# Properties

 Label 27T28 Order $$81$$ n $$27$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $C_3.C_3^3$

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## Group action invariants

 Degree $n$ : $27$ Transitive number $t$ : $28$ Group : $C_3.C_3^3$ Parity: $1$ Primitive: No Nilpotency class: $2$ Generators: (1,16,19)(2,17,20)(3,18,21)(4,10,23)(5,11,24)(6,12,22)(7,14,25)(8,15,26)(9,13,27), (1,4,7,2,5,8,3,6,9)(10,13,18,11,14,16,12,15,17)(19,24,27,20,22,25,21,23,26), (10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,24,23)(25,27,26) $|\Aut(F/K)|$: $9$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$ x 13
9:  $C_3^2$ x 13
27:  27T4

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $C_3$ x 4

Degree 9: $C_3^2$

## Low degree siblings

27T28 x 3

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $3$ $3$ $(10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,24,23)(25,27,26)$ $3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $3$ $3$ $(10,12,11)(13,15,14)(16,18,17)(19,20,21)(22,23,24)(25,26,27)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $1$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $1$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)$ $9, 9, 9$ $3$ $9$ $( 1, 4, 7, 2, 5, 8, 3, 6, 9)(10,13,18,11,14,16,12,15,17)(19,24,27,20,22,25,21, 23,26)$ $9, 9, 9$ $1$ $9$ $( 1, 4, 7, 2, 5, 8, 3, 6, 9)(10,14,17,11,15,18,12,13,16)(19,23,25,20,24,26,21, 22,27)$ $9, 9, 9$ $3$ $9$ $( 1, 4, 7, 2, 5, 8, 3, 6, 9)(10,15,16,11,13,17,12,14,18)(19,22,26,20,23,27,21, 24,25)$ $9, 9, 9$ $1$ $9$ $( 1, 5, 9, 2, 6, 7, 3, 4, 8)(10,15,16,11,13,17,12,14,18)(19,24,27,20,22,25,21, 23,26)$ $9, 9, 9$ $1$ $9$ $( 1, 6, 8, 2, 4, 9, 3, 5, 7)(10,13,18,11,14,16,12,15,17)(19,22,26,20,23,27,21, 24,25)$ $9, 9, 9$ $3$ $9$ $( 1, 7, 5, 3, 9, 4, 2, 8, 6)(10,16,13,12,18,15,11,17,14)(19,26,23,21,25,22,20, 27,24)$ $9, 9, 9$ $1$ $9$ $( 1, 7, 5, 3, 9, 4, 2, 8, 6)(10,17,15,12,16,14,11,18,13)(19,25,24,21,27,23,20, 26,22)$ $9, 9, 9$ $3$ $9$ $( 1, 7, 5, 3, 9, 4, 2, 8, 6)(10,18,14,12,17,13,11,16,15)(19,27,22,21,26,24,20, 25,23)$ $9, 9, 9$ $1$ $9$ $( 1, 8, 4, 3, 7, 6, 2, 9, 5)(10,18,14,12,17,13,11,16,15)(19,26,23,21,25,22,20, 27,24)$ $9, 9, 9$ $1$ $9$ $( 1, 9, 6, 3, 8, 5, 2, 7, 4)(10,16,13,12,18,15,11,17,14)(19,27,22,21,26,24,20, 25,23)$ $9, 9, 9$ $3$ $9$ $( 1,10,25, 2,11,26, 3,12,27)( 4,14,20, 5,15,21, 6,13,19)( 7,17,24, 8,18,22, 9, 16,23)$ $9, 9, 9$ $3$ $9$ $( 1,10,26, 2,11,27, 3,12,25)( 4,14,21, 5,15,19, 6,13,20)( 7,17,22, 8,18,23, 9, 16,24)$ $9, 9, 9$ $3$ $9$ $( 1,10,27, 2,11,25, 3,12,26)( 4,14,19, 5,15,20, 6,13,21)( 7,17,23, 8,18,24, 9, 16,22)$ $9, 9, 9$ $3$ $9$ $( 1,13,22, 3,15,24, 2,14,23)( 4,16,27, 6,18,26, 5,17,25)( 7,10,19, 9,12,21, 8, 11,20)$ $9, 9, 9$ $3$ $9$ $( 1,13,23, 3,15,22, 2,14,24)( 4,16,25, 6,18,27, 5,17,26)( 7,10,20, 9,12,19, 8, 11,21)$ $9, 9, 9$ $3$ $9$ $( 1,13,24, 3,15,23, 2,14,22)( 4,16,26, 6,18,25, 5,17,27)( 7,10,21, 9,12,20, 8, 11,19)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $3$ $3$ $( 1,16,21)( 2,17,19)( 3,18,20)( 4,10,22)( 5,11,23)( 6,12,24)( 7,14,27) ( 8,15,25)( 9,13,26)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $3$ $3$ $( 1,16,19)( 2,17,20)( 3,18,21)( 4,10,23)( 5,11,24)( 6,12,22)( 7,14,25) ( 8,15,26)( 9,13,27)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $3$ $3$ $( 1,16,20)( 2,17,21)( 3,18,19)( 4,10,24)( 5,11,22)( 6,12,23)( 7,14,26) ( 8,15,27)( 9,13,25)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $3$ $3$ $( 1,19,16)( 2,20,17)( 3,21,18)( 4,23,10)( 5,24,11)( 6,22,12)( 7,25,14) ( 8,26,15)( 9,27,13)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $3$ $3$ $( 1,19,18)( 2,20,16)( 3,21,17)( 4,23,12)( 5,24,10)( 6,22,11)( 7,25,13) ( 8,26,14)( 9,27,15)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $3$ $3$ $( 1,19,17)( 2,20,18)( 3,21,16)( 4,23,11)( 5,24,12)( 6,22,10)( 7,25,15) ( 8,26,13)( 9,27,14)$ $9, 9, 9$ $3$ $9$ $( 1,22,14, 2,23,15, 3,24,13)( 4,27,17, 5,25,18, 6,26,16)( 7,19,11, 8,20,12, 9, 21,10)$ $9, 9, 9$ $3$ $9$ $( 1,22,13, 2,23,14, 3,24,15)( 4,27,16, 5,25,17, 6,26,18)( 7,19,10, 8,20,11, 9, 21,12)$ $9, 9, 9$ $3$ $9$ $( 1,22,15, 2,23,13, 3,24,14)( 4,27,18, 5,25,16, 6,26,17)( 7,19,12, 8,20,10, 9, 21,11)$ $9, 9, 9$ $3$ $9$ $( 1,25,12, 3,27,11, 2,26,10)( 4,20,13, 6,19,15, 5,21,14)( 7,24,16, 9,23,18, 8, 22,17)$ $9, 9, 9$ $3$ $9$ $( 1,25,11, 3,27,10, 2,26,12)( 4,20,15, 6,19,14, 5,21,13)( 7,24,18, 9,23,17, 8, 22,16)$ $9, 9, 9$ $3$ $9$ $( 1,25,10, 3,27,12, 2,26,11)( 4,20,14, 6,19,13, 5,21,15)( 7,24,17, 9,23,16, 8, 22,18)$

## Group invariants

 Order: $81=3^{4}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [81, 14]
 Character table: Data not available.