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Magma
magma: G := TransitiveGroup(24, 50);
Group action invariants
Degree $n$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $50$ | 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: | (1,22,18,13,10,6)(2,21,17,14,9,5)(3,23,20,15,11,8)(4,24,19,16,12,7), (1,4,10,23,5,8)(2,3,9,24,6,7)(11,17,20,13,16,22)(12,18,19,14,15,21) | 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: $C_2$ x 3
Degree 3: $C_3$
Degree 4: $C_2^2$
Degree 6: $C_6$ x 3, $A_4$, $A_4\times C_2$ x 3
Degree 8: None
Degree 12: $C_6\times C_2$, $A_4 \times C_2$ x 3, $C_2^2 \times A_4$ x 3
Low degree siblings
12T25 x 3, 12T26 x 2, 16T58, 24T49 x 3Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Label | 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 $ | $3$ | $2$ | $( 3,16)( 4,15)( 5,18)( 6,17)( 7,20)( 8,19)( 9,22)(10,21)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7,19)( 8,20)( 9,21)(10,22)(11,12)(13,14)(15,16)(17,18) (23,24)$ | |
$ 6, 6, 6, 6 $ | $4$ | $6$ | $( 1, 3,10,11,18,20)( 2, 4, 9,12,17,19)( 5, 7,14,16,21,24)( 6, 8,13,15,22,23)$ | |
$ 6, 6, 6, 6 $ | $4$ | $6$ | $( 1, 4,10,23, 5, 8)( 2, 3, 9,24, 6, 7)(11,17,20,13,16,22)(12,18,19,14,15,21)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 5,10)( 2, 6, 9)( 3, 7,24)( 4, 8,23)(11,16,20)(12,15,19)(13,17,22) (14,18,21)$ | |
$ 6, 6, 6, 6 $ | $4$ | $6$ | $( 1, 6,10,13,18,22)( 2, 5, 9,14,17,21)( 3, 8,11,15,20,23)( 4, 7,12,16,19,24)$ | |
$ 6, 6, 6, 6 $ | $4$ | $6$ | $( 1, 7,18,11,21, 3)( 2, 8,17,12,22, 4)( 5,24,10,16,14,20)( 6,23, 9,15,13,19)$ | |
$ 6, 6, 6, 6 $ | $4$ | $6$ | $( 1, 8, 5,23,10, 4)( 2, 7, 6,24, 9, 3)(11,22,16,13,20,17)(12,21,15,14,19,18)$ | |
$ 6, 6, 6, 6 $ | $4$ | $6$ | $( 1, 9, 5,13,21,17)( 2,10, 6,14,22,18)( 3,12,20,15,24, 8)( 4,11,19,16,23, 7)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1,10,18)( 2, 9,17)( 3,11,20)( 4,12,19)( 5,14,21)( 6,13,22)( 7,16,24) ( 8,15,23)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1,11)( 2,12)( 3, 5)( 4, 6)( 7,10)( 8, 9)(13,23)(14,24)(15,17)(16,18)(19,22) (20,21)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,11)( 2,12)( 3,18)( 4,17)( 5,16)( 6,15)( 7,21)( 8,22)( 9,19)(10,20)(13,23) (14,24)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1,12)( 2,11)( 3, 6)( 4, 5)( 7,22)( 8,21)( 9,20)(10,19)(13,24)(14,23)(15,18) (16,17)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,13)( 2,14)( 3,15)( 4,16)( 5,17)( 6,18)( 7,19)( 8,20)( 9,21)(10,22)(11,23) (12,24)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,23)( 2,24)( 3, 6)( 4, 5)( 7, 9)( 8,10)(11,13)(12,14)(15,18)(16,17)(19,21) (20,22)$ |
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: |
1A | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A1 | 3A-1 | 6A1 | 6A-1 | 6B1 | 6B-1 | 6C1 | 6C-1 | ||
Size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
2 P | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 3A-1 | 3A1 | 3A1 | 3A-1 | 3A-1 | 3A1 | 3A1 | 3A-1 | |
3 P | 1A | 2C | 2A | 2B | 2G | 2D | 2F | 2E | 1A | 1A | 2B | 2A | 2B | 2A | 2C | 2C | |
Type | |||||||||||||||||
48.49.1a | R | ||||||||||||||||
48.49.1b | R | ||||||||||||||||
48.49.1c | R | ||||||||||||||||
48.49.1d | R | ||||||||||||||||
48.49.1e1 | C | ||||||||||||||||
48.49.1e2 | C | ||||||||||||||||
48.49.1f1 | C | ||||||||||||||||
48.49.1f2 | C | ||||||||||||||||
48.49.1g1 | C | ||||||||||||||||
48.49.1g2 | C | ||||||||||||||||
48.49.1h1 | C | ||||||||||||||||
48.49.1h2 | C | ||||||||||||||||
48.49.3a | R | ||||||||||||||||
48.49.3b | R | ||||||||||||||||
48.49.3c | R | ||||||||||||||||
48.49.3d | R |
magma: CharacterTable(G);