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Magma
magma: G := TransitiveGroup(24, 12);
Group action invariants
Degree $n$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_4\times S_3$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $24$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,6)(2,5)(3,15)(4,16)(7,12)(8,11)(9,21)(10,22)(13,17)(14,18)(19,23)(20,24), (1,16,5,20,9,23,13,3,18,7,22,11)(2,15,6,19,10,24,14,4,17,8,21,12) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $6$: $S_3$ $8$: $C_4\times C_2$ $12$: $D_{6}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 3: $S_3$
Degree 8: $C_4\times C_2$
Degree 12: $D_6$, $S_3 \times C_4$ x 2
Low degree siblings
12T11 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, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3,12)( 4,11)( 5,21)( 6,22)( 7, 8)( 9,17)(10,18)(13,14)(15,23)(16,24) (19,20)$ |
$ 12, 12 $ | $2$ | $12$ | $( 1, 3, 5, 7, 9,11,13,16,18,20,22,23)( 2, 4, 6, 8,10,12,14,15,17,19,21,24)$ |
$ 4, 4, 4, 4, 4, 4 $ | $3$ | $4$ | $( 1, 4,13,15)( 2, 3,14,16)( 5,24,18,12)( 6,23,17,11)( 7,10,20,21)( 8, 9,19,22)$ |
$ 6, 6, 6, 6 $ | $2$ | $6$ | $( 1, 5, 9,13,18,22)( 2, 6,10,14,17,21)( 3, 7,11,16,20,23)( 4, 8,12,15,19,24)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 6)( 2, 5)( 3,15)( 4,16)( 7,12)( 8,11)( 9,21)(10,22)(13,17)(14,18)(19,23) (20,24)$ |
$ 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 7,13,20)( 2, 8,14,19)( 3, 9,16,22)( 4,10,15,21)( 5,11,18,23)( 6,12,17,24)$ |
$ 4, 4, 4, 4, 4, 4 $ | $3$ | $4$ | $( 1, 8,13,19)( 2, 7,14,20)( 3,17,16, 6)( 4,18,15, 5)( 9,24,22,12)(10,23,21,11)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 9,18)( 2,10,17)( 3,11,20)( 4,12,19)( 5,13,22)( 6,14,21)( 7,16,23) ( 8,15,24)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,13)( 2,14)( 3,16)( 4,15)( 5,18)( 6,17)( 7,20)( 8,19)( 9,22)(10,21)(11,23) (12,24)$ |
$ 12, 12 $ | $2$ | $12$ | $( 1,16, 5,20, 9,23,13, 3,18, 7,22,11)( 2,15, 6,19,10,24,14, 4,17, 8,21,12)$ |
$ 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,20,13, 7)( 2,19,14, 8)( 3,22,16, 9)( 4,21,15,10)( 5,23,18,11)( 6,24,17,12)$ |
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: | 24.5 | magma: IdentifyGroup(G);
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Character table: |
2 3 3 2 3 2 3 3 3 2 3 2 3 3 1 . 1 . 1 . 1 . 1 1 1 1 1a 2a 12a 4a 6a 2b 4b 4c 3a 2c 12b 4d 2P 1a 1a 6a 2c 3a 1a 2c 2c 3a 1a 6a 2c 3P 1a 2a 4b 4c 2c 2b 4d 4a 1a 2c 4d 4b 5P 1a 2a 12a 4a 6a 2b 4b 4c 3a 2c 12b 4d 7P 1a 2a 12b 4c 6a 2b 4d 4a 3a 2c 12a 4b 11P 1a 2a 12b 4c 6a 2b 4d 4a 3a 2c 12a 4b X.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 X.3 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 X.5 1 -1 A -A -1 1 -A A 1 -1 -A A X.6 1 -1 -A A -1 1 A -A 1 -1 A -A X.7 1 1 A A -1 -1 -A -A 1 -1 -A A X.8 1 1 -A -A -1 -1 A A 1 -1 A -A X.9 2 . 1 . -1 . -2 . -1 2 1 -2 X.10 2 . -1 . -1 . 2 . -1 2 -1 2 X.11 2 . A . 1 . B . -1 -2 -A -B X.12 2 . -A . 1 . -B . -1 -2 A B A = -E(4) = -Sqrt(-1) = -i B = -2*E(4) = -2*Sqrt(-1) = -2i |
magma: CharacterTable(G);