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Magma
magma: G := TransitiveGroup(24, 65);
Group action invariants
| Degree $n$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $65$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $S_3\times C_{12}$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $12$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,22,17,13,9,5,2,21,18,14,10,6)(3,16,4,15)(7,20,8,19)(11,24,12,23), (1,23,17,16,9,7,2,24,18,15,10,8)(3,21,19,14,11,6,4,22,20,13,12,5) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_4$ x 2, $C_2^2$ $6$: $S_3$, $C_6$ x 3 $8$: $C_4\times C_2$ $12$: $D_{6}$, $C_{12}$ x 2, $C_6\times C_2$ $18$: $S_3\times C_3$ $24$: $S_3 \times C_4$, 24T2 $36$: $C_6\times S_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 3: None
Degree 6: $S_3\times C_3$
Degree 8: $C_4\times C_2$
Degree 12: $C_6\times S_3$
Low degree siblings
36T27 x 2Siblings 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 $ | $1$ | $1$ | $()$ | |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $3$ | $( 3,11,20)( 4,12,19)( 7,15,23)( 8,16,24)$ | |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $3$ | $( 3,20,11)( 4,19,12)( 7,23,15)( 8,24,16)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)$ | |
| $ 6, 6, 2, 2, 2, 2, 2, 2 $ | $2$ | $6$ | $( 1, 2)( 3,12,20, 4,11,19)( 5, 6)( 7,16,23, 8,15,24)( 9,10)(13,14)(17,18) (21,22)$ | |
| $ 6, 6, 2, 2, 2, 2, 2, 2 $ | $2$ | $6$ | $( 1, 2)( 3,19,11, 4,20,12)( 5, 6)( 7,24,15, 8,23,16)( 9,10)(13,14)(17,18) (21,22)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23) (22,24)$ | |
| $ 6, 6, 6, 6 $ | $3$ | $6$ | $( 1, 3, 9,11,18,20)( 2, 4,10,12,17,19)( 5, 8,14,16,22,24)( 6, 7,13,15,21,23)$ | |
| $ 6, 6, 6, 6 $ | $3$ | $6$ | $( 1, 3,18,20, 9,11)( 2, 4,17,19,10,12)( 5, 8,22,24,14,16)( 6, 7,21,23,13,15)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24) (22,23)$ | |
| $ 6, 6, 6, 6 $ | $3$ | $6$ | $( 1, 4, 9,12,18,19)( 2, 3,10,11,17,20)( 5, 7,14,15,22,23)( 6, 8,13,16,21,24)$ | |
| $ 6, 6, 6, 6 $ | $3$ | $6$ | $( 1, 4,18,19, 9,12)( 2, 3,17,20,10,11)( 5, 7,22,23,14,15)( 6, 8,21,24,13,16)$ | |
| $ 12, 12 $ | $1$ | $12$ | $( 1, 5,10,13,18,22, 2, 6, 9,14,17,21)( 3, 8,12,15,20,24, 4, 7,11,16,19,23)$ | |
| $ 12, 4, 4, 4 $ | $2$ | $12$ | $( 1, 5,10,13,18,22, 2, 6, 9,14,17,21)( 3,16, 4,15)( 7,20, 8,19)(11,24,12,23)$ | |
| $ 12, 12 $ | $2$ | $12$ | $( 1, 5,10,13,18,22, 2, 6, 9,14,17,21)( 3,24,19,15,11, 8, 4,23,20,16,12, 7)$ | |
| $ 12, 12 $ | $1$ | $12$ | $( 1, 6,10,14,18,21, 2, 5, 9,13,17,22)( 3, 7,12,16,20,23, 4, 8,11,15,19,24)$ | |
| $ 12, 4, 4, 4 $ | $2$ | $12$ | $( 1, 6,10,14,18,21, 2, 5, 9,13,17,22)( 3,15, 4,16)( 7,19, 8,20)(11,23,12,24)$ | |
| $ 12, 12 $ | $2$ | $12$ | $( 1, 6,10,14,18,21, 2, 5, 9,13,17,22)( 3,23,19,16,11, 7, 4,24,20,15,12, 8)$ | |
| $ 12, 12 $ | $3$ | $12$ | $( 1, 7,10,16,18,23, 2, 8, 9,15,17,24)( 3, 6,12,14,20,21, 4, 5,11,13,19,22)$ | |
| $ 12, 12 $ | $3$ | $12$ | $( 1, 7,17,24, 9,15, 2, 8,18,23,10,16)( 3,13,19, 5,11,21, 4,14,20, 6,12,22)$ | |
| $ 4, 4, 4, 4, 4, 4 $ | $3$ | $4$ | $( 1, 7, 2, 8)( 3,21, 4,22)( 5,11, 6,12)( 9,15,10,16)(13,19,14,20)(17,24,18,23)$ | |
| $ 12, 12 $ | $3$ | $12$ | $( 1, 8,10,15,18,24, 2, 7, 9,16,17,23)( 3, 5,12,13,20,22, 4, 6,11,14,19,21)$ | |
| $ 12, 12 $ | $3$ | $12$ | $( 1, 8,17,23, 9,16, 2, 7,18,24,10,15)( 3,14,19, 6,11,22, 4,13,20, 5,12,21)$ | |
| $ 4, 4, 4, 4, 4, 4 $ | $3$ | $4$ | $( 1, 8, 2, 7)( 3,22, 4,21)( 5,12, 6,11)( 9,16,10,15)(13,20,14,19)(17,23,18,24)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 9,18)( 2,10,17)( 3,11,20)( 4,12,19)( 5,14,22)( 6,13,21)( 7,15,23) ( 8,16,24)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 9,18)( 2,10,17)( 3,20,11)( 4,19,12)( 5,14,22)( 6,13,21)( 7,23,15) ( 8,24,16)$ | |
| $ 6, 6, 6, 6 $ | $1$ | $6$ | $( 1,10,18, 2, 9,17)( 3,12,20, 4,11,19)( 5,13,22, 6,14,21)( 7,16,23, 8,15,24)$ | |
| $ 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,10,18, 2, 9,17)( 3,19,11, 4,20,12)( 5,13,22, 6,14,21)( 7,24,15, 8,23,16)$ | |
| $ 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,13, 2,14)( 3,15, 4,16)( 5,18, 6,17)( 7,19, 8,20)( 9,21,10,22)(11,23,12,24)$ | |
| $ 12, 4, 4, 4 $ | $2$ | $12$ | $( 1,13, 2,14)( 3,23,19,16,11, 7, 4,24,20,15,12, 8)( 5,18, 6,17)( 9,21,10,22)$ | |
| $ 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,14, 2,13)( 3,16, 4,15)( 5,17, 6,18)( 7,20, 8,19)( 9,22,10,21)(11,24,12,23)$ | |
| $ 12, 4, 4, 4 $ | $2$ | $12$ | $( 1,14, 2,13)( 3,24,19,15,11, 8, 4,23,20,16,12, 7)( 5,17, 6,18)( 9,22,10,21)$ | |
| $ 6, 6, 6, 6 $ | $1$ | $6$ | $( 1,17, 9, 2,18,10)( 3,19,11, 4,20,12)( 5,21,14, 6,22,13)( 7,24,15, 8,23,16)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,18, 9)( 2,17,10)( 3,20,11)( 4,19,12)( 5,22,14)( 6,21,13)( 7,23,15) ( 8,24,16)$ | |
| $ 12, 12 $ | $1$ | $12$ | $( 1,21,17,14, 9, 6, 2,22,18,13,10, 5)( 3,23,19,16,11, 7, 4,24,20,15,12, 8)$ | |
| $ 12, 12 $ | $1$ | $12$ | $( 1,22,17,13, 9, 5, 2,21,18,14,10, 6)( 3,24,19,15,11, 8, 4,23,20,16,12, 7)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $72=2^{3} \cdot 3^{2}$ | 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: | 72.27 | magma: IdentifyGroup(G);
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| Character table: | 36 x 36 character table |
magma: CharacterTable(G);