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Magma
magma: G := TransitiveGroup(24, 114);
Group action invariants
| Degree $n$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $114$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $D_8:C_6$ | ||
| 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: | (1,18,6,22,9,13)(2,17,5,21,10,14)(3,19,7,23,11,16)(4,20,8,24,12,15), (1,23,11,22,10,20,8,17,6,16,3,13,2,24,12,21,9,19,7,18,5,15,4,14), (1,14,3,16,5,18,8,20,9,21,11,23,2,13,4,15,6,17,7,19,10,22,12,24) | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ 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 $8$: $D_{4}$ x 2, $C_2^3$ $12$: $C_6\times C_2$ x 7 $16$: $D_4\times C_2$ $24$: $D_4 \times C_3$ x 2, 24T3 $32$: $Z_8 : Z_8^\times$ $48$: 24T38 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $C_3$
Degree 4: $D_{4}$
Degree 6: $C_6$
Degree 8: $Z_8 : Z_8^\times$
Degree 12: $D_4 \times C_3$
Low degree siblings
24T114Siblings are shown with degree $\leq 47$
A number field with this Galois group has exactly one arithmetically equivalent field.
Conjugacy classes
| Label | Cycle Type | Size | Order | Index | Representative |
| 1A | $1^{24}$ | $1$ | $1$ | $0$ | $()$ |
| 2A | $2^{12}$ | $1$ | $2$ | $12$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)$ |
| 2B | $2^{6},1^{12}$ | $2$ | $2$ | $6$ | $(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)$ |
| 2C | $2^{12}$ | $4$ | $2$ | $12$ | $( 1,16)( 2,15)( 3,17)( 4,18)( 5,20)( 6,19)( 7,21)( 8,22)( 9,23)(10,24)(11,14)(12,13)$ |
| 2D | $2^{9},1^{6}$ | $4$ | $2$ | $9$ | $( 3, 4)( 7, 8)(11,12)(13,20)(14,19)(15,22)(16,21)(17,23)(18,24)$ |
| 2E | $2^{9},1^{6}$ | $4$ | $2$ | $9$ | $( 3, 4)( 7, 8)(11,12)(13,19)(14,20)(15,21)(16,22)(17,24)(18,23)$ |
| 3A1 | $3^{8}$ | $1$ | $3$ | $16$ | $( 1, 9, 6)( 2,10, 5)( 3,11, 7)( 4,12, 8)(13,22,18)(14,21,17)(15,24,20)(16,23,19)$ |
| 3A-1 | $3^{8}$ | $1$ | $3$ | $16$ | $( 1, 6, 9)( 2, 5,10)( 3, 7,11)( 4, 8,12)(13,18,22)(14,17,21)(15,20,24)(16,19,23)$ |
| 4A | $4^{6}$ | $2$ | $4$ | $18$ | $( 1, 7, 2, 8)( 3,10, 4, 9)( 5,12, 6,11)(13,19,14,20)(15,22,16,21)(17,24,18,23)$ |
| 4B | $4^{6}$ | $2$ | $4$ | $18$ | $( 1, 7, 2, 8)( 3,10, 4, 9)( 5,12, 6,11)(13,20,14,19)(15,21,16,22)(17,23,18,24)$ |
| 4C | $4^{6}$ | $4$ | $4$ | $18$ | $( 1,15, 2,16)( 3,18, 4,17)( 5,19, 6,20)( 7,22, 8,21)( 9,24,10,23)(11,13,12,14)$ |
| 6A1 | $6^{4}$ | $1$ | $6$ | $20$ | $( 1, 5, 9, 2, 6,10)( 3, 8,11, 4, 7,12)(13,17,22,14,18,21)(15,19,24,16,20,23)$ |
| 6A-1 | $6^{4}$ | $1$ | $6$ | $20$ | $( 1,10, 6, 2, 9, 5)( 3,12, 7, 4,11, 8)(13,21,18,14,22,17)(15,23,20,16,24,19)$ |
| 6B1 | $6^{2},3^{4}$ | $2$ | $6$ | $18$ | $( 1, 6, 9)( 2, 5,10)( 3, 7,11)( 4, 8,12)(13,17,22,14,18,21)(15,19,24,16,20,23)$ |
| 6B-1 | $6^{2},3^{4}$ | $2$ | $6$ | $18$ | $( 1, 9, 6)( 2,10, 5)( 3,11, 7)( 4,12, 8)(13,21,18,14,22,17)(15,23,20,16,24,19)$ |
| 6C1 | $6^{3},3^{2}$ | $4$ | $6$ | $19$ | $( 1, 9, 6)( 2,10, 5)( 3,12, 7, 4,11, 8)(13,15,18,20,22,24)(14,16,17,19,21,23)$ |
| 6C-1 | $6^{3},3^{2}$ | $4$ | $6$ | $19$ | $( 1, 6, 9)( 2, 5,10)( 3, 8,11, 4, 7,12)(13,23,22,19,18,16)(14,24,21,20,17,15)$ |
| 6D1 | $6^{3},3^{2}$ | $4$ | $6$ | $19$ | $( 1, 6, 9)( 2, 5,10)( 3, 8,11, 4, 7,12)(13,24,22,20,18,15)(14,23,21,19,17,16)$ |
| 6D-1 | $6^{3},3^{2}$ | $4$ | $6$ | $19$ | $( 1, 9, 6)( 2,10, 5)( 3,12, 7, 4,11, 8)(13,16,18,19,22,23)(14,15,17,20,21,24)$ |
| 6E1 | $6^{4}$ | $4$ | $6$ | $20$ | $( 1,23, 6,16, 9,19)( 2,24, 5,15,10,20)( 3,14, 7,17,11,21)( 4,13, 8,18,12,22)$ |
| 6E-1 | $6^{4}$ | $4$ | $6$ | $20$ | $( 1,19, 9,16, 6,23)( 2,20,10,15, 5,24)( 3,21,11,17, 7,14)( 4,22,12,18, 8,13)$ |
| 8A | $8^{3}$ | $4$ | $8$ | $21$ | $( 1,21, 8,15, 2,22, 7,16)( 3,23, 9,17, 4,24,10,18)( 5,13,11,19, 6,14,12,20)$ |
| 8B | $8^{3}$ | $4$ | $8$ | $21$ | $( 1,22, 7,15, 2,21, 8,16)( 3,24,10,17, 4,23, 9,18)( 5,14,12,19, 6,13,11,20)$ |
| 12A1 | $12^{2}$ | $2$ | $12$ | $22$ | $( 1, 3, 5, 8, 9,11, 2, 4, 6, 7,10,12)(13,15,17,19,22,24,14,16,18,20,21,23)$ |
| 12A-1 | $12^{2}$ | $2$ | $12$ | $22$ | $( 1,11,10, 8, 6, 3, 2,12, 9, 7, 5, 4)(13,24,21,19,18,15,14,23,22,20,17,16)$ |
| 12B1 | $12^{2}$ | $2$ | $12$ | $22$ | $( 1, 3, 5, 8, 9,11, 2, 4, 6, 7,10,12)(13,16,17,20,22,23,14,15,18,19,21,24)$ |
| 12B-1 | $12^{2}$ | $2$ | $12$ | $22$ | $( 1,11,10, 8, 6, 3, 2,12, 9, 7, 5, 4)(13,23,21,20,18,16,14,24,22,19,17,15)$ |
| 12C1 | $12^{2}$ | $4$ | $12$ | $22$ | $( 1,24, 5,16, 9,20, 2,23, 6,15,10,19)( 3,13, 8,17,11,22, 4,14, 7,18,12,21)$ |
| 12C-1 | $12^{2}$ | $4$ | $12$ | $22$ | $( 1,20,10,16, 6,24, 2,19, 9,15, 5,23)( 3,22,12,17, 7,13, 4,21,11,18, 8,14)$ |
| 24A1 | $24$ | $4$ | $24$ | $23$ | $( 1,13, 3,15, 5,17, 8,19, 9,22,11,24, 2,14, 4,16, 6,18, 7,20,10,21,12,23)$ |
| 24A-1 | $24$ | $4$ | $24$ | $23$ | $( 1,18,11,15,10,14, 8,23, 6,22, 3,20, 2,17,12,16, 9,13, 7,24, 5,21, 4,19)$ |
| 24B1 | $24$ | $4$ | $24$ | $23$ | $( 1,14, 4,15, 5,18, 7,19, 9,21,12,24, 2,13, 3,16, 6,17, 8,20,10,22,11,23)$ |
| 24B-1 | $24$ | $4$ | $24$ | $23$ | $( 1,17,12,15,10,13, 7,23, 6,21, 4,20, 2,18,11,16, 9,14, 8,24, 5,22, 3,19)$ |
Malle's constant $a(G)$: $1/6$
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: | 96.183 | magma: IdentifyGroup(G);
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| Character table: | 33 x 33 character table |
magma: CharacterTable(G);