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Magma
magma: G := TransitiveGroup(24, 135);
Group action invariants
| Degree $n$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $135$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_2^3\times A_4$ | ||
| 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: | (1,12)(2,11)(3,18)(4,17)(5,15)(6,16)(7,9)(8,10)(13,23)(14,24)(19,21)(20,22), (1,22,17,14,9,5)(2,21,18,13,10,6)(3,23,19,16,11,8)(4,24,20,15,12,7), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 7 $3$: $C_3$ $4$: $C_2^2$ x 7 $6$: $C_6$ x 7 $8$: $C_2^3$ $12$: $A_4$, $C_6\times C_2$ x 7 $24$: $A_4\times C_2$ x 7, 24T3 $48$: $C_2^2 \times A_4$ x 7 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 3: $C_3$
Degree 4: $C_2^2$
Degree 6: $C_6$ x 3, $A_4\times C_2$ x 4
Degree 8: None
Degree 12: $C_6\times C_2$, $C_2^2 \times A_4$ x 6
Low degree siblings
24T135 x 6, 24T136 x 14, 32T404Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 7,20)( 8,19)( 9,22)(10,21)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 3,16)( 4,15)( 5,17)( 6,18)( 7,20)( 8,19)( 9,22)(10,21)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7,19)( 8,20)( 9,21)(10,22)(11,12)(13,14)(15,16)(17,18) (23,24)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3,15)( 4,16)( 5,18)( 6,17)( 7,19)( 8,20)( 9,21)(10,22)(11,12)(13,14) (23,24)$ | |
| $ 6, 6, 6, 6 $ | $4$ | $6$ | $( 1, 3, 9,23, 5, 8)( 2, 4,10,24, 6, 7)(11,17,19,14,16,22)(12,18,20,13,15,21)$ | |
| $ 6, 6, 6, 6 $ | $4$ | $6$ | $( 1, 3, 9,11,17,19)( 2, 4,10,12,18,20)( 5, 8,14,16,22,23)( 6, 7,13,15,21,24)$ | |
| $ 6, 6, 6, 6 $ | $4$ | $6$ | $( 1, 4, 9,24, 5, 7)( 2, 3,10,23, 6, 8)(11,18,19,13,16,21)(12,17,20,14,15,22)$ | |
| $ 6, 6, 6, 6 $ | $4$ | $6$ | $( 1, 4, 9,12,17,20)( 2, 3,10,11,18,19)( 5, 7,14,15,22,24)( 6, 8,13,16,21,23)$ | |
| $ 6, 6, 6, 6 $ | $4$ | $6$ | $( 1, 5, 9,14,17,22)( 2, 6,10,13,18,21)( 3, 8,11,16,19,23)( 4, 7,12,15,20,24)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 5, 9)( 2, 6,10)( 3, 8,23)( 4, 7,24)(11,16,19)(12,15,20)(13,18,21) (14,17,22)$ | |
| $ 6, 6, 6, 6 $ | $4$ | $6$ | $( 1, 6, 9,13,17,21)( 2, 5,10,14,18,22)( 3, 7,11,15,19,24)( 4, 8,12,16,20,23)$ | |
| $ 6, 6, 6, 6 $ | $4$ | $6$ | $( 1, 6, 9, 2, 5,10)( 3, 7,23, 4, 8,24)(11,15,19,12,16,20)(13,17,21,14,18,22)$ | |
| $ 6, 6, 6, 6 $ | $4$ | $6$ | $( 1, 7, 5,24, 9, 4)( 2, 8, 6,23,10, 3)(11,21,16,13,19,18)(12,22,15,14,20,17)$ | |
| $ 6, 6, 6, 6 $ | $4$ | $6$ | $( 1, 7,17,12,22, 4)( 2, 8,18,11,21, 3)( 5,24, 9,15,14,20)( 6,23,10,16,13,19)$ | |
| $ 6, 6, 6, 6 $ | $4$ | $6$ | $( 1, 8, 5,23, 9, 3)( 2, 7, 6,24,10, 4)(11,22,16,14,19,17)(12,21,15,13,20,18)$ | |
| $ 6, 6, 6, 6 $ | $4$ | $6$ | $( 1, 8,17,11,22, 3)( 2, 7,18,12,21, 4)( 5,23, 9,16,14,19)( 6,24,10,15,13,20)$ | |
| $ 6, 6, 6, 6 $ | $4$ | $6$ | $( 1, 9, 5,14,22,17)( 2,10, 6,13,21,18)( 3,11,19,16,23, 8)( 4,12,20,15,24, 7)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 9,17)( 2,10,18)( 3,11,19)( 4,12,20)( 5,14,22)( 6,13,21)( 7,15,24) ( 8,16,23)$ | |
| $ 6, 6, 6, 6 $ | $4$ | $6$ | $( 1,10, 5,13,22,18)( 2, 9, 6,14,21,17)( 3,12,19,15,23, 7)( 4,11,20,16,24, 8)$ | |
| $ 6, 6, 6, 6 $ | $4$ | $6$ | $( 1,10,17, 2, 9,18)( 3,12,19, 4,11,20)( 5,13,22, 6,14,21)( 7,16,24, 8,15,23)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1,11)( 2,12)( 3, 5)( 4, 6)( 7,10)( 8, 9)(13,24)(14,23)(15,18)(16,17)(19,22) (20,21)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1,11)( 2,12)( 3, 5)( 4, 6)( 7,21)( 8,22)( 9,19)(10,20)(13,24)(14,23)(15,18) (16,17)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,11)( 2,12)( 3,17)( 4,18)( 5,16)( 6,15)( 7,21)( 8,22)( 9,19)(10,20)(13,24) (14,23)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1,12)( 2,11)( 3, 6)( 4, 5)( 7, 9)( 8,10)(13,23)(14,24)(15,17)(16,18)(19,21) (20,22)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1,12)( 2,11)( 3, 6)( 4, 5)( 7,22)( 8,21)( 9,20)(10,19)(13,23)(14,24)(15,17) (16,18)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,12)( 2,11)( 3,18)( 4,17)( 5,15)( 6,16)( 7,22)( 8,21)( 9,20)(10,19)(13,23) (14,24)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,13)( 2,14)( 3,15)( 4,16)( 5,18)( 6,17)( 7,19)( 8,20)( 9,21)(10,22)(11,24) (12,23)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,14)( 2,13)( 3,16)( 4,15)( 5,17)( 6,18)( 7,20)( 8,19)( 9,22)(10,21)(11,23) (12,24)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,23)( 2,24)( 3, 5)( 4, 6)( 7,10)( 8, 9)(11,14)(12,13)(15,18)(16,17)(19,22) (20,21)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,24)( 2,23)( 3, 6)( 4, 5)( 7, 9)( 8,10)(11,13)(12,14)(15,17)(16,18)(19,21) (20,22)$ |
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: | 96.228 | magma: IdentifyGroup(G);
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| Character table: | 32 x 32 character table |
magma: CharacterTable(G);