# Properties

 Label 27T34 Degree $27$ Order $108$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $S_3\times C_3:S_3$

## Group action invariants

 Degree $n$: $27$ Transitive number $t$: $34$ Group: $S_3\times C_3:S_3$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,19,11)(2,20,12)(3,21,10)(4,17,15,26,23,8)(5,18,13,27,24,9)(6,16,14,25,22,7), (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), (1,26,4)(2,25,5,3,27,6)(7,24,10,18,14,20)(8,23,11,17,15,19)(9,22,12,16,13,21)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$ x 5
$12$:  $D_{6}$ x 5
$18$:  $C_3^2:C_2$
$36$:  $S_3^2$ x 4, 18T12

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $S_3$ x 5

Degree 9: $C_3^2:C_2$, $S_3^2$ x 4

## Low degree siblings

18T58 x 4, 36T91 x 4

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

## Group invariants

 Order: $108=2^{2} \cdot 3^{3}$ Cyclic: no Abelian: no Solvable: yes GAP id: [108, 39]
 Character table:  2 2 2 2 2 1 1 1 1 . . 1 . 1 . 1 1 1 1 3 3 2 1 . 3 2 3 1 3 3 2 3 2 3 2 3 3 3 1a 2a 2b 2c 3a 6a 3b 6b 3c 3d 6c 3e 6d 3f 6e 3g 3h 3i 2P 1a 1a 1a 1a 3a 3a 3b 3b 3c 3d 3g 3e 3h 3f 3i 3g 3h 3i 3P 1a 2a 2b 2c 1a 2a 1a 2b 1a 1a 2a 1a 2a 1a 2a 1a 1a 1a 5P 1a 2a 2b 2c 3a 6a 3b 6b 3c 3d 6c 3e 6d 3f 6e 3g 3h 3i X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 1 1 X.3 1 -1 1 -1 1 -1 1 1 1 1 -1 1 -1 1 -1 1 1 1 X.4 1 1 -1 -1 1 1 1 -1 1 1 1 1 1 1 1 1 1 1 X.5 2 -2 . . 2 -2 2 . 2 -1 1 -1 1 -1 1 -1 -1 -1 X.6 2 2 . . 2 2 2 . 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 X.7 2 -2 . . -1 1 2 . -1 2 -2 -1 1 -1 1 2 -1 -1 X.8 2 . -2 . 2 . -1 1 -1 -1 . -1 . -1 . 2 2 2 X.9 2 . 2 . 2 . -1 -1 -1 -1 . -1 . -1 . 2 2 2 X.10 2 2 . . -1 -1 2 . -1 2 2 -1 -1 -1 -1 2 -1 -1 X.11 2 -2 . . -1 1 2 . -1 -1 1 -1 1 2 -2 -1 -1 2 X.12 2 -2 . . -1 1 2 . -1 -1 1 2 -2 -1 1 -1 2 -1 X.13 2 2 . . -1 -1 2 . -1 -1 -1 -1 -1 2 2 -1 -1 2 X.14 2 2 . . -1 -1 2 . -1 -1 -1 2 2 -1 -1 -1 2 -1 X.15 4 . . . 4 . -2 . -2 1 . 1 . 1 . -2 -2 -2 X.16 4 . . . -2 . -2 . 1 -2 . 1 . 1 . 4 -2 -2 X.17 4 . . . -2 . -2 . 1 1 . -2 . 1 . -2 4 -2 X.18 4 . . . -2 . -2 . 1 1 . 1 . -2 . -2 -2 4