Show commands:
Magma
magma: G := TransitiveGroup(12, 10);
Group action invariants
Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
Transitive number $t$: | $10$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
Group: | $S_3 \times C_2^2$ | ||
CHM label: | $S(3)[x]E(4)$ | ||
Parity: | $1$ | magma: IsEven(G);
| |
Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
$\card{\Aut(F/K)}$: | $4$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
Generators: | (1,10)(2,5)(3,12)(4,7)(6,9)(8,11), (1,5)(2,10)(4,8)(7,11), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_2^2$ x 7 $6$: $S_3$ $8$: $C_2^3$ $12$: $D_{6}$ x 3 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 3: $S_3$
Degree 4: $C_2^2$
Degree 6: $D_{6}$ x 3
Low degree siblings
12T10 x 3, 24T11Siblings 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 $ | $1$ | $1$ | $()$ | |
$ 2, 2, 2, 2, 1, 1, 1, 1 $ | $3$ | $2$ | $( 2, 6)( 3,11)( 5, 9)( 8,12)$ | |
$ 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$ | |
$ 6, 6 $ | $2$ | $6$ | $( 1, 2, 9,10, 5, 6)( 3, 4,11,12, 7, 8)$ | |
$ 6, 6 $ | $2$ | $6$ | $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$ | |
$ 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 3)( 2, 8)( 4, 6)( 5,11)( 7, 9)(10,12)$ | |
$ 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9)$ | |
$ 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$ | |
$ 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$ | |
$ 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$ | |
$ 6, 6 $ | $2$ | $6$ | $( 1, 8, 9, 4, 5,12)( 2, 3,10,11, 6, 7)$ | |
$ 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,10)( 2, 5)( 3,12)( 4, 7)( 6, 9)( 8,11)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $24=2^{3} \cdot 3$ | magma: Order(G);
| |
Cyclic: | no | magma: IsCyclic(G);
| |
Abelian: | no | magma: IsAbelian(G);
| |
Solvable: | yes | magma: IsSolvable(G);
| |
Nilpotency class: | not nilpotent | ||
Label: | 24.14 | magma: IdentifyGroup(G);
| |
Character table: |
1A | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 6A | 6B | 6C | ||
Size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 2 | |
2 P | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 3A | 3A | 3A | 3A | |
3 P | 1A | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 1A | 2A | 2B | 2C | |
Type | |||||||||||||
24.14.1a | R | ||||||||||||
24.14.1b | R | ||||||||||||
24.14.1c | R | ||||||||||||
24.14.1d | R | ||||||||||||
24.14.1e | R | ||||||||||||
24.14.1f | R | ||||||||||||
24.14.1g | R | ||||||||||||
24.14.1h | R | ||||||||||||
24.14.2a | R | ||||||||||||
24.14.2b | R | ||||||||||||
24.14.2c | R | ||||||||||||
24.14.2d | R |
magma: CharacterTable(G);