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Magma
magma: G := TransitiveGroup(24, 22);
Group action invariants
Degree $n$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $22$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $\GL(2,3)$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,12,17,15,2,11,18,16)(3,10,7,24,4,9,8,23)(5,20,21,13,6,19,22,14), (1,6,14,2,5,13)(3,17,9,4,18,10)(7,11,22,8,12,21)(15,23,19,16,24,20) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $6$: $S_3$ $24$: $S_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $S_3$
Degree 4: $S_4$
Degree 6: $S_4$
Degree 8: $\textrm{GL(2,3)}$ x 2
Degree 12: $S_4$
Low degree siblings
8T23 x 2, 16T66Siblings 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, 2, 2, 2, 1, 1 $ | $12$ | $2$ | $( 3,21)( 4,22)( 5, 7)( 6, 8)( 9,20)(10,19)(11,16)(12,15)(13,24)(14,23)(17,18)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 3,21)( 2, 4,22)( 5,12,20)( 6,11,19)( 7, 9,15)( 8,10,16)(13,23,18) (14,24,17)$ |
$ 6, 6, 6, 6 $ | $8$ | $6$ | $( 1, 4,21, 2, 3,22)( 5,11,20, 6,12,19)( 7,10,15, 8, 9,16)(13,24,18,14,23,17)$ |
$ 8, 8, 8 $ | $6$ | $8$ | $( 1, 5,12,24, 2, 6,11,23)( 3, 8,13,19, 4, 7,14,20)( 9,16,21,18,10,15,22,17)$ |
$ 8, 8, 8 $ | $6$ | $8$ | $( 1, 6,12,23, 2, 5,11,24)( 3, 7,13,20, 4, 8,14,19)( 9,15,21,17,10,16,22,18)$ |
$ 4, 4, 4, 4, 4, 4 $ | $6$ | $4$ | $( 1,11, 2,12)( 3,14, 4,13)( 5,23, 6,24)( 7,19, 8,20)( 9,22,10,21)(15,18,16,17)$ |
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: | 48.29 | magma: IdentifyGroup(G);
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Character table: |
2 4 2 4 1 1 3 3 3 3 1 . 1 1 1 . . . 1a 2a 2b 3a 6a 8a 8b 4a 2P 1a 1a 1a 3a 3a 4a 4a 2b 3P 1a 2a 2b 1a 2b 8a 8b 4a 5P 1a 2a 2b 3a 6a 8b 8a 4a 7P 1a 2a 2b 3a 6a 8b 8a 4a X.1 1 1 1 1 1 1 1 1 X.2 1 -1 1 1 1 -1 -1 1 X.3 2 . 2 -1 -1 . . 2 X.4 2 . -2 -1 1 A -A . X.5 2 . -2 -1 1 -A A . X.6 3 1 3 . . -1 -1 -1 X.7 3 -1 3 . . 1 1 -1 X.8 4 . -4 1 -1 . . . A = E(8)+E(8)^3 = Sqrt(-2) = i2 |
magma: CharacterTable(G);