Show commands:
Magma
magma: G := TransitiveGroup(12, 19);
Group action invariants
Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
Transitive number $t$: | $19$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
Group: | $C_3\times (C_3 : C_4)$ | ||
CHM label: | $[3^{2}]4$ | ||
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: | (2,6,10)(4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $4$: $C_4$ $6$: $S_3$, $C_6$ $12$: $C_{12}$, $C_3 : C_4$ $18$: $S_3\times C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 4: $C_4$
Degree 6: $S_3\times C_3$
Low degree siblings
36T5Siblings 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 $ | $1$ | $1$ | $()$ |
$ 3, 3, 1, 1, 1, 1, 1, 1 $ | $2$ | $3$ | $( 2, 6,10)( 4, 8,12)$ |
$ 3, 3, 1, 1, 1, 1, 1, 1 $ | $2$ | $3$ | $( 2,10, 6)( 4,12, 8)$ |
$ 12 $ | $3$ | $12$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12)$ |
$ 4, 4, 4 $ | $3$ | $4$ | $( 1, 2, 7, 8)( 3, 4, 9,10)( 5, 6,11,12)$ |
$ 12 $ | $3$ | $12$ | $( 1, 2,11,12, 9,10, 7, 8, 5, 6, 3, 4)$ |
$ 6, 6 $ | $1$ | $6$ | $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$ |
$ 6, 2, 2, 2 $ | $2$ | $6$ | $( 1, 3, 5, 7, 9,11)( 2, 8)( 4,10)( 6,12)$ |
$ 6, 6 $ | $2$ | $6$ | $( 1, 3, 5, 7, 9,11)( 2,12,10, 8, 6, 4)$ |
$ 12 $ | $3$ | $12$ | $( 1, 4, 3, 6, 5, 8, 7,10, 9,12,11, 2)$ |
$ 4, 4, 4 $ | $3$ | $4$ | $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12)$ |
$ 12 $ | $3$ | $12$ | $( 1, 4,11, 2, 9,12, 7,10, 5, 8, 3, 6)$ |
$ 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$ |
$ 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 5, 9)( 2,10, 6)( 3, 7,11)( 4,12, 8)$ |
$ 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$ |
$ 6, 2, 2, 2 $ | $2$ | $6$ | $( 1, 7)( 2,12,10, 8, 6, 4)( 3, 9)( 5,11)$ |
$ 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$ |
$ 6, 6 $ | $1$ | $6$ | $( 1,11, 9, 7, 5, 3)( 2,12,10, 8, 6, 4)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $36=2^{2} \cdot 3^{2}$ | magma: Order(G);
| |
Cyclic: | no | magma: IsCyclic(G);
| |
Abelian: | no | magma: IsAbelian(G);
| |
Solvable: | yes | magma: IsSolvable(G);
| |
Nilpotency class: | not nilpotent | ||
Label: | 36.6 | magma: IdentifyGroup(G);
|
Character table: |
2 2 1 1 2 2 2 2 1 1 2 2 2 2 1 2 1 2 2 3 2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 2 2 2 1a 3a 3b 12a 4a 12b 6a 6b 6c 12c 4b 12d 3c 3d 2a 6d 3e 6e 2P 1a 3b 3a 6a 2a 6e 3c 3a 3d 6a 2a 6e 3e 3d 1a 3b 3c 3e 3P 1a 1a 1a 4b 4b 4b 2a 2a 2a 4a 4a 4a 1a 1a 2a 2a 1a 2a 5P 1a 3b 3a 12b 4a 12a 6e 6d 6c 12d 4b 12c 3e 3d 2a 6b 3c 6a 7P 1a 3a 3b 12c 4b 12d 6a 6b 6c 12a 4a 12b 3c 3d 2a 6d 3e 6e 11P 1a 3b 3a 12d 4b 12c 6e 6d 6c 12b 4a 12a 3e 3d 2a 6b 3c 6a 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 B B B -1 -1 -1 -B -B -B 1 1 -1 -1 1 -1 X.4 1 1 1 -B -B -B -1 -1 -1 B B B 1 1 -1 -1 1 -1 X.5 1 A /A -/A -1 -A A /A 1 -/A -1 -A /A 1 1 A A /A X.6 1 /A A -A -1 -/A /A A 1 -A -1 -/A A 1 1 /A /A A X.7 1 A /A /A 1 A A /A 1 /A 1 A /A 1 1 A A /A X.8 1 /A A A 1 /A /A A 1 A 1 /A A 1 1 /A /A A X.9 1 A /A C B -/C -A -/A -1 -C -B /C /A 1 -1 -A A -/A X.10 1 A /A -C -B /C -A -/A -1 C B -/C /A 1 -1 -A A -/A X.11 1 /A A -/C B C -/A -A -1 /C -B -C A 1 -1 -/A /A -A X.12 1 /A A /C -B -C -/A -A -1 -/C B C A 1 -1 -/A /A -A X.13 2 -1 -1 . . . -2 1 1 . . . 2 -1 -2 1 2 -2 X.14 2 -1 -1 . . . 2 -1 -1 . . . 2 -1 2 -1 2 2 X.15 2 -/A -A . . . D A 1 . . . -/D -1 -2 /A -D /D X.16 2 -A -/A . . . /D /A 1 . . . -D -1 -2 A -/D D X.17 2 -/A -A . . . -D -A -1 . . . -/D -1 2 -/A -D -/D X.18 2 -A -/A . . . -/D -/A -1 . . . -D -1 2 -A -/D -D A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = -E(4) = -Sqrt(-1) = -i C = -E(12)^7 D = -2*E(3) = 1-Sqrt(-3) = 1-i3 |
magma: CharacterTable(G);