Show commands:
Magma
magma: G := TransitiveGroup(18, 85);
Group action invariants
Degree $n$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
Transitive number $t$: | $85$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
Group: | $C_3^3:C_6$ | ||
Parity: | $-1$ | magma: IsEven(G);
| |
Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
$\card{\Aut(F/K)}$: | $6$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
Generators: | (1,3)(2,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,16)(17,18), (1,14,8,4,15,9,17,11,5)(2,13,7,3,16,10,18,12,6) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $18$: $S_3\times C_3$ $54$: $C_3^2 : C_6$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $C_3$
Degree 6: $C_6$
Degree 9: $(C_3^3:C_3):C_2$
Low degree siblings
9T22 x 3, 18T85 x 2, 27T53 x 3, 27T62, 27T63Siblings 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 $ | $1$ | $1$ | $()$ | |
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $3$ | $(11,14,15)(12,13,16)$ | |
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $6$ | $3$ | $( 5, 8, 9)( 6, 7,10)(11,14,15)(12,13,16)$ | |
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $6$ | $3$ | $( 5, 8, 9)( 6, 7,10)(11,15,14)(12,16,13)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $27$ | $2$ | $( 1, 2)( 3,17)( 4,18)( 5, 6)( 7, 9)( 8,10)(11,12)(13,15)(14,16)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 4,17)( 2, 3,18)( 5, 8, 9)( 6, 7,10)(11,14,15)(12,13,16)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $6$ | $3$ | $( 1, 4,17)( 2, 3,18)( 5, 8, 9)( 6, 7,10)(11,15,14)(12,16,13)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $9$ | $3$ | $( 1, 5,11)( 2, 6,12)( 3, 7,13)( 4, 8,14)( 9,15,17)(10,16,18)$ | |
$ 9, 9 $ | $18$ | $9$ | $( 1, 5,11, 4, 8,14,17, 9,15)( 2, 6,12, 3, 7,13,18,10,16)$ | |
$ 6, 6, 6 $ | $27$ | $6$ | $( 1, 6,11, 2, 5,12)( 3, 9,13,17, 7,15)( 4,10,14,18, 8,16)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $9$ | $3$ | $( 1,11, 5)( 2,12, 6)( 3,13, 7)( 4,14, 8)( 9,17,15)(10,18,16)$ | |
$ 9, 9 $ | $18$ | $9$ | $( 1,11, 8, 4,14, 9,17,15, 5)( 2,12, 7, 3,13,10,18,16, 6)$ | |
$ 6, 6, 6 $ | $27$ | $6$ | $( 1,12, 5, 2,11, 6)( 3,15, 7,17,13, 9)( 4,16, 8,18,14,10)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $162=2 \cdot 3^{4}$ | magma: Order(G);
| |
Cyclic: | no | magma: IsCyclic(G);
| |
Abelian: | no | magma: IsAbelian(G);
| |
Solvable: | yes | magma: IsSolvable(G);
| |
Nilpotency class: | not nilpotent | ||
Label: | 162.11 | magma: IdentifyGroup(G);
| |
Character table: |
1A | 2A | 3A | 3B | 3C | 3D | 3E | 3F1 | 3F-1 | 6A1 | 6A-1 | 9A1 | 9A-1 | ||
Size | 1 | 27 | 2 | 6 | 6 | 6 | 6 | 9 | 9 | 27 | 27 | 18 | 18 | |
2 P | 1A | 1A | 3A | 3D | 3B | 3C | 3E | 3F-1 | 3F1 | 3F1 | 3F-1 | 9A-1 | 9A1 | |
3 P | 1A | 2A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 2A | 2A | 3A | 3A | |
Type | ||||||||||||||
162.11.1a | R | |||||||||||||
162.11.1b | R | |||||||||||||
162.11.1c1 | C | |||||||||||||
162.11.1c2 | C | |||||||||||||
162.11.1d1 | C | |||||||||||||
162.11.1d2 | C | |||||||||||||
162.11.2a | R | |||||||||||||
162.11.2b1 | C | |||||||||||||
162.11.2b2 | C | |||||||||||||
162.11.6a | R | |||||||||||||
162.11.6b | R | |||||||||||||
162.11.6c | R | |||||||||||||
162.11.6d | R |
magma: CharacterTable(G);