Group action invariants
Degree $n$: | $18$ | |
Transitive number $t$: | $41$ | |
Group: | $C_2\times He_3:C_2$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $2$ | |
Generators: | (1,11,17,13,4,16)(2,12,18,14,3,15)(5,9)(6,10)(7,8), (1,17)(2,18)(5,16,8,13,10,11)(6,15,7,14,9,12) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $S_3$, $C_6$ x 3 $12$: $D_{6}$, $C_6\times C_2$ $18$: $S_3\times C_3$ $36$: $C_6\times S_3$ $54$: $C_3^2 : C_6$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 6: $D_{6}$
Degree 9: $C_3^2 : C_6$
Low degree siblings
18T41, 18T42 x 2Siblings 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 $ | $1$ | $1$ | $()$ |
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $3$ | $3$ | $( 5, 8,10)( 6, 7, 9)(11,16,13)(12,15,14)$ |
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $3$ | $3$ | $( 5,10, 8)( 6, 9, 7)(11,13,16)(12,14,15)$ |
$ 6, 6, 2, 2, 1, 1 $ | $9$ | $6$ | $( 3,18)( 4,17)( 5,11, 8,16,10,13)( 6,12, 7,15, 9,14)$ |
$ 6, 6, 2, 2, 1, 1 $ | $9$ | $6$ | $( 3,18)( 4,17)( 5,13,10,16, 8,11)( 6,14, 9,15, 7,12)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $9$ | $2$ | $( 3,18)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ |
$ 6, 6, 2, 2, 2 $ | $3$ | $6$ | $( 1, 2)( 3, 4)( 5, 7,10, 6, 8, 9)(11,15,13,12,16,14)(17,18)$ |
$ 6, 6, 2, 2, 2 $ | $3$ | $6$ | $( 1, 2)( 3, 4)( 5, 9, 8, 6,10, 7)(11,14,16,12,13,15)(17,18)$ |
$ 6, 6, 2, 2, 2 $ | $9$ | $6$ | $( 1, 2)( 3,17)( 4,18)( 5,12, 8,15,10,14)( 6,11, 7,16, 9,13)$ |
$ 6, 6, 2, 2, 2 $ | $9$ | $6$ | $( 1, 2)( 3,17)( 4,18)( 5,14,10,15, 8,12)( 6,13, 9,16, 7,11)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1, 2)( 3,17)( 4,18)( 5,15)( 6,16)( 7,13)( 8,14)( 9,11)(10,12)$ |
$ 6, 6, 6 $ | $2$ | $6$ | $( 1, 3,17, 2, 4,18)( 5, 7,10, 6, 8, 9)(11,14,16,12,13,15)$ |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 4,17)( 2, 3,18)( 5, 8,10)( 6, 7, 9)(11,13,16)(12,14,15)$ |
$ 6, 6, 6 $ | $6$ | $6$ | $( 1, 5,12, 2, 6,11)( 3, 7,13, 4, 8,14)( 9,16,17,10,15,18)$ |
$ 6, 6, 6 $ | $6$ | $6$ | $( 1, 5,14, 2, 6,13)( 3, 7,16, 4, 8,15)( 9,11,17,10,12,18)$ |
$ 6, 6, 6 $ | $6$ | $6$ | $( 1, 5,15, 2, 6,16)( 3, 7,11, 4, 8,12)( 9,13,17,10,14,18)$ |
$ 3, 3, 3, 3, 3, 3 $ | $6$ | $3$ | $( 1, 6,12)( 2, 5,11)( 3, 8,13)( 4, 7,14)( 9,15,17)(10,16,18)$ |
$ 3, 3, 3, 3, 3, 3 $ | $6$ | $3$ | $( 1, 6,14)( 2, 5,13)( 3, 8,16)( 4, 7,15)( 9,12,17)(10,11,18)$ |
$ 3, 3, 3, 3, 3, 3 $ | $6$ | $3$ | $( 1, 6,15)( 2, 5,16)( 3, 8,11)( 4, 7,12)( 9,14,17)(10,13,18)$ |
Group invariants
Order: | $108=2^{2} \cdot 3^{3}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [108, 25] |
Character table: |
2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 3 3 2 2 1 1 1 3 2 2 1 1 1 3 3 2 2 2 2 2 2 1a 3a 3b 6a 6b 2a 2b 6c 6d 6e 6f 2c 6g 3c 6h 6i 6j 3d 3e 3f 2P 1a 3b 3a 3a 3b 1a 1a 3b 3a 3a 3b 1a 3c 3c 3e 3d 3f 3e 3d 3f 3P 1a 1a 1a 2a 2a 2a 2b 2b 2b 2c 2c 2c 2b 1a 2b 2b 2b 1a 1a 1a 5P 1a 3b 3a 6b 6a 2a 2b 6d 6c 6f 6e 2c 6g 3c 6i 6h 6j 3e 3d 3f X.1 1 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 1 1 X.3 1 1 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 1 1 X.5 1 A /A -/A -A -1 -1 -A -/A /A A 1 -1 1 -A -/A -1 A /A 1 X.6 1 /A A -A -/A -1 -1 -/A -A A /A 1 -1 1 -/A -A -1 /A A 1 X.7 1 A /A -/A -A -1 1 A /A -/A -A -1 1 1 A /A 1 A /A 1 X.8 1 /A A -A -/A -1 1 /A A -A -/A -1 1 1 /A A 1 /A A 1 X.9 1 A /A /A A 1 -1 -A -/A -/A -A -1 -1 1 -A -/A -1 A /A 1 X.10 1 /A A A /A 1 -1 -/A -A -A -/A -1 -1 1 -/A -A -1 /A A 1 X.11 1 A /A /A A 1 1 A /A /A A 1 1 1 A /A 1 A /A 1 X.12 1 /A A A /A 1 1 /A A A /A 1 1 1 /A A 1 /A A 1 X.13 2 2 2 . . . 2 2 2 . . . 2 2 -1 -1 -1 -1 -1 -1 X.14 2 2 2 . . . -2 -2 -2 . . . -2 2 1 1 1 -1 -1 -1 X.15 2 B /B . . . 2 B /B . . . 2 2 -/A -A -1 -/A -A -1 X.16 2 /B B . . . 2 /B B . . . 2 2 -A -/A -1 -A -/A -1 X.17 2 B /B . . . -2 -B -/B . . . -2 2 /A A 1 -/A -A -1 X.18 2 /B B . . . -2 -/B -B . . . -2 2 A /A 1 -A -/A -1 X.19 6 . . . . . -6 . . . . . 3 -3 . . . . . . X.20 6 . . . . . 6 . . . . . -3 -3 . . . . . . A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 2*E(3) = -1+Sqrt(-3) = 2b3 |