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Magma
magma: G := TransitiveGroup(24, 25);
Group action invariants
Degree $n$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $25$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_6:D_4$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $4$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (3,12)(4,11)(5,22)(6,21)(7,8)(9,17)(10,18)(15,23)(16,24)(19,20), (1,23,2,24)(3,22,4,21)(5,20,6,19)(7,17,8,18)(9,16,10,15)(11,14,12,13), (1,20)(2,19)(3,10)(4,9)(5,23)(6,24)(7,13)(8,14)(11,18)(12,17)(15,21)(16,22) | 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 $6$: $S_3$ $8$: $D_{4}$ x 2, $C_2^3$ $12$: $D_{6}$ x 3 $16$: $D_4\times C_2$ $24$: $S_3 \times C_2^2$, $(C_6\times C_2):C_2$ x 2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 3: $S_3$
Degree 4: $C_2^2$, $D_{4}$ x 2
Degree 6: $D_{6}$ x 3
Degree 8: $D_4\times C_2$
Degree 12: $S_3 \times C_2^2$, $(C_6\times C_2):C_2$ x 2
Low degree siblings
24T25, 24T45 x 2Siblings 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, 1, 1, 1, 1 $ | $6$ | $2$ | $( 3,12)( 4,11)( 5,22)( 6,21)( 7, 8)( 9,17)(10,18)(15,23)(16,24)(19,20)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)$ |
$ 4, 4, 4, 4, 4, 4 $ | $6$ | $4$ | $( 1, 3, 2, 4)( 5,24, 6,23)( 7,21, 8,22)( 9,19,10,20)(11,18,12,17)(13,16,14,15)$ |
$ 6, 6, 6, 6 $ | $2$ | $6$ | $( 1, 3,17,19, 9,11)( 2, 4,18,20,10,12)( 5, 7,22,24,14,15)( 6, 8,21,23,13,16)$ |
$ 6, 6, 6, 6 $ | $2$ | $6$ | $( 1, 4,17,20, 9,12)( 2, 3,18,19,10,11)( 5, 8,22,23,14,16)( 6, 7,21,24,13,15)$ |
$ 6, 6, 6, 6 $ | $2$ | $6$ | $( 1, 5, 9,14,17,22)( 2, 6,10,13,18,21)( 3, 7,11,15,19,24)( 4, 8,12,16,20,23)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $6$ | $2$ | $( 1, 5)( 2, 6)( 3,16)( 4,15)( 7,12)( 8,11)( 9,22)(10,21)(13,18)(14,17)(19,23) (20,24)$ |
$ 6, 6, 6, 6 $ | $2$ | $6$ | $( 1, 6, 9,13,17,21)( 2, 5,10,14,18,22)( 3, 8,11,16,19,23)( 4, 7,12,15,20,24)$ |
$ 4, 4, 4, 4, 4, 4 $ | $6$ | $4$ | $( 1, 7, 2, 8)( 3, 6, 4, 5)( 9,24,10,23)(11,21,12,22)(13,20,14,19)(15,18,16,17)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 7)( 2, 8)( 3,22)( 4,21)( 5,11)( 6,12)( 9,15)(10,16)(13,20)(14,19)(17,24) (18,23)$ |
$ 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 $ | $2$ | $6$ | $( 1,10,17, 2, 9,18)( 3,12,19, 4,11,20)( 5,13,22, 6,14,21)( 7,16,24, 8,15,23)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,13)( 2,14)( 3,16)( 4,15)( 5,18)( 6,17)( 7,20)( 8,19)( 9,21)(10,22)(11,23) (12,24)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,14)( 2,13)( 3,15)( 4,16)( 5,17)( 6,18)( 7,19)( 8,20)( 9,22)(10,21)(11,24) (12,23)$ |
$ 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,15,17, 7, 9,24)( 2,16,18, 8,10,23)( 3, 5,19,22,11,14)( 4, 6,20,21,12,13)$ |
$ 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,16,17, 8, 9,23)( 2,15,18, 7,10,24)( 3, 6,19,21,11,13)( 4, 5,20,22,12,14)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1,19)( 2,20)( 3, 9)( 4,10)( 5,24)( 6,23)( 7,14)( 8,13)(11,17)(12,18)(15,22) (16,21)$ |
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: | 48.43 | magma: IdentifyGroup(G);
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Character table: |
2 4 3 4 3 3 3 3 3 3 3 3 3 3 4 4 3 3 3 3 1 . 1 . 1 1 1 . 1 . 1 1 1 1 1 1 1 1 1a 2a 2b 4a 6a 6b 6c 2c 6d 4b 2d 3a 6e 2e 2f 6f 6g 2g 2P 1a 1a 1a 2b 3a 3a 3a 1a 3a 2b 1a 3a 3a 1a 1a 3a 3a 1a 3P 1a 2a 2b 4a 2g 2g 2f 2c 2e 4b 2d 1a 2b 2e 2f 2d 2d 2g 5P 1a 2a 2b 4a 6b 6a 6c 2c 6d 4b 2d 3a 6e 2e 2f 6g 6f 2g X.1 1 1 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 -1 -1 1 X.3 1 -1 1 -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 1 1 -1 X.5 1 -1 1 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 1 1 -1 X.7 1 1 1 -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 -1 -1 1 X.9 2 . -2 . . . -2 . 2 . . 2 -2 2 -2 . . . X.10 2 . -2 . . . 2 . -2 . . 2 -2 -2 2 . . . X.11 2 . 2 . -1 -1 -1 . -1 . 2 -1 -1 2 2 -1 -1 2 X.12 2 . 2 . -1 -1 1 . 1 . -2 -1 -1 -2 -2 1 1 2 X.13 2 . 2 . 1 1 -1 . -1 . -2 -1 -1 2 2 1 1 -2 X.14 2 . 2 . 1 1 1 . 1 . 2 -1 -1 -2 -2 -1 -1 -2 X.15 2 . -2 . A -A -1 . 1 . . -1 1 -2 2 A -A . X.16 2 . -2 . -A A -1 . 1 . . -1 1 -2 2 -A A . X.17 2 . -2 . A -A 1 . -1 . . -1 1 2 -2 -A A . X.18 2 . -2 . -A A 1 . -1 . . -1 1 2 -2 A -A . A = -E(3)+E(3)^2 = -Sqrt(-3) = -i3 |
magma: CharacterTable(G);