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Magma
magma: G := TransitiveGroup(24, 9);
Group action invariants
Degree $n$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $9$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2\times A_4$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $24$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,2)(3,9)(4,10)(5,11)(6,12)(7,8)(13,14)(15,21)(16,22)(17,23)(18,24)(19,20), (1,3,11,14,22,17)(2,4,12,13,21,18)(5,8,10,23,19,15)(6,7,9,24,20,16) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ $12$: $A_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $C_3$
Degree 4: $A_4$
Degree 6: $C_6$, $A_4$, $A_4\times C_2$
Degree 8: $A_4\times C_2$
Degree 12: $A_4$, $A_4\times C_2$, $A_4 \times C_2$
Low degree siblings
6T6, 8T13, 12T6, 12T7Siblings 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, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3, 9)( 4,10)( 5,11)( 6,12)( 7, 8)(13,14)(15,21)(16,22)(17,23)(18,24) (19,20)$ | |
$ 6, 6, 6, 6 $ | $4$ | $6$ | $( 1, 3,11,14,22,17)( 2, 4,12,13,21,18)( 5, 8,10,23,19,15)( 6, 7, 9,24,20,16)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 4,23)( 2, 3,24)( 5,14,21)( 6,13,22)( 7,10,12)( 8, 9,11)(15,18,20) (16,17,19)$ | |
$ 6, 6, 6, 6 $ | $4$ | $6$ | $( 1, 5, 4,14,23,21)( 2, 6, 3,13,24,22)( 7,18,10,20,12,15)( 8,17, 9,19,11,16)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 6,15)( 2, 5,16)( 3,19,12)( 4,20,11)( 7,17,21)( 8,18,22)( 9,13,23) (10,14,24)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 8)( 2, 7)( 3,21)( 4,22)( 5,12)( 6,11)( 9,15)(10,16)(13,20)(14,19)(17,24) (18,23)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,14)( 2,13)( 3,22)( 4,21)( 5,23)( 6,24)( 7,20)( 8,19)( 9,16)(10,15)(11,17) (12,18)$ |
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: | 24.13 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 3A1 | 3A-1 | 6A1 | 6A-1 | ||
Size | 1 | 1 | 3 | 3 | 4 | 4 | 4 | 4 | |
2 P | 1A | 1A | 1A | 1A | 3A-1 | 3A1 | 3A1 | 3A-1 | |
3 P | 1A | 2A | 2B | 2C | 1A | 1A | 2A | 2A | |
Type | |||||||||
24.13.1a | R | ||||||||
24.13.1b | R | ||||||||
24.13.1c1 | C | ||||||||
24.13.1c2 | C | ||||||||
24.13.1d1 | C | ||||||||
24.13.1d2 | C | ||||||||
24.13.3a | R | ||||||||
24.13.3b | R |
magma: CharacterTable(G);