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Magma
magma: G := TransitiveGroup(28, 31);
Group action invariants
| Degree $n$: | $28$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $31$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_7\times S_4$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $7$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,24,15,6,25,20,11,2,21,16,7,26,17,12,3,22,13,8,27,18,9,4,23,14,5,28,19,10), (1,27,24,17,15,12,5,3,28,21,19,16,9,7,4,25,23,20,13,11,8)(2,26,22,18,14,10,6) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $6$: $S_3$ $7$: $C_7$ $14$: $C_{14}$ $24$: $S_4$ $42$: 21T6 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: $S_4$
Degree 7: $C_7$
Degree 14: None
Low degree siblings
42T34, 42T35Siblings 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, 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 $ | $6$ | $2$ | $( 3, 4)( 7, 8)(11,12)(15,16)(19,20)(23,24)(27,28)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $3$ | $( 2, 3, 4)( 6, 7, 8)(10,11,12)(14,15,16)(18,19,20)(22,23,24)(26,27,28)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)$ | |
| $ 4, 4, 4, 4, 4, 4, 4 $ | $6$ | $4$ | $( 1, 2, 3, 4)( 5, 6, 7, 8)( 9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24) (25,26,27,28)$ | |
| $ 7, 7, 7, 7 $ | $1$ | $7$ | $( 1, 5, 9,13,17,21,25)( 2, 6,10,14,18,22,26)( 3, 7,11,15,19,23,27) ( 4, 8,12,16,20,24,28)$ | |
| $ 14, 7, 7 $ | $6$ | $14$ | $( 1, 5, 9,13,17,21,25)( 2, 6,10,14,18,22,26)( 3, 8,11,16,19,24,27, 4, 7,12,15, 20,23,28)$ | |
| $ 21, 7 $ | $8$ | $21$ | $( 1, 5, 9,13,17,21,25)( 2, 7,12,14,19,24,26, 3, 8,10,15,20,22,27, 4, 6,11,16, 18,23,28)$ | |
| $ 14, 14 $ | $3$ | $14$ | $( 1, 6, 9,14,17,22,25, 2, 5,10,13,18,21,26)( 3, 8,11,16,19,24,27, 4, 7,12,15, 20,23,28)$ | |
| $ 28 $ | $6$ | $28$ | $( 1, 6,11,16,17,22,27, 4, 5,10,15,20,21,26, 3, 8, 9,14,19,24,25, 2, 7,12,13, 18,23,28)$ | |
| $ 7, 7, 7, 7 $ | $1$ | $7$ | $( 1, 9,17,25, 5,13,21)( 2,10,18,26, 6,14,22)( 3,11,19,27, 7,15,23) ( 4,12,20,28, 8,16,24)$ | |
| $ 14, 7, 7 $ | $6$ | $14$ | $( 1, 9,17,25, 5,13,21)( 2,10,18,26, 6,14,22)( 3,12,19,28, 7,16,23, 4,11,20,27, 8,15,24)$ | |
| $ 21, 7 $ | $8$ | $21$ | $( 1, 9,17,25, 5,13,21)( 2,11,20,26, 7,16,22, 3,12,18,27, 8,14,23, 4,10,19,28, 6,15,24)$ | |
| $ 14, 14 $ | $3$ | $14$ | $( 1,10,17,26, 5,14,21, 2, 9,18,25, 6,13,22)( 3,12,19,28, 7,16,23, 4,11,20,27, 8,15,24)$ | |
| $ 28 $ | $6$ | $28$ | $( 1,10,19,28, 5,14,23, 4, 9,18,27, 8,13,22, 3,12,17,26, 7,16,21, 2,11,20,25, 6,15,24)$ | |
| $ 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,13,25, 9,21, 5,17)( 2,14,26,10,22, 6,18)( 3,15,27,11,23, 7,19) ( 4,16,28,12,24, 8,20)$ | |
| $ 14, 7, 7 $ | $6$ | $14$ | $( 1,13,25, 9,21, 5,17)( 2,14,26,10,22, 6,18)( 3,16,27,12,23, 8,19, 4,15,28,11, 24, 7,20)$ | |
| $ 21, 7 $ | $8$ | $21$ | $( 1,13,25, 9,21, 5,17)( 2,15,28,10,23, 8,18, 3,16,26,11,24, 6,19, 4,14,27,12, 22, 7,20)$ | |
| $ 14, 14 $ | $3$ | $14$ | $( 1,14,25,10,21, 6,17, 2,13,26, 9,22, 5,18)( 3,16,27,12,23, 8,19, 4,15,28,11, 24, 7,20)$ | |
| $ 28 $ | $6$ | $28$ | $( 1,14,27,12,21, 6,19, 4,13,26,11,24, 5,18, 3,16,25,10,23, 8,17, 2,15,28, 9, 22, 7,20)$ | |
| $ 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,17, 5,21, 9,25,13)( 2,18, 6,22,10,26,14)( 3,19, 7,23,11,27,15) ( 4,20, 8,24,12,28,16)$ | |
| $ 14, 7, 7 $ | $6$ | $14$ | $( 1,17, 5,21, 9,25,13)( 2,18, 6,22,10,26,14)( 3,20, 7,24,11,28,15, 4,19, 8,23, 12,27,16)$ | |
| $ 21, 7 $ | $8$ | $21$ | $( 1,17, 5,21, 9,25,13)( 2,19, 8,22,11,28,14, 3,20, 6,23,12,26,15, 4,18, 7,24, 10,27,16)$ | |
| $ 14, 14 $ | $3$ | $14$ | $( 1,18, 5,22, 9,26,13, 2,17, 6,21,10,25,14)( 3,20, 7,24,11,28,15, 4,19, 8,23, 12,27,16)$ | |
| $ 28 $ | $6$ | $28$ | $( 1,18, 7,24, 9,26,15, 4,17, 6,23,12,25,14, 3,20, 5,22,11,28,13, 2,19, 8,21, 10,27,16)$ | |
| $ 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,21,13, 5,25,17, 9)( 2,22,14, 6,26,18,10)( 3,23,15, 7,27,19,11) ( 4,24,16, 8,28,20,12)$ | |
| $ 14, 7, 7 $ | $6$ | $14$ | $( 1,21,13, 5,25,17, 9)( 2,22,14, 6,26,18,10)( 3,24,15, 8,27,20,11, 4,23,16, 7, 28,19,12)$ | |
| $ 21, 7 $ | $8$ | $21$ | $( 1,21,13, 5,25,17, 9)( 2,23,16, 6,27,20,10, 3,24,14, 7,28,18,11, 4,22,15, 8, 26,19,12)$ | |
| $ 14, 14 $ | $3$ | $14$ | $( 1,22,13, 6,25,18, 9, 2,21,14, 5,26,17,10)( 3,24,15, 8,27,20,11, 4,23,16, 7, 28,19,12)$ | |
| $ 28 $ | $6$ | $28$ | $( 1,22,15, 8,25,18,11, 4,21,14, 7,28,17,10, 3,24,13, 6,27,20, 9, 2,23,16, 5, 26,19,12)$ | |
| $ 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,25,21,17,13, 9, 5)( 2,26,22,18,14,10, 6)( 3,27,23,19,15,11, 7) ( 4,28,24,20,16,12, 8)$ | |
| $ 14, 7, 7 $ | $6$ | $14$ | $( 1,25,21,17,13, 9, 5)( 2,26,22,18,14,10, 6)( 3,28,23,20,15,12, 7, 4,27,24,19, 16,11, 8)$ | |
| $ 21, 7 $ | $8$ | $21$ | $( 1,25,21,17,13, 9, 5)( 2,27,24,18,15,12, 6, 3,28,22,19,16,10, 7, 4,26,23,20, 14,11, 8)$ | |
| $ 14, 14 $ | $3$ | $14$ | $( 1,26,21,18,13,10, 5, 2,25,22,17,14, 9, 6)( 3,28,23,20,15,12, 7, 4,27,24,19, 16,11, 8)$ | |
| $ 28 $ | $6$ | $28$ | $( 1,26,23,20,13,10, 7, 4,25,22,19,16, 9, 6, 3,28,21,18,15,12, 5, 2,27,24,17, 14,11, 8)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $168=2^{3} \cdot 3 \cdot 7$ | magma: Order(G);
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| Cyclic: | no | magma: IsCyclic(G);
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| Abelian: | no | magma: IsAbelian(G);
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| Solvable: | yes | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 168.45 | magma: IdentifyGroup(G);
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| Character table: | 35 x 35 character table |
magma: CharacterTable(G);