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Magma
magma: G := TransitiveGroup(32, 33);
Group action invariants
| Degree $n$: | $32$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $33$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{32}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $32$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,29,26,24,18,15,11,7,4,32,28,22,19,13,9,6,2,30,25,23,17,16,12,8,3,31,27,21,20,14,10,5), (1,19,3,18,2,20,4,17)(5,22,8,24,6,21,7,23)(9,27,11,25,10,28,12,26)(13,31,15,30,14,32,16,29) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $8$: $C_8$ $16$: $C_{16}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $C_4$
Degree 8: $C_8$
Degree 16: $C_{16}$
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, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,15,14,16)(17,19,18,20)(21,23,22,24) (25,28,26,27)(29,31,30,32)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,12,10,11)(13,16,14,15)(17,20,18,19)(21,24,22,23) (25,27,26,28)(29,32,30,31)$ |
| $ 32 $ | $1$ | $32$ | $( 1, 5,10,14,20,21,27,31, 3, 8,12,16,17,23,25,30, 2, 6, 9,13,19,22,28,32, 4, 7,11,15,18,24,26,29)$ |
| $ 32 $ | $1$ | $32$ | $( 1, 6,10,13,20,22,27,32, 3, 7,12,15,17,24,25,29, 2, 5, 9,14,19,21,28,31, 4, 8,11,16,18,23,26,30)$ |
| $ 32 $ | $1$ | $32$ | $( 1, 7, 9,16,20,24,28,30, 3, 5,11,13,17,21,26,32, 2, 8,10,15,19,23,27,29, 4, 6,12,14,18,22,25,31)$ |
| $ 32 $ | $1$ | $32$ | $( 1, 8, 9,15,20,23,28,29, 3, 6,11,14,17,22,26,31, 2, 7,10,16,19,24,27,30, 4, 5,12,13,18,21,25,32)$ |
| $ 16, 16 $ | $1$ | $16$ | $( 1, 9,20,28, 3,11,17,26, 2,10,19,27, 4,12,18,25)( 5,13,21,32, 8,15,23,29, 6, 14,22,31, 7,16,24,30)$ |
| $ 16, 16 $ | $1$ | $16$ | $( 1,10,20,27, 3,12,17,25, 2, 9,19,28, 4,11,18,26)( 5,14,21,31, 8,16,23,30, 6, 13,22,32, 7,15,24,29)$ |
| $ 16, 16 $ | $1$ | $16$ | $( 1,11,19,25, 3,10,18,28, 2,12,20,26, 4, 9,17,27)( 5,15,22,30, 8,14,24,32, 6, 16,21,29, 7,13,23,31)$ |
| $ 16, 16 $ | $1$ | $16$ | $( 1,12,19,26, 3, 9,18,27, 2,11,20,25, 4,10,17,28)( 5,16,22,29, 8,13,24,31, 6, 15,21,30, 7,14,23,32)$ |
| $ 32 $ | $1$ | $32$ | $( 1,13,27, 7,17,29, 9,21, 4,16,26, 6,20,32,12,24, 2,14,28, 8,18,30,10,22, 3, 15,25, 5,19,31,11,23)$ |
| $ 32 $ | $1$ | $32$ | $( 1,14,27, 8,17,30, 9,22, 4,15,26, 5,20,31,12,23, 2,13,28, 7,18,29,10,21, 3, 16,25, 6,19,32,11,24)$ |
| $ 32 $ | $1$ | $32$ | $( 1,15,28, 6,17,31,10,24, 4,13,25, 8,20,29,11,22, 2,16,27, 5,18,32, 9,23, 3, 14,26, 7,19,30,12,21)$ |
| $ 32 $ | $1$ | $32$ | $( 1,16,28, 5,17,32,10,23, 4,14,25, 7,20,30,11,21, 2,15,27, 6,18,31, 9,24, 3, 13,26, 8,19,29,12,22)$ |
| $ 8, 8, 8, 8 $ | $1$ | $8$ | $( 1,17, 4,20, 2,18, 3,19)( 5,23, 7,21, 6,24, 8,22)( 9,26,12,28,10,25,11,27) (13,29,16,32,14,30,15,31)$ |
| $ 8, 8, 8, 8 $ | $1$ | $8$ | $( 1,18, 4,19, 2,17, 3,20)( 5,24, 7,22, 6,23, 8,21)( 9,25,12,27,10,26,11,28) (13,30,16,31,14,29,15,32)$ |
| $ 8, 8, 8, 8 $ | $1$ | $8$ | $( 1,19, 3,18, 2,20, 4,17)( 5,22, 8,24, 6,21, 7,23)( 9,27,11,25,10,28,12,26) (13,31,15,30,14,32,16,29)$ |
| $ 8, 8, 8, 8 $ | $1$ | $8$ | $( 1,20, 3,17, 2,19, 4,18)( 5,21, 8,23, 6,22, 7,24)( 9,28,11,26,10,27,12,25) (13,32,15,29,14,31,16,30)$ |
| $ 32 $ | $1$ | $32$ | $( 1,21,12,30,19, 7,26,14, 3,23, 9,32,18, 5,27,16, 2,22,11,29,20, 8,25,13, 4, 24,10,31,17, 6,28,15)$ |
| $ 32 $ | $1$ | $32$ | $( 1,22,12,29,19, 8,26,13, 3,24, 9,31,18, 6,27,15, 2,21,11,30,20, 7,25,14, 4, 23,10,32,17, 5,28,16)$ |
| $ 32 $ | $1$ | $32$ | $( 1,23,11,31,19, 5,25,15, 3,22,10,30,18, 8,28,14, 2,24,12,32,20, 6,26,16, 4, 21, 9,29,17, 7,27,13)$ |
| $ 32 $ | $1$ | $32$ | $( 1,24,11,32,19, 6,25,16, 3,21,10,29,18, 7,28,13, 2,23,12,31,20, 5,26,15, 4, 22, 9,30,17, 8,27,14)$ |
| $ 16, 16 $ | $1$ | $16$ | $( 1,25,18,12, 4,27,19,10, 2,26,17,11, 3,28,20, 9)( 5,30,24,16, 7,31,22,14, 6, 29,23,15, 8,32,21,13)$ |
| $ 16, 16 $ | $1$ | $16$ | $( 1,26,18,11, 4,28,19, 9, 2,25,17,12, 3,27,20,10)( 5,29,24,15, 7,32,22,13, 6, 30,23,16, 8,31,21,14)$ |
| $ 16, 16 $ | $1$ | $16$ | $( 1,27,17, 9, 4,26,20,12, 2,28,18,10, 3,25,19,11)( 5,31,23,13, 7,29,21,16, 6, 32,24,14, 8,30,22,15)$ |
| $ 16, 16 $ | $1$ | $16$ | $( 1,28,17,10, 4,25,20,11, 2,27,18, 9, 3,26,19,12)( 5,32,23,14, 7,30,21,15, 6, 31,24,13, 8,29,22,16)$ |
| $ 32 $ | $1$ | $32$ | $( 1,29,26,24,18,15,11, 7, 4,32,28,22,19,13, 9, 6, 2,30,25,23,17,16,12, 8, 3, 31,27,21,20,14,10, 5)$ |
| $ 32 $ | $1$ | $32$ | $( 1,30,26,23,18,16,11, 8, 4,31,28,21,19,14, 9, 5, 2,29,25,24,17,15,12, 7, 3, 32,27,22,20,13,10, 6)$ |
| $ 32 $ | $1$ | $32$ | $( 1,31,25,22,18,14,12, 6, 4,29,27,23,19,15,10, 8, 2,32,26,21,17,13,11, 5, 3, 30,28,24,20,16, 9, 7)$ |
| $ 32 $ | $1$ | $32$ | $( 1,32,25,21,18,13,12, 5, 4,30,27,24,19,16,10, 7, 2,31,26,22,17,14,11, 6, 3, 29,28,23,20,15, 9, 8)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $32=2^{5}$ | 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: | 32.1 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);