Show commands:
Magma
magma: G := TransitiveGroup(18, 93);
Group action invariants
Degree $n$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
Transitive number $t$: | $93$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
Group: | $S_3^2:C_6$ | ||
Parity: | $-1$ | magma: IsEven(G);
| |
Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
$\card{\Aut(F/K)}$: | $3$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
Generators: | (1,16,14,11,8,4)(2,17,15,12,9,5)(3,18,13,10,7,6), (1,14,2,15,3,13)(4,16,11)(5,17,12)(6,18,10)(7,9,8) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $C_6$ x 3 $8$: $D_{4}$ $12$: $C_6\times C_2$ $24$: $D_4 \times C_3$ $72$: $C_3^2:D_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $C_3$
Degree 6: $C_6$, $C_3^2:D_4$
Degree 9: None
Low degree siblings
12T121 x 2, 18T93, 24T561 x 2, 27T84, 36T258 x 2, 36T259 x 2, 36T260 x 2, 36T292 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Label | Cycle Type | Size | Order | Representative |
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ | |
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $2$ | $(10,17)(11,18)(12,16)$ | |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 7,15)( 8,13)( 9,14)(10,17)(11,18)(12,16)$ | |
$ 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $3$ | $( 4,10,17)( 5,11,18)( 6,12,16)$ | |
$ 3, 3, 3, 2, 2, 2, 1, 1, 1 $ | $12$ | $6$ | $( 4,10,17)( 5,11,18)( 6,12,16)( 7,15)( 8,13)( 9,14)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)$ | |
$ 6, 3, 3, 3, 3 $ | $6$ | $6$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,18,12,17,11,16)(13,14,15)$ | |
$ 6, 6, 3, 3 $ | $9$ | $6$ | $( 1, 2, 3)( 4, 5, 6)( 7,13, 9,15, 8,14)(10,18,12,17,11,16)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 2, 3)( 4,11,16)( 5,12,17)( 6,10,18)( 7, 8, 9)(13,14,15)$ | |
$ 6, 3, 3, 3, 3 $ | $12$ | $6$ | $( 1, 2, 3)( 4,11,16)( 5,12,17)( 6,10,18)( 7,13, 9,15, 8,14)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)$ | |
$ 6, 3, 3, 3, 3 $ | $6$ | $6$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,16,11,17,12,18)(13,15,14)$ | |
$ 6, 6, 3, 3 $ | $9$ | $6$ | $( 1, 3, 2)( 4, 6, 5)( 7,14, 8,15, 9,13)(10,16,11,17,12,18)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 3, 2)( 4,12,18)( 5,10,16)( 6,11,17)( 7, 9, 8)(13,15,14)$ | |
$ 6, 3, 3, 3, 3 $ | $12$ | $6$ | $( 1, 3, 2)( 4,12,18)( 5,10,16)( 6,11,17)( 7,14, 8,15, 9,13)$ | |
$ 6, 6, 6 $ | $6$ | $6$ | $( 1, 4, 2, 5, 3, 6)( 7,10, 8,11, 9,12)(13,18,14,16,15,17)$ | |
$ 12, 6 $ | $18$ | $12$ | $( 1, 4, 2, 5, 3, 6)( 7,10,13,18, 9,12,15,17, 8,11,14,16)$ | |
$ 6, 6, 6 $ | $12$ | $6$ | $( 1, 4, 8,11,14,16)( 2, 5, 9,12,15,17)( 3, 6, 7,10,13,18)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $6$ | $2$ | $( 1, 5)( 2, 6)( 3, 4)( 7,11)( 8,12)( 9,10)(13,16)(14,17)(15,18)$ | |
$ 4, 4, 4, 2, 2, 2 $ | $18$ | $4$ | $( 1, 5)( 2, 6)( 3, 4)( 7,11,15,18)( 8,12,13,16)( 9,10,14,17)$ | |
$ 6, 6, 6 $ | $12$ | $6$ | $( 1, 5, 7,11,15,18)( 2, 6, 8,12,13,16)( 3, 4, 9,10,14,17)$ | |
$ 6, 6, 6 $ | $6$ | $6$ | $( 1, 6, 3, 5, 2, 4)( 7,12, 9,11, 8,10)(13,17,15,16,14,18)$ | |
$ 12, 6 $ | $18$ | $12$ | $( 1, 6, 3, 5, 2, 4)( 7,12,14,18, 8,10,15,16, 9,11,13,17)$ | |
$ 6, 6, 6 $ | $12$ | $6$ | $( 1, 6, 9,11,13,17)( 2, 4, 7,12,14,18)( 3, 5, 8,10,15,16)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 7,15)( 2, 8,13)( 3, 9,14)( 4,10,17)( 5,11,18)( 6,12,16)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 8,14)( 2, 9,15)( 3, 7,13)( 4,11,16)( 5,12,17)( 6,10,18)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 9,13)( 2, 7,14)( 3, 8,15)( 4,12,18)( 5,10,16)( 6,11,17)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $216=2^{3} \cdot 3^{3}$ | magma: Order(G);
| |
Cyclic: | no | magma: IsCyclic(G);
| |
Abelian: | no | magma: IsAbelian(G);
| |
Solvable: | yes | magma: IsSolvable(G);
| |
Nilpotency class: | not nilpotent | ||
Label: | 216.157 | magma: IdentifyGroup(G);
| |
Character table: |
1A | 2A | 2B | 2C | 3A1 | 3A-1 | 3B | 3C | 3D1 | 3D-1 | 3E1 | 3E-1 | 4A | 6A1 | 6A-1 | 6B1 | 6B-1 | 6C1 | 6C-1 | 6D | 6E | 6F1 | 6F-1 | 6G1 | 6G-1 | 12A1 | 12A-1 | ||
Size | 1 | 6 | 6 | 9 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 18 | 6 | 6 | 6 | 6 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | |
2 P | 1A | 1A | 1A | 1A | 3A-1 | 3A1 | 3D1 | 3E-1 | 3D-1 | 3B | 3C | 3E1 | 2C | 3A1 | 3A-1 | 3A1 | 3A-1 | 3A-1 | 3A1 | 3B | 3D1 | 3D-1 | 3C | 3E1 | 3E-1 | 6C1 | 6C-1 | |
3 P | 1A | 2A | 2B | 2C | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 4A | 2A | 2A | 2B | 2B | 2C | 2C | 2B | 2B | 2B | 2A | 2A | 2A | 4A | 4A | |
Type | ||||||||||||||||||||||||||||
216.157.1a | R | |||||||||||||||||||||||||||
216.157.1b | R | |||||||||||||||||||||||||||
216.157.1c | R | |||||||||||||||||||||||||||
216.157.1d | R | |||||||||||||||||||||||||||
216.157.1e1 | C | |||||||||||||||||||||||||||
216.157.1e2 | C | |||||||||||||||||||||||||||
216.157.1f1 | C | |||||||||||||||||||||||||||
216.157.1f2 | C | |||||||||||||||||||||||||||
216.157.1g1 | C | |||||||||||||||||||||||||||
216.157.1g2 | C | |||||||||||||||||||||||||||
216.157.1h1 | C | |||||||||||||||||||||||||||
216.157.1h2 | C | |||||||||||||||||||||||||||
216.157.2a | R | |||||||||||||||||||||||||||
216.157.2b1 | C | |||||||||||||||||||||||||||
216.157.2b2 | C | |||||||||||||||||||||||||||
216.157.4a | R | |||||||||||||||||||||||||||
216.157.4b | R | |||||||||||||||||||||||||||
216.157.4c | R | |||||||||||||||||||||||||||
216.157.4d | R | |||||||||||||||||||||||||||
216.157.4e1 | C | |||||||||||||||||||||||||||
216.157.4e2 | C | |||||||||||||||||||||||||||
216.157.4f1 | C | |||||||||||||||||||||||||||
216.157.4f2 | C | |||||||||||||||||||||||||||
216.157.4g1 | C | |||||||||||||||||||||||||||
216.157.4g2 | C | |||||||||||||||||||||||||||
216.157.4h1 | C | |||||||||||||||||||||||||||
216.157.4h2 | C |
magma: CharacterTable(G);