $C_4$

Degree 6: None

## Low degree siblings

12T170 x 3, 18T195 x 4, 24T1522 x 4, 36T1090 x 4, 36T1091 x 4, 36T1092 x 4, 36T1149 x 4, 36T1196 x 4, 36T1199 x 4, 36T1228 x 4, 36T1229 x 8

Siblings are shown with degree $\leq 47$

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

## Conjugacy Classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $8$ $3$ $( 4,12, 8)$ $3, 3, 1, 1, 1, 1, 1, 1$ $8$ $3$ $( 3,11, 7)( 4,12, 8)$ $3, 3, 1, 1, 1, 1, 1, 1$ $8$ $3$ $( 3,11, 7)( 4, 8,12)$ $3, 3, 1, 1, 1, 1, 1, 1$ $4$ $3$ $( 2,10, 6)( 4,12, 8)$ $3, 3, 1, 1, 1, 1, 1, 1$ $4$ $3$ $( 2,10, 6)( 4, 8,12)$ $3, 3, 3, 1, 1, 1$ $8$ $3$ $( 2,10, 6)( 3,11, 7)( 4,12, 8)$ $3, 3, 3, 1, 1, 1$ $8$ $3$ $( 2,10, 6)( 3,11, 7)( 4, 8,12)$ $3, 3, 3, 1, 1, 1$ $8$ $3$ $( 2,10, 6)( 3, 7,11)( 4,12, 8)$ $3, 3, 3, 1, 1, 1$ $8$ $3$ $( 2,10, 6)( 3, 7,11)( 4, 8,12)$ $3, 3, 3, 3$ $2$ $3$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$ $3, 3, 3, 3$ $8$ $3$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4, 8,12)$ $3, 3, 3, 3$ $4$ $3$ $( 1, 9, 5)( 2,10, 6)( 3, 7,11)( 4, 8,12)$ $3, 3, 3, 3$ $2$ $3$ $( 1, 9, 5)( 2, 6,10)( 3,11, 7)( 4, 8,12)$ $2, 2, 2, 2, 1, 1, 1, 1$ $81$ $2$ $( 5, 9)( 6,10)( 7,11)( 8,12)$ $2, 2, 2, 2, 2, 2$ $9$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$ $6, 2, 2, 2$ $36$ $6$ $( 1, 7)( 2, 4,10,12, 6, 8)( 3, 9)( 5,11)$ $6, 6$ $18$ $6$ $( 1, 3, 9,11, 5, 7)( 2, 4,10,12, 6, 8)$ $6, 6$ $18$ $6$ $( 1, 3, 9,11, 5, 7)( 2,12, 6, 4,10, 8)$ $6, 6$ $36$ $6$ $( 1,11, 9, 3, 5, 7)( 2,12,10, 4, 6, 8)$ $6, 2, 2, 2$ $36$ $6$ $( 1,11, 9, 3, 5, 7)( 2, 8)( 4, 6)(10,12)$ $2, 2, 2, 2, 2, 2$ $9$ $2$ $( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,11)(10,12)$ $4, 4, 4$ $27$ $4$ $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12)$ $12$ $54$ $12$ $( 1,12, 3, 6, 9, 8,11, 2, 5, 4, 7,10)$ $12$ $54$ $12$ $( 1, 4,11, 2, 9, 8, 7, 6, 5,12, 3,10)$ $4, 4, 4$ $27$ $4$ $( 1,12, 3,10)( 2, 9, 4,11)( 5, 8, 7, 6)$ $4, 4, 4$ $27$ $4$ $( 1,10, 7, 4)( 2,11, 8, 5)( 3,12, 9, 6)$ $12$ $54$ $12$ $( 1,10, 7,12, 9, 6, 3, 8, 5, 2,11, 4)$ $12$ $54$ $12$ $( 1, 6, 3, 8, 9,10,11,12, 5, 2, 7, 4)$ $4, 4, 4$ $27$ $4$ $( 1, 6, 3, 4)( 2, 7,12, 5)( 8, 9,10,11)$

## Group invariants

 Order: $648=2^{3} \cdot 3^{4}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [648, 716]
 Character table: Data not available.