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Magma
magma: G := TransitiveGroup(24, 91);
Group action invariants
| Degree $n$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $91$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_2^3:C_{12}$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $6$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,11,17,23,6,10,15,21,3,7,14,20)(2,12,18,24,5,9,16,22,4,8,13,19), (1,3,6)(2,4,5)(7,10,11)(8,9,12)(13,15,18,14,16,17)(19,21,24,20,22,23) | 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, $C_2^2$ $6$: $C_6$ x 3 $8$: $D_{4}$ x 2, $C_4\times C_2$ $12$: $C_{12}$ x 2, $C_6\times C_2$ $16$: $C_2^2:C_4$ $24$: $D_4 \times C_3$ x 2, 24T2 $32$: $C_2^3 : C_4 $ $48$: 24T39 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $C_3$
Degree 4: $C_4$
Degree 6: $C_6$
Degree 8: $C_2^3: C_4$
Degree 12: $C_{12}$
Low degree siblings
24T93, 24T115 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 $ | $1$ | $1$ | $()$ | |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)$ | |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 7, 8)( 9,10)(11,12)(19,20)(21,22)(23,24)$ | |
| $ 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, 3, 3, 3, 3 $ | $4$ | $6$ | $( 1, 3, 6)( 2, 4, 5)( 7, 9,11, 8,10,12)(13,15,18,14,16,17)(19,22,24)(20,21,23)$ | |
| $ 6, 6, 3, 3, 3, 3 $ | $2$ | $6$ | $( 1, 3, 6)( 2, 4, 5)( 7, 9,11, 8,10,12)(13,16,18)(14,15,17)(19,21,24,20,22,23)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 6)( 2, 4, 5)( 7,10,11)( 8, 9,12)(13,16,18)(14,15,17)(19,22,24) (20,21,23)$ | |
| $ 6, 6, 6, 6 $ | $1$ | $6$ | $( 1, 4, 6, 2, 3, 5)( 7, 9,11, 8,10,12)(13,15,18,14,16,17)(19,21,24,20,22,23)$ | |
| $ 6, 6, 3, 3, 3, 3 $ | $2$ | $6$ | $( 1, 5, 3, 2, 6, 4)( 7,11,10)( 8,12, 9)(13,17,16,14,18,15)(19,24,22)(20,23,21)$ | |
| $ 6, 6, 3, 3, 3, 3 $ | $4$ | $6$ | $( 1, 5, 3, 2, 6, 4)( 7,11,10)( 8,12, 9)(13,18,16)(14,17,15)(19,23,22,20,24,21)$ | |
| $ 6, 6, 6, 6 $ | $1$ | $6$ | $( 1, 5, 3, 2, 6, 4)( 7,12,10, 8,11, 9)(13,17,16,14,18,15)(19,23,22,20,24,21)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 6, 3)( 2, 5, 4)( 7,11,10)( 8,12, 9)(13,18,16)(14,17,15)(19,24,22) (20,23,21)$ | |
| $ 4, 4, 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 7,15,24)( 2, 8,16,23)( 3,10,17,19)( 4, 9,18,20)( 5,12,13,21)( 6,11,14,22)$ | |
| $ 4, 4, 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 7,15,23)( 2, 8,16,24)( 3,10,17,20)( 4, 9,18,19)( 5,12,13,22)( 6,11,14,21)$ | |
| $ 12, 12 $ | $4$ | $12$ | $( 1, 9,13,23, 3,12,16,20, 6, 8,18,21)( 2,10,14,24, 4,11,15,19, 5, 7,17,22)$ | |
| $ 12, 12 $ | $4$ | $12$ | $( 1, 9,13,24, 3,12,16,19, 6, 8,18,22)( 2,10,14,23, 4,11,15,20, 5, 7,17,21)$ | |
| $ 12, 12 $ | $4$ | $12$ | $( 1,11,18,24, 6,10,16,22, 3, 7,13,19)( 2,12,17,23, 5, 9,15,21, 4, 8,14,20)$ | |
| $ 12, 12 $ | $4$ | $12$ | $( 1,11,18,23, 6,10,16,21, 3, 7,13,20)( 2,12,17,24, 5, 9,15,22, 4, 8,14,19)$ | |
| $ 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,13, 3,16, 6,18)( 2,14, 4,15, 5,17)( 7,21,10,23,11,20)( 8,22, 9,24,12,19)$ | |
| $ 12, 12 $ | $4$ | $12$ | $( 1,13, 4,15, 6,18, 2,14, 3,16, 5,17)( 7,21, 9,24,11,20, 8,22,10,23,12,19)$ | |
| $ 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,13, 3,16, 6,18)( 2,14, 4,15, 5,17)( 7,22,10,24,11,19)( 8,21, 9,23,12,20)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1,15)( 2,16)( 3,17)( 4,18)( 5,13)( 6,14)( 7,23)( 8,24)( 9,19)(10,20)(11,21) (12,22)$ | |
| $ 4, 4, 4, 4, 4, 4 $ | $4$ | $4$ | $( 1,15, 2,16)( 3,17, 4,18)( 5,13, 6,14)( 7,23, 8,24)( 9,19,10,20)(11,21,12,22)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1,15)( 2,16)( 3,17)( 4,18)( 5,13)( 6,14)( 7,24)( 8,23)( 9,20)(10,19)(11,22) (12,21)$ | |
| $ 12, 12 $ | $4$ | $12$ | $( 1,17, 5,16, 3,14, 2,18, 6,15, 4,13)( 7,19,12,23,10,22, 8,20,11,24, 9,21)$ | |
| $ 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,17, 6,15, 3,14)( 2,18, 5,16, 4,13)( 7,19,11,24,10,22)( 8,20,12,23, 9,21)$ | |
| $ 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,17, 6,15, 3,14)( 2,18, 5,16, 4,13)( 7,20,11,23,10,21)( 8,19,12,24, 9,22)$ | |
| $ 12, 12 $ | $4$ | $12$ | $( 1,19,13, 7, 3,22,16,10, 6,24,18,11)( 2,20,14, 8, 4,21,15, 9, 5,23,17,12)$ | |
| $ 12, 12 $ | $4$ | $12$ | $( 1,19,14, 7, 3,22,15,10, 6,24,17,11)( 2,20,13, 8, 4,21,16, 9, 5,23,18,12)$ | |
| $ 12, 12 $ | $4$ | $12$ | $( 1,21,18, 8, 6,20,16,12, 3,23,13, 9)( 2,22,17, 7, 5,19,15,11, 4,24,14,10)$ | |
| $ 12, 12 $ | $4$ | $12$ | $( 1,21,17, 8, 6,20,15,12, 3,23,14, 9)( 2,22,18, 7, 5,19,16,11, 4,24,13,10)$ | |
| $ 4, 4, 4, 4, 4, 4 $ | $4$ | $4$ | $( 1,23,16, 7)( 2,24,15, 8)( 3,20,18,10)( 4,19,17, 9)( 5,22,14,12)( 6,21,13,11)$ | |
| $ 4, 4, 4, 4, 4, 4 $ | $4$ | $4$ | $( 1,23,15, 7)( 2,24,16, 8)( 3,20,17,10)( 4,19,18, 9)( 5,22,13,12)( 6,21,14,11)$ |
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: | $3$ | ||
| Label: | 96.49 | magma: IdentifyGroup(G);
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| Character table: | 33 x 33 character table |
magma: CharacterTable(G);