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Magma
magma: G := TransitiveGroup(32, 402);
Group action invariants
| Degree $n$: | $32$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $402$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_8.A_4$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $8$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,15,29,23,9,26,3,14,31,22,11,27,2,16,30,24,10,25,4,13,32,21,12,28)(5,18,8,20,6,17,7,19), (1,17,11,21,8,14,4,19,9,24,6,15,2,18,12,22,7,13,3,20,10,23,5,16)(25,31,28,30,26,32,27,29) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $4$: $C_4$ $6$: $C_6$ $12$: $A_4$, $C_{12}$ $24$: $A_4\times C_2$ $48$: 12T29 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 8: $A_4\times C_2$
Degree 16: 16T57
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 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $3$ | $( 5,11,30)( 6,12,29)( 7,10,32)( 8, 9,31)(13,28,18)(14,27,17)(15,26,19) (16,25,20)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $3$ | $( 5,30,11)( 6,29,12)( 7,32,10)( 8,31, 9)(13,18,28)(14,17,27)(15,19,26) (16,20,25)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)$ |
| $ 6, 6, 6, 6, 2, 2, 2, 2 $ | $4$ | $6$ | $( 1, 2)( 3, 4)( 5,12,30, 6,11,29)( 7, 9,32, 8,10,31)(13,27,18,14,28,17) (15,25,19,16,26,20)(21,22)(23,24)$ |
| $ 6, 6, 6, 6, 2, 2, 2, 2 $ | $4$ | $6$ | $( 1, 2)( 3, 4)( 5,29,11, 6,30,12)( 7,31,10, 8,32, 9)(13,17,28,14,18,27) (15,20,26,16,19,25)(21,22)(23,24)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)(17,20,18,19)(21,23,22,24) (25,28,26,27)(29,31,30,32)$ |
| $ 12, 12, 4, 4 $ | $4$ | $12$ | $( 1, 3, 2, 4)( 5,10,29, 8,11,32, 6, 9,30, 7,12,31)(13,26,17,16,28,19,14,25,18, 15,27,20)(21,23,22,24)$ |
| $ 12, 12, 4, 4 $ | $4$ | $12$ | $( 1, 3, 2, 4)( 5,32,12, 8,30,10, 6,31,11, 7,29, 9)(13,19,27,16,18,26,14,20,28, 15,17,25)(21,23,22,24)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,12,10,11)(13,16,14,15)(17,19,18,20)(21,24,22,23) (25,27,26,28)(29,32,30,31)$ |
| $ 12, 12, 4, 4 $ | $4$ | $12$ | $( 1, 4, 2, 3)( 5, 9,29, 7,11,31, 6,10,30, 8,12,32)(13,25,17,15,28,20,14,26,18, 16,27,19)(21,24,22,23)$ |
| $ 12, 12, 4, 4 $ | $4$ | $12$ | $( 1, 4, 2, 3)( 5,31,12, 7,30, 9, 6,32,11, 8,29,10)(13,20,27,15,18,25,14,19,28, 16,17,26)(21,24,22,23)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $6$ | $2$ | $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,29)(10,30)(11,31)(12,32)(13,25)(14,26)(15,28) (16,27)(17,21)(18,22)(19,24)(20,23)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $6$ | $4$ | $( 1, 7, 2, 8)( 3, 6, 4, 5)( 9,31,10,32)(11,30,12,29)(13,28,14,27)(15,26,16,25) (17,23,18,24)(19,21,20,22)$ |
| $ 24, 8 $ | $4$ | $24$ | $( 1,13,30,22, 9,28, 4,16,31,23,12,25, 2,14,29,21,10,27, 3,15,32,24,11,26) ( 5,20, 7,18, 6,19, 8,17)$ |
| $ 24, 8 $ | $4$ | $24$ | $( 1,13, 6,21,10,18, 4,16, 7,24,11,20, 2,14, 5,22, 9,17, 3,15, 8,23,12,19) (25,31,28,30,26,32,27,29)$ |
| $ 8, 8, 8, 8 $ | $6$ | $8$ | $( 1,13, 4,16, 2,14, 3,15)( 5,26, 8,28, 6,25, 7,27)( 9,23,12,21,10,24,11,22) (17,29,19,32,18,30,20,31)$ |
| $ 24, 8 $ | $4$ | $24$ | $( 1,14,30,21, 9,27, 4,15,31,24,12,26, 2,13,29,22,10,28, 3,16,32,23,11,25) ( 5,19, 7,17, 6,20, 8,18)$ |
| $ 24, 8 $ | $4$ | $24$ | $( 1,14, 6,22,10,17, 4,15, 7,23,11,19, 2,13, 5,21, 9,18, 3,16, 8,24,12,20) (25,32,28,29,26,31,27,30)$ |
| $ 24, 8 $ | $4$ | $24$ | $( 1,15,29,23, 9,26, 3,14,31,22,11,27, 2,16,30,24,10,25, 4,13,32,21,12,28) ( 5,18, 8,20, 6,17, 7,19)$ |
| $ 24, 8 $ | $4$ | $24$ | $( 1,15, 5,24,10,19, 3,14, 7,21,12,17, 2,16, 6,23, 9,20, 4,13, 8,22,11,18) (25,30,27,31,26,29,28,32)$ |
| $ 8, 8, 8, 8 $ | $6$ | $8$ | $( 1,15, 3,14, 2,16, 4,13)( 5,27, 7,25, 6,28, 8,26)( 9,22,11,24,10,21,12,23) (17,31,20,30,18,32,19,29)$ |
| $ 24, 8 $ | $4$ | $24$ | $( 1,16,29,24, 9,25, 3,13,31,21,11,28, 2,15,30,23,10,26, 4,14,32,22,12,27) ( 5,17, 8,19, 6,18, 7,20)$ |
| $ 24, 8 $ | $4$ | $24$ | $( 1,16, 5,23,10,20, 3,13, 7,22,12,18, 2,15, 6,24, 9,19, 4,14, 8,21,11,17) (25,29,27,32,26,30,28,31)$ |
| $ 8, 8, 8, 8 $ | $1$ | $8$ | $( 1,21, 4,24, 2,22, 3,23)( 5,17, 8,19, 6,18, 7,20)( 9,15,12,13,10,16,11,14) (25,30,27,31,26,29,28,32)$ |
| $ 8, 8, 8, 8 $ | $1$ | $8$ | $( 1,22, 4,23, 2,21, 3,24)( 5,18, 8,20, 6,17, 7,19)( 9,16,12,14,10,15,11,13) (25,29,27,32,26,30,28,31)$ |
| $ 8, 8, 8, 8 $ | $1$ | $8$ | $( 1,23, 3,22, 2,24, 4,21)( 5,20, 7,18, 6,19, 8,17)( 9,14,11,16,10,13,12,15) (25,32,28,29,26,31,27,30)$ |
| $ 8, 8, 8, 8 $ | $1$ | $8$ | $( 1,24, 3,21, 2,23, 4,22)( 5,19, 7,17, 6,20, 8,18)( 9,13,11,15,10,14,12,16) (25,31,28,30,26,32,27,29)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $96=2^{5} \cdot 3$ | 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: | 96.74 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);