Properties

 Label 27T13 Order $$54$$ n $$27$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_3\times C_3:S_3$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$ x 4, $C_6$
18:  $S_3\times C_3$ x 4, $C_3^2:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$, $S_3$ x 4

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

Low degree siblings

18T23 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1$ $9$ $2$ $( 2, 3)( 4,26)( 5,25)( 6,27)( 7,15)( 8,14)( 9,13)(11,12)(16,23)(17,22)(18,24) (20,21)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1, 4,26)( 2, 5,27)( 3, 6,25)( 7,12,13)( 8,10,14)( 9,11,15)(16,21,24) (17,19,22)(18,20,23)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1, 5,25)( 2, 6,26)( 3, 4,27)( 7,10,15)( 8,11,13)( 9,12,14)(16,19,23) (17,20,24)(18,21,22)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1, 6,27)( 2, 4,25)( 3, 5,26)( 7,11,14)( 8,12,15)( 9,10,13)(16,20,22) (17,21,23)(18,19,24)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1, 7,23)( 2, 8,24)( 3, 9,22)( 4,12,18)( 5,10,16)( 6,11,17)(13,20,26) (14,21,27)(15,19,25)$ $6, 6, 6, 6, 3$ $9$ $6$ $( 1, 7,19,25,10,16)( 2, 9,20,27,11,18)( 3, 8,21,26,12,17)( 4,13,22, 6,14,24) ( 5,15,23)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1, 8,22)( 2, 9,23)( 3, 7,24)( 4,10,17)( 5,11,18)( 6,12,16)(13,21,25) (14,19,26)(15,20,27)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1, 9,24)( 2, 7,22)( 3, 8,23)( 4,11,16)( 5,12,17)( 6,10,18)(13,19,27) (14,20,25)(15,21,26)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $1$ $3$ $( 1,10,19)( 2,11,20)( 3,12,21)( 4,14,22)( 5,15,23)( 6,13,24)( 7,16,25) ( 8,17,26)( 9,18,27)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1,11,21)( 2,12,19)( 3,10,20)( 4,15,24)( 5,13,22)( 6,14,23)( 7,17,27) ( 8,18,25)( 9,16,26)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1,16,15)( 2,17,13)( 3,18,14)( 4,21, 9)( 5,19, 7)( 6,20, 8)(10,25,23) (11,26,24)(12,27,22)$ $6, 6, 6, 6, 3$ $9$ $6$ $( 1,16,10,25,19, 7)( 2,18,11,27,20, 9)( 3,17,12,26,21, 8)( 4,24,14, 6,22,13) ( 5,23,15)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1,17,14)( 2,18,15)( 3,16,13)( 4,19, 8)( 5,20, 9)( 6,21, 7)(10,26,22) (11,27,23)(12,25,24)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1,18,13)( 2,16,14)( 3,17,15)( 4,20, 7)( 5,21, 8)( 6,19, 9)(10,27,24) (11,25,22)(12,26,23)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $1$ $3$ $( 1,19,10)( 2,20,11)( 3,21,12)( 4,22,14)( 5,23,15)( 6,24,13)( 7,25,16) ( 8,26,17)( 9,27,18)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1,20,12)( 2,21,10)( 3,19,11)( 4,23,13)( 5,24,14)( 6,22,15)( 7,26,18) ( 8,27,16)( 9,25,17)$

Group invariants

 Order: $54=2 \cdot 3^{3}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [54, 13]
 Character table:  2 1 1 . . . . . 1 . . 1 . . 1 . . 1 . 3 3 1 3 3 3 3 3 1 3 3 3 3 3 1 3 3 3 3 1a 2a 3a 3b 3c 3d 3e 6a 3f 3g 3h 3i 3j 6b 3k 3l 3m 3n 2P 1a 1a 3a 3b 3c 3d 3j 3m 3k 3l 3m 3n 3e 3h 3f 3g 3h 3i 3P 1a 2a 1a 1a 1a 1a 1a 2a 1a 1a 1a 1a 1a 2a 1a 1a 1a 1a 5P 1a 2a 3a 3b 3c 3d 3j 6b 3k 3l 3m 3n 3e 6a 3f 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 A -A A A A A /A -/A /A /A /A /A X.4 1 -1 1 1 1 1 /A -/A /A /A /A /A A -A A A A A X.5 1 1 1 1 1 1 A A A A A A /A /A /A /A /A /A X.6 1 1 1 1 1 1 /A /A /A /A /A /A A A A A A A X.7 2 . 2 -1 -1 -1 -1 . -1 -1 2 2 -1 . -1 -1 2 2 X.8 2 . -1 2 -1 -1 -1 . 2 -1 2 -1 -1 . 2 -1 2 -1 X.9 2 . -1 -1 -1 2 -1 . -1 2 2 -1 -1 . -1 2 2 -1 X.10 2 . -1 -1 2 -1 2 . -1 -1 2 -1 2 . -1 -1 2 -1 X.11 2 . 2 -1 -1 -1 -A . -A -A B B -/A . -/A -/A /B /B X.12 2 . 2 -1 -1 -1 -/A . -/A -/A /B /B -A . -A -A B B X.13 2 . -1 2 -1 -1 -A . B -A B -A -/A . /B -/A /B -/A X.14 2 . -1 2 -1 -1 -/A . /B -/A /B -/A -A . B -A B -A X.15 2 . -1 -1 -1 2 -/A . -/A /B /B -/A -A . -A B B -A X.16 2 . -1 -1 -1 2 -A . -A B B -A -/A . -/A /B /B -/A X.17 2 . -1 -1 2 -1 B . -A -A B -A /B . -/A -/A /B -/A X.18 2 . -1 -1 2 -1 /B . -/A -/A /B -/A B . -A -A B -A A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 2*E(3)^2 = -1-Sqrt(-3) = -1-i3