# Properties

 Label 27T31 Degree $27$ Order $108$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_3^2:(C_3:C_4)$

## Group action invariants

 Degree $n$: $27$ Transitive number $t$: $31$ Group: $C_3^2:(C_3:C_4)$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,5,8,14)(2,4,9,13)(3,6,7,15)(10,16,25,21)(11,18,26,20)(12,17,27,19)(23,24), (1,21,27,7)(2,20,25,9)(3,19,26,8)(4,5)(10,15,18,23)(11,14,16,22)(12,13,17,24)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$6$:  $S_3$
$12$:  $C_3 : C_4$
$36$:  $C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $S_3$

Degree 9: $C_3^2:C_4$

## Low degree siblings

12T72 x 2, 18T54 x 2, 36T89 x 2, 36T94 x 2

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

## Group invariants

 Order: $108=2^{2} \cdot 3^{3}$ Cyclic: no Abelian: no Solvable: yes GAP id: [108, 37]
 Character table:  2 2 2 2 2 1 1 . . . . . . 3 3 1 . . 3 1 3 3 3 3 3 3 1a 2a 4a 4b 3a 6a 3b 3c 3d 3e 3f 3g 2P 1a 1a 2a 2a 3a 3a 3d 3c 3b 3g 3f 3e 3P 1a 2a 4b 4a 1a 2a 1a 1a 1a 1a 1a 1a 5P 1a 2a 4a 4b 3a 6a 3d 3c 3b 3g 3f 3e X.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 X.3 1 -1 A -A 1 -1 1 1 1 1 1 1 X.4 1 -1 -A A 1 -1 1 1 1 1 1 1 X.5 2 -2 . . -1 1 -1 2 -1 -1 2 -1 X.6 2 2 . . -1 -1 -1 2 -1 -1 2 -1 X.7 4 . . . 4 . -2 -2 -2 1 1 1 X.8 4 . . . 4 . 1 1 1 -2 -2 -2 X.9 4 . . . -2 . B 1 /B 1 -2 1 X.10 4 . . . -2 . /B 1 B 1 -2 1 X.11 4 . . . -2 . 1 -2 1 /B 1 B X.12 4 . . . -2 . 1 -2 1 B 1 /B A = -E(4) = -Sqrt(-1) = -i B = 2*E(3)-E(3)^2 = (-1+3*Sqrt(-3))/2 = 1+3b3