# Properties

 Label 27T35 Degree $27$ Order $108$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_3^3:C_2^2$

## Group action invariants

 Degree $n$: $27$ Transitive number $t$: $35$ Group: $C_3^3:C_2^2$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,19,3,21,2,20)(4,16,6,18,5,17)(7,14,9,13,8,15)(10,11,12)(22,25,24,27,23,26), (1,26,4)(2,25,5,3,27,6)(7,23,10,17,13,20)(8,22,11,16,14,19)(9,24,12,18,15,21)

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $S_3$ x 3

Degree 9: $S_3^2$ x 3

## Low degree siblings

12T71, 18T53 x 3, 36T88 x 3, 36T93

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

## Group invariants

 Order: $108=2^{2} \cdot 3^{3}$ Cyclic: no Abelian: no Solvable: yes GAP id: [108, 40]
 Character table:  2 2 2 2 2 1 1 1 1 . . 1 . . . 1 3 3 1 1 1 3 1 3 1 3 3 1 3 3 3 3 1a 2a 2b 2c 3a 6a 3b 6b 3c 3d 6c 3e 3f 3g 3h 2P 1a 1a 1a 1a 3a 3a 3b 3b 3c 3e 3h 3d 3f 3g 3h 3P 1a 2a 2b 2c 1a 2a 1a 2b 1a 1a 2c 1a 1a 1a 1a 5P 1a 2a 2b 2c 3a 6a 3b 6b 3c 3e 6c 3d 3f 3g 3h X.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 X.3 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 X.5 2 . . -2 2 . 2 . 2 -1 1 -1 -1 -1 -1 X.6 2 . . 2 2 . 2 . 2 -1 -1 -1 -1 -1 -1 X.7 2 -2 . . -1 1 2 . -1 -1 . -1 2 -1 2 X.8 2 . -2 . 2 . -1 1 -1 -1 . -1 -1 2 2 X.9 2 . 2 . 2 . -1 -1 -1 -1 . -1 -1 2 2 X.10 2 2 . . -1 -1 2 . -1 -1 . -1 2 -1 2 X.11 4 . . . 4 . -2 . -2 1 . 1 1 -2 -2 X.12 4 . . . -2 . 4 . -2 1 . 1 -2 1 -2 X.13 4 . . . -2 . -2 . 1 1 . 1 -2 -2 4 X.14 4 . . . -2 . -2 . 1 A . /A 1 1 -2 X.15 4 . . . -2 . -2 . 1 /A . A 1 1 -2 A = -E(3)+2*E(3)^2 = (-1-3*Sqrt(-3))/2 = -2-3b3