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, 36T93Siblings 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 |