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Magma
magma: G := TransitiveGroup(24, 14);
Group action invariants
Degree $n$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $14$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_3:D_4$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $24$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,9)(2,10)(3,8)(4,7)(5,6)(11,23)(12,24)(13,22)(14,21)(15,20)(16,19)(17,18), (1,19,14,7)(2,20,13,8)(3,6,15,18)(4,5,16,17)(9,11,21,24)(10,12,22,23) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ $8$: $D_{4}$ $12$: $D_{6}$ 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 8: $D_4$
Degree 12: $D_6$, $(C_6\times C_2):C_2$, $(C_6\times C_2):C_2$
Low degree siblings
12T13, 12T15Siblings 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, 2 $ | $6$ | $2$ | $( 1, 2)( 3,24)( 4,23)( 5,22)( 6,21)( 7,20)( 8,19)( 9,18)(10,17)(11,15)(12,16) (13,14)$ |
$ 4, 4, 4, 4, 4, 4 $ | $6$ | $4$ | $( 1, 3,14,15)( 2, 4,13,16)( 5,24,17,11)( 6,23,18,12)( 7,10,19,22)( 8, 9,20,21)$ |
$ 6, 6, 6, 6 $ | $2$ | $6$ | $( 1, 4,18,20,10,11)( 2, 3,17,19, 9,12)( 5, 7,21,23,13,15)( 6, 8,22,24,14,16)$ |
$ 6, 6, 6, 6 $ | $2$ | $6$ | $( 1, 6,10,14,18,22)( 2, 5, 9,13,17,21)( 3, 7,12,15,19,23)( 4, 8,11,16,20,24)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 8)( 2, 7)( 3,21)( 4,22)( 5,12)( 6,11)( 9,15)(10,16)(13,19)(14,20)(17,23) (18,24)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,10,18)( 2, 9,17)( 3,12,19)( 4,11,20)( 5,13,21)( 6,14,22)( 7,15,23) ( 8,16,24)$ |
$ 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,11,10,20,18, 4)( 2,12, 9,19,17, 3)( 5,15,13,23,21, 7)( 6,16,14,24,22, 8)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,14)( 2,13)( 3,15)( 4,16)( 5,17)( 6,18)( 7,19)( 8,20)( 9,21)(10,22)(11,24) (12,23)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $24=2^{3} \cdot 3$ | 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: | 24.8 | magma: IdentifyGroup(G);
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Character table: |
2 3 2 2 2 2 2 2 2 3 3 1 . . 1 1 1 1 1 1 1a 2a 4a 6a 6b 2b 3a 6c 2c 2P 1a 1a 2c 3a 3a 1a 3a 3a 1a 3P 1a 2a 4a 2b 2c 2b 1a 2b 2c 5P 1a 2a 4a 6c 6b 2b 3a 6a 2c X.1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 1 1 1 1 X.3 1 -1 1 -1 1 -1 1 -1 1 X.4 1 1 -1 -1 1 -1 1 -1 1 X.5 2 . . . -2 . 2 . -2 X.6 2 . . 1 -1 -2 -1 1 2 X.7 2 . . -1 -1 2 -1 -1 2 X.8 2 . . A 1 . -1 -A -2 X.9 2 . . -A 1 . -1 A -2 A = -E(3)+E(3)^2 = -Sqrt(-3) = -i3 |
magma: CharacterTable(G);