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Magma
magma: G := TransitiveGroup(28, 1);
Group action invariants
| Degree $n$: | $28$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $1$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{28}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $28$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,20,10,28,17,7,26,15,6,24,13,4,22,11,2,19,9,27,18,8,25,16,5,23,14,3,21,12) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $7$: $C_7$ $14$: $C_{14}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $C_4$
Degree 7: $C_7$
Degree 14: $C_{14}$
Low degree siblings
There are no siblings 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, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)$ |
| $ 28 $ | $1$ | $28$ | $( 1, 3, 5, 8, 9,11,13,15,17,20,21,23,25,27, 2, 4, 6, 7,10,12,14,16,18,19,22, 24,26,28)$ |
| $ 28 $ | $1$ | $28$ | $( 1, 4, 5, 7, 9,12,13,16,17,19,21,24,25,28, 2, 3, 6, 8,10,11,14,15,18,20,22, 23,26,27)$ |
| $ 14, 14 $ | $1$ | $14$ | $( 1, 5, 9,13,17,21,25, 2, 6,10,14,18,22,26)( 3, 8,11,15,20,23,27, 4, 7,12,16, 19,24,28)$ |
| $ 7, 7, 7, 7 $ | $1$ | $7$ | $( 1, 6, 9,14,17,22,25)( 2, 5,10,13,18,21,26)( 3, 7,11,16,20,24,27) ( 4, 8,12,15,19,23,28)$ |
| $ 28 $ | $1$ | $28$ | $( 1, 7,13,19,25, 3,10,15,22,27, 5,12,17,24, 2, 8,14,20,26, 4, 9,16,21,28, 6, 11,18,23)$ |
| $ 28 $ | $1$ | $28$ | $( 1, 8,13,20,25, 4,10,16,22,28, 5,11,17,23, 2, 7,14,19,26, 3, 9,15,21,27, 6, 12,18,24)$ |
| $ 7, 7, 7, 7 $ | $1$ | $7$ | $( 1, 9,17,25, 6,14,22)( 2,10,18,26, 5,13,21)( 3,11,20,27, 7,16,24) ( 4,12,19,28, 8,15,23)$ |
| $ 14, 14 $ | $1$ | $14$ | $( 1,10,17,26, 6,13,22, 2, 9,18,25, 5,14,21)( 3,12,20,28, 7,15,24, 4,11,19,27, 8,16,23)$ |
| $ 28 $ | $1$ | $28$ | $( 1,11,21, 4,14,24, 5,15,25, 7,18,28, 9,20, 2,12,22, 3,13,23, 6,16,26, 8,17, 27,10,19)$ |
| $ 28 $ | $1$ | $28$ | $( 1,12,21, 3,14,23, 5,16,25, 8,18,27, 9,19, 2,11,22, 4,13,24, 6,15,26, 7,17, 28,10,20)$ |
| $ 14, 14 $ | $1$ | $14$ | $( 1,13,25,10,22, 5,17, 2,14,26, 9,21, 6,18)( 3,15,27,12,24, 8,20, 4,16,28,11, 23, 7,19)$ |
| $ 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,14,25, 9,22, 6,17)( 2,13,26,10,21, 5,18)( 3,16,27,11,24, 7,20) ( 4,15,28,12,23, 8,19)$ |
| $ 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,15, 2,16)( 3,17, 4,18)( 5,20, 6,19)( 7,22, 8,21)( 9,23,10,24)(11,25,12,26) (13,27,14,28)$ |
| $ 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,16, 2,15)( 3,18, 4,17)( 5,19, 6,20)( 7,21, 8,22)( 9,24,10,23)(11,26,12,25) (13,28,14,27)$ |
| $ 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,17, 6,22, 9,25,14)( 2,18, 5,21,10,26,13)( 3,20, 7,24,11,27,16) ( 4,19, 8,23,12,28,15)$ |
| $ 14, 14 $ | $1$ | $14$ | $( 1,18, 6,21, 9,26,14, 2,17, 5,22,10,25,13)( 3,19, 7,23,11,28,16, 4,20, 8,24, 12,27,15)$ |
| $ 28 $ | $1$ | $28$ | $( 1,19,10,27,17, 8,26,16, 6,23,13, 3,22,12, 2,20, 9,28,18, 7,25,15, 5,24,14, 4,21,11)$ |
| $ 28 $ | $1$ | $28$ | $( 1,20,10,28,17, 7,26,15, 6,24,13, 4,22,11, 2,19, 9,27,18, 8,25,16, 5,23,14, 3,21,12)$ |
| $ 14, 14 $ | $1$ | $14$ | $( 1,21,14, 5,25,18, 9, 2,22,13, 6,26,17,10)( 3,23,16, 8,27,19,11, 4,24,15, 7, 28,20,12)$ |
| $ 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,22,14, 6,25,17, 9)( 2,21,13, 5,26,18,10)( 3,24,16, 7,27,20,11) ( 4,23,15, 8,28,19,12)$ |
| $ 28 $ | $1$ | $28$ | $( 1,23,18,11, 6,28,21,16, 9, 4,26,20,14, 8, 2,24,17,12, 5,27,22,15,10, 3,25, 19,13, 7)$ |
| $ 28 $ | $1$ | $28$ | $( 1,24,18,12, 6,27,21,15, 9, 3,26,19,14, 7, 2,23,17,11, 5,28,22,16,10, 4,25, 20,13, 8)$ |
| $ 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,25,22,17,14, 9, 6)( 2,26,21,18,13,10, 5)( 3,27,24,20,16,11, 7) ( 4,28,23,19,15,12, 8)$ |
| $ 14, 14 $ | $1$ | $14$ | $( 1,26,22,18,14,10, 6, 2,25,21,17,13, 9, 5)( 3,28,24,19,16,12, 7, 4,27,23,20, 15,11, 8)$ |
| $ 28 $ | $1$ | $28$ | $( 1,27,26,23,22,20,18,15,14,11,10, 8, 6, 3, 2,28,25,24,21,19,17,16,13,12, 9, 7, 5, 4)$ |
| $ 28 $ | $1$ | $28$ | $( 1,28,26,24,22,19,18,16,14,12,10, 7, 6, 4, 2,27,25,23,21,20,17,15,13,11, 9, 8, 5, 3)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $28=2^{2} \cdot 7$ | magma: Order(G);
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| Cyclic: | yes | magma: IsCyclic(G);
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| Abelian: | yes | magma: IsAbelian(G);
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| Solvable: | yes | magma: IsSolvable(G);
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| Nilpotency class: | $1$ | ||
| Label: | 28.2 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);