Show commands:
Magma
magma: G := TransitiveGroup(28, 18);
Group action invariants
Degree $n$: | $28$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
Transitive number $t$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
Group: | $D_4\times D_7$ | ||
Parity: | $-1$ | magma: IsEven(G);
| |
Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
$\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
Generators: | (1,25,2,26)(3,23,4,24)(5,21,6,22)(7,20,8,19)(9,18,10,17)(11,16,12,15)(13,27,14,28), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,14)(15,24)(16,23)(17,22)(18,21)(19,20)(25,28)(26,27), (1,14,11,9,7,6,4,2,13,12,10,8,5,3)(15,28,26,23,22,19,18)(16,27,25,24,21,20,17) | 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 $8$: $D_{4}$ x 2, $C_2^3$ $14$: $D_{7}$ $16$: $D_4\times C_2$ $28$: $D_{14}$ x 3 $56$: 28T9 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $D_{4}$
Degree 7: $D_{7}$
Degree 14: $D_{14}$
Low degree siblings
28T18 x 3Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $14$ | $2$ | $( 3,14)( 4,13)( 5,11)( 6,12)( 7,10)( 8, 9)(15,27)(16,28)(17,26)(18,25)(19,24) (20,23)(21,22)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $7$ | $2$ | $( 3,14)( 4,13)( 5,11)( 6,12)( 7,10)( 8, 9)(15,28)(16,27)(17,25)(18,26)(19,23) (20,24)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $7$ | $2$ | $( 1, 2)( 3,13)( 4,14)( 5,12)( 6,11)( 7, 9)( 8,10)(15,27)(16,28)(17,26)(18,25) (19,24)(20,23)(21,22)$ |
$ 14, 14 $ | $2$ | $14$ | $( 1, 3, 5, 8,10,12,13, 2, 4, 6, 7, 9,11,14)(15,17,19,21,23,25,28,16,18,20,22, 24,26,27)$ |
$ 14, 7, 7 $ | $4$ | $14$ | $( 1, 3, 5, 8,10,12,13, 2, 4, 6, 7, 9,11,14)(15,18,19,22,23,26,28) (16,17,20,21,24,25,27)$ |
$ 7, 7, 7, 7 $ | $2$ | $7$ | $( 1, 4, 5, 7,10,11,13)( 2, 3, 6, 8, 9,12,14)(15,18,19,22,23,26,28) (16,17,20,21,24,25,27)$ |
$ 7, 7, 7, 7 $ | $2$ | $7$ | $( 1, 5,10,13, 4, 7,11)( 2, 6, 9,14, 3, 8,12)(15,19,23,28,18,22,26) (16,20,24,27,17,21,25)$ |
$ 14, 7, 7 $ | $4$ | $14$ | $( 1, 5,10,13, 4, 7,11)( 2, 6, 9,14, 3, 8,12)(15,20,23,27,18,21,26,16,19,24,28, 17,22,25)$ |
$ 14, 14 $ | $2$ | $14$ | $( 1, 6,10,14, 4, 8,11, 2, 5, 9,13, 3, 7,12)(15,20,23,27,18,21,26,16,19,24,28, 17,22,25)$ |
$ 14, 7, 7 $ | $4$ | $14$ | $( 1, 7,13, 5,11, 4,10)( 2, 8,14, 6,12, 3, 9)(15,21,28,20,26,17,23,16,22,27,19, 25,18,24)$ |
$ 7, 7, 7, 7 $ | $2$ | $7$ | $( 1, 7,13, 5,11, 4,10)( 2, 8,14, 6,12, 3, 9)(15,22,28,19,26,18,23) (16,21,27,20,25,17,24)$ |
$ 14, 14 $ | $2$ | $14$ | $( 1, 8,13, 6,11, 3,10, 2, 7,14, 5,12, 4, 9)(15,21,28,20,26,17,23,16,22,27,19, 25,18,24)$ |
$ 28 $ | $4$ | $28$ | $( 1,15, 3,17, 5,19, 8,21,10,23,12,25,13,28, 2,16, 4,18, 6,20, 7,22, 9,24,11, 26,14,27)$ |
$ 14, 14 $ | $4$ | $14$ | $( 1,15, 4,18, 5,19, 7,22,10,23,11,26,13,28)( 2,16, 3,17, 6,20, 8,21, 9,24,12, 25,14,27)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $14$ | $2$ | $( 1,15)( 2,16)( 3,27)( 4,28)( 5,26)( 6,25)( 7,23)( 8,24)( 9,21)(10,22)(11,19) (12,20)(13,18)(14,17)$ |
$ 4, 4, 4, 4, 4, 4, 4 $ | $14$ | $4$ | $( 1,15, 2,16)( 3,27, 4,28)( 5,26, 6,25)( 7,23, 8,24)( 9,21,10,22)(11,19,12,20) (13,18,14,17)$ |
$ 28 $ | $4$ | $28$ | $( 1,17, 8,23,13,16, 6,22,11,27, 3,19,10,25, 2,18, 7,24,14,15, 5,21,12,28, 4, 20, 9,26)$ |
$ 14, 14 $ | $4$ | $14$ | $( 1,17, 7,24,13,16, 5,21,11,27, 4,20,10,25)( 2,18, 8,23,14,15, 6,22,12,28, 3, 19, 9,26)$ |
$ 14, 14 $ | $4$ | $14$ | $( 1,19,11,15, 7,26, 4,22,13,18,10,28, 5,23)( 2,20,12,16, 8,25, 3,21,14,17, 9, 27, 6,24)$ |
$ 28 $ | $4$ | $28$ | $( 1,19,12,16, 7,26, 3,21,13,18, 9,27, 5,23, 2,20,11,15, 8,25, 4,22,14,17,10, 28, 6,24)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1,21)( 2,22)( 3,23)( 4,24)( 5,25)( 6,26)( 7,27)( 8,28)( 9,15)(10,16)(11,17) (12,18)(13,20)(14,19)$ |
$ 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,21, 2,22)( 3,23, 4,24)( 5,25, 6,26)( 7,27, 8,28)( 9,15,10,16)(11,17,12,18) (13,20,14,19)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $112=2^{4} \cdot 7$ | magma: Order(G);
| |
Cyclic: | no | magma: IsCyclic(G);
| |
Abelian: | no | magma: IsAbelian(G);
| |
Solvable: | yes | magma: IsSolvable(G);
| |
Nilpotency class: | not nilpotent | ||
Label: | 112.31 | magma: IdentifyGroup(G);
|
Character table: not available. |
magma: CharacterTable(G);