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Magma
magma: G := TransitiveGroup(24, 26);
Group action invariants
Degree $n$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $26$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $S_3\times Q_8$ | ||
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: | (3,11)(4,12)(5,22)(6,21)(9,17)(10,18)(15,24)(16,23), (1,14,2,13)(3,24,4,23)(5,10,6,9)(7,20,8,19)(11,15,12,16)(17,22,18,21), (1,24,10,8,17,15,2,23,9,7,18,16)(3,13,12,22,19,6,4,14,11,21,20,5) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_2^2$ x 7 $6$: $S_3$ $8$: $C_2^3$, $Q_8$ x 2 $12$: $D_{6}$ x 3 $16$: $Q_8\times C_2$ $24$: $S_3 \times C_2^2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 3: $S_3$
Degree 4: $C_2^2$
Degree 6: $D_{6}$ x 3
Degree 8: $Q_8$
Degree 12: $S_3 \times C_2^2$
Low degree siblings
24T26Siblings are shown 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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 3,11)( 4,12)( 5,22)( 6,21)( 9,17)(10,18)(15,24)(16,23)$ |
$ 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 $ | $3$ | $2$ | $( 1, 2)( 3,12)( 4,11)( 5,21)( 6,22)( 7, 8)( 9,18)(10,17)(13,14)(15,23)(16,24) (19,20)$ |
$ 4, 4, 4, 4, 4, 4 $ | $6$ | $4$ | $( 1, 3, 2, 4)( 5,23, 6,24)( 7,22, 8,21)( 9,19,10,20)(11,18,12,17)(13,15,14,16)$ |
$ 12, 12 $ | $4$ | $12$ | $( 1, 3,18,20, 9,11, 2, 4,17,19,10,12)( 5, 8,21,24,14,16, 6, 7,22,23,13,15)$ |
$ 12, 12 $ | $4$ | $12$ | $( 1, 5,10,13,17,22, 2, 6, 9,14,18,21)( 3, 7,12,16,19,24, 4, 8,11,15,20,23)$ |
$ 4, 4, 4, 4, 4, 4 $ | $6$ | $4$ | $( 1, 5, 2, 6)( 3,15, 4,16)( 7,12, 8,11)( 9,22,10,21)(13,17,14,18)(19,24,20,23)$ |
$ 4, 4, 4, 4, 4, 4 $ | $6$ | $4$ | $( 1, 7, 2, 8)( 3, 6, 4, 5)( 9,24,10,23)(11,21,12,22)(13,20,14,19)(15,18,16,17)$ |
$ 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 7, 2, 8)( 3,21, 4,22)( 5,11, 6,12)( 9,15,10,16)(13,20,14,19)(17,24,18,23)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 9,17)( 2,10,18)( 3,11,19)( 4,12,20)( 5,14,22)( 6,13,21)( 7,15,24) ( 8,16,23)$ |
$ 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,10,17, 2, 9,18)( 3,12,19, 4,11,20)( 5,13,22, 6,14,21)( 7,16,24, 8,15,23)$ |
$ 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,13, 2,14)( 3,16, 4,15)( 5,17, 6,18)( 7,19, 8,20)( 9,21,10,22)(11,23,12,24)$ |
$ 12, 12 $ | $4$ | $12$ | $( 1,15,18, 8, 9,24, 2,16,17, 7,10,23)( 3, 6,20,22,11,13, 4, 5,19,21,12,14)$ |
$ 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,19, 2,20)( 3,10, 4, 9)( 5,23, 6,24)( 7,14, 8,13)(11,18,12,17)(15,22,16,21)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $48=2^{4} \cdot 3$ | 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: | 48.40 | magma: IdentifyGroup(G);
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Character table: |
2 4 4 4 4 3 2 2 3 3 3 3 3 3 2 3 3 1 . 1 . . 1 1 . . 1 1 1 1 1 1 1a 2a 2b 2c 4a 12a 12b 4b 4c 4d 3a 6a 4e 12c 4f 2P 1a 1a 1a 1a 2b 6a 6a 2b 2b 2b 3a 3a 2b 6a 2b 3P 1a 2a 2b 2c 4a 4f 4e 4b 4c 4d 1a 2b 4e 4d 4f 5P 1a 2a 2b 2c 4a 12a 12b 4b 4c 4d 3a 6a 4e 12c 4f 7P 1a 2a 2b 2c 4a 12a 12b 4b 4c 4d 3a 6a 4e 12c 4f 11P 1a 2a 2b 2c 4a 12a 12b 4b 4c 4d 3a 6a 4e 12c 4f X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 -1 1 -1 1 1 -1 1 1 -1 -1 1 X.3 1 -1 1 -1 -1 1 1 -1 -1 1 1 1 1 1 1 X.4 1 -1 1 -1 1 -1 -1 1 -1 1 1 1 -1 1 -1 X.5 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 -1 X.6 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 X.7 1 1 1 1 -1 -1 1 1 -1 -1 1 1 1 -1 -1 X.8 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 X.9 2 . 2 . . -1 -1 . . 2 -1 -1 2 -1 2 X.10 2 . 2 . . -1 1 . . -2 -1 -1 -2 1 2 X.11 2 . 2 . . 1 -1 . . -2 -1 -1 2 1 -2 X.12 2 . 2 . . 1 1 . . 2 -1 -1 -2 -1 -2 X.13 2 -2 -2 2 . . . . . . 2 -2 . . . X.14 2 2 -2 -2 . . . . . . 2 -2 . . . X.15 4 . -4 . . . . . . . -2 2 . . . |
magma: CharacterTable(G);