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Magma
magma: G := TransitiveGroup(24, 49);
Group action invariants
Degree $n$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $49$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2^2\times A_4$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $8$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,15,21,23,17,20)(2,16,22,24,18,19)(3,9,11,5,8,13)(4,10,12,6,7,14), (1,19,18,12,9,3)(2,20,17,11,10,4)(5,23,22,16,14,8)(6,24,21,15,13,7) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $C_6$ x 3 $12$: $A_4$, $C_6\times C_2$ $24$: $A_4\times C_2$ x 3 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $C_3$
Degree 4: None
Degree 6: $C_6$, $A_4$, $A_4\times C_2$ x 3
Degree 8: None
Degree 12: $A_4\times C_2$ x 2, $A_4 \times C_2$, $C_2^2 \times A_4$, $C_2^2 \times A_4$ x 2
Low degree siblings
12T25 x 3, 12T26 x 2, 16T58, 24T49 x 2, 24T50Siblings 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 $ | $3$ | $2$ | $( 3, 4)( 5, 6)( 7,19)( 8,20)( 9,21)(10,22)(15,16)(17,18)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3,16)( 4,15)( 5,18)( 6,17)( 7,19)( 8,20)( 9,21)(10,22)(11,12)(13,14) (23,24)$ |
$ 6, 6, 6, 6 $ | $4$ | $6$ | $( 1, 3, 9,12,18,19)( 2, 4,10,11,17,20)( 5, 8,14,16,22,23)( 6, 7,13,15,21,24)$ |
$ 6, 6, 6, 6 $ | $4$ | $6$ | $( 1, 3,10,23, 5, 7)( 2, 4, 9,24, 6, 8)(11,17,19,13,15,22)(12,18,20,14,16,21)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 5,10)( 2, 6, 9)( 3, 7,23)( 4, 8,24)(11,15,19)(12,16,20)(13,17,22) (14,18,21)$ |
$ 6, 6, 6, 6 $ | $4$ | $6$ | $( 1, 5, 9,14,18,22)( 2, 6,10,13,17,21)( 3, 8,12,16,19,23)( 4, 7,11,15,20,24)$ |
$ 6, 6, 6, 6 $ | $4$ | $6$ | $( 1, 7, 5,23,10, 3)( 2, 8, 6,24, 9, 4)(11,22,15,13,19,17)(12,21,16,14,20,18)$ |
$ 6, 6, 6, 6 $ | $4$ | $6$ | $( 1, 7,17,12,21, 4)( 2, 8,18,11,22, 3)( 5,24, 9,16,13,19)( 6,23,10,15,14,20)$ |
$ 6, 6, 6, 6 $ | $4$ | $6$ | $( 1, 9, 6,14,22,17)( 2,10, 5,13,21,18)( 3,11,20,16,24, 7)( 4,12,19,15,23, 8)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 9,18)( 2,10,17)( 3,12,19)( 4,11,20)( 5,14,22)( 6,13,21)( 7,15,24) ( 8,16,23)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1,11)( 2,12)( 3, 5)( 4, 6)( 7, 9)( 8,10)(13,23)(14,24)(15,17)(16,18)(19,21) (20,22)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1,11)( 2,12)( 3, 6)( 4, 5)( 7,21)( 8,22)( 9,19)(10,20)(13,23)(14,24)(15,18) (16,17)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,12)( 2,11)( 3,18)( 4,17)( 5,16)( 6,15)( 7,21)( 8,22)( 9,19)(10,20)(13,24) (14,23)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,14)( 2,13)( 3,16)( 4,15)( 5,18)( 6,17)( 7,20)( 8,19)( 9,22)(10,21)(11,24) (12,23)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,23)( 2,24)( 3, 5)( 4, 6)( 7,10)( 8, 9)(11,13)(12,14)(15,17)(16,18)(19,22) (20,21)$ |
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: | 48.49 | magma: IdentifyGroup(G);
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Character table: |
2 4 4 4 2 2 2 2 2 2 2 2 4 4 4 4 4 3 1 . . 1 1 1 1 1 1 1 1 . . 1 1 1 1a 2a 2b 6a 6b 3a 6c 6d 6e 6f 3b 2c 2d 2e 2f 2g 2P 1a 1a 1a 3b 3b 3b 3b 3a 3a 3a 3a 1a 1a 1a 1a 1a 3P 1a 2a 2b 2e 2g 1a 2f 2g 2e 2f 1a 2c 2d 2e 2f 2g 5P 1a 2a 2b 6e 6d 3b 6f 6b 6a 6c 3a 2c 2d 2e 2f 2g X.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 X.3 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 X.5 1 -1 1 A -A -A A -/A /A /A -/A -1 1 -1 -1 1 X.6 1 -1 1 /A -/A -/A /A -A A A -A -1 1 -1 -1 1 X.7 1 -1 1 -/A /A -/A /A A -A A -A 1 -1 1 -1 -1 X.8 1 -1 1 -A A -A A /A -/A /A -/A 1 -1 1 -1 -1 X.9 1 1 1 A A -A -A /A /A -/A -/A -1 -1 -1 1 -1 X.10 1 1 1 /A /A -/A -/A A A -A -A -1 -1 -1 1 -1 X.11 1 1 1 -/A -/A -/A -/A -A -A -A -A 1 1 1 1 1 X.12 1 1 1 -A -A -A -A -/A -/A -/A -/A 1 1 1 1 1 X.13 3 -1 -1 . . . . . . . . -1 -1 3 3 3 X.14 3 -1 -1 . . . . . . . . 1 1 -3 3 -3 X.15 3 1 -1 . . . . . . . . -1 1 3 -3 -3 X.16 3 1 -1 . . . . . . . . 1 -1 -3 -3 3 A = -E(3) = (1-Sqrt(-3))/2 = -b3 |
magma: CharacterTable(G);