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
$\card{(G/N)}$ 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
Label | Cycle Type | Size | Order | Index | Representative |
1A | $1^{12}$ | $1$ | $1$ | $0$ | $()$ |
2A | $2^{6}$ | $1$ | $2$ | $6$ | $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$ |
3A1 | $3^{4}$ | $1$ | $3$ | $8$ | $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$ |
3A-1 | $3^{4}$ | $1$ | $3$ | $8$ | $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$ |
3B | $3^{2},1^{6}$ | $2$ | $3$ | $4$ | $( 1, 9, 5)( 3,11, 7)$ |
3C1 | $3^{4}$ | $2$ | $3$ | $8$ | $( 1, 5, 9)( 2,10, 6)( 3, 7,11)( 4,12, 8)$ |
3C-1 | $3^{2},1^{6}$ | $2$ | $3$ | $4$ | $( 2, 6,10)( 4, 8,12)$ |
4A1 | $4^{3}$ | $3$ | $4$ | $9$ | $( 1,10, 7, 4)( 2,11, 8, 5)( 3,12, 9, 6)$ |
4A-1 | $4^{3}$ | $3$ | $4$ | $9$ | $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12)$ |
6A1 | $6^{2}$ | $1$ | $6$ | $10$ | $( 1,11, 9, 7, 5, 3)( 2,12,10, 8, 6, 4)$ |
6A-1 | $6^{2}$ | $1$ | $6$ | $10$ | $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$ |
6B | $6^{2}$ | $2$ | $6$ | $10$ | $( 1,11, 9, 7, 5, 3)( 2, 4, 6, 8,10,12)$ |
6C1 | $6,2^{3}$ | $2$ | $6$ | $8$ | $( 1, 3, 5, 7, 9,11)( 2, 8)( 4,10)( 6,12)$ |
6C-1 | $6,2^{3}$ | $2$ | $6$ | $8$ | $( 1, 7)( 2,12,10, 8, 6, 4)( 3, 9)( 5,11)$ |
12A1 | $12$ | $3$ | $12$ | $11$ | $( 1,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$ |
12A-1 | $12$ | $3$ | $12$ | $11$ | $( 1, 6,11, 4, 9, 2, 7,12, 5,10, 3, 8)$ |
12A5 | $12$ | $3$ | $12$ | $11$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12)$ |
12A-5 | $12$ | $3$ | $12$ | $11$ | $( 1, 8, 3,10, 5,12, 7, 2, 9, 4,11, 6)$ |
Malle's constant $a(G)$: $1/4$
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: |
1A | 2A | 3A1 | 3A-1 | 3B | 3C1 | 3C-1 | 4A1 | 4A-1 | 6A1 | 6A-1 | 6B | 6C1 | 6C-1 | 12A1 | 12A-1 | 12A5 | 12A-5 | ||
Size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | |
2 P | 1A | 1A | 3A-1 | 3A1 | 3C1 | 3B | 3C-1 | 2A | 2A | 3A1 | 3A-1 | 3B | 3C1 | 3C-1 | 6A1 | 6A1 | 6A-1 | 6A-1 | |
3 P | 1A | 2A | 1A | 1A | 1A | 1A | 1A | 4A-1 | 4A1 | 2A | 2A | 2A | 2A | 2A | 4A1 | 4A-1 | 4A-1 | 4A1 | |
Type | |||||||||||||||||||
36.6.1a | R | ||||||||||||||||||
36.6.1b | R | ||||||||||||||||||
36.6.1c1 | C | ||||||||||||||||||
36.6.1c2 | C | ||||||||||||||||||
36.6.1d1 | C | ||||||||||||||||||
36.6.1d2 | C | ||||||||||||||||||
36.6.1e1 | C | ||||||||||||||||||
36.6.1e2 | C | ||||||||||||||||||
36.6.1f1 | C | ||||||||||||||||||
36.6.1f2 | C | ||||||||||||||||||
36.6.1f3 | C | ||||||||||||||||||
36.6.1f4 | C | ||||||||||||||||||
36.6.2a | R | ||||||||||||||||||
36.6.2b | S | ||||||||||||||||||
36.6.2c1 | C | ||||||||||||||||||
36.6.2c2 | C | ||||||||||||||||||
36.6.2d1 | C | ||||||||||||||||||
36.6.2d2 | C |
magma: CharacterTable(G);