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Magma
magma: G := TransitiveGroup(24, 17);
Group action invariants
Degree $n$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $17$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $D_4:C_6$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $12$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,7,5,12,9,3,2,8,6,11,10,4)(13,19,17,24,22,16,14,20,18,23,21,15), (1,24,5,16,9,20,2,23,6,15,10,19)(3,17,8,22,11,14,4,18,7,21,12,13), (1,3,10,12,6,7,2,4,9,11,5,8)(13,15,21,23,18,20,14,16,22,24,17,19) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 7 $3$: $C_3$ $4$: $C_2^2$ x 7 $6$: $C_6$ x 7 $8$: $C_2^3$ $12$: $C_6\times C_2$ x 7 $16$: $Q_8:C_2$ $24$: 24T3 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
Degree 8: $Q_8:C_2$
Degree 12: $C_6\times C_2$
Low degree siblings
24T17 x 2Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)$ | |
$ 12, 12 $ | $1$ | $12$ | $( 1, 3,10,12, 6, 7, 2, 4, 9,11, 5, 8)(13,15,21,23,18,20,14,16,22,24,17,19)$ | |
$ 12, 12 $ | $2$ | $12$ | $( 1, 3,10,12, 6, 7, 2, 4, 9,11, 5, 8)(13,16,21,24,18,19,14,15,22,23,17,20)$ | |
$ 12, 12 $ | $1$ | $12$ | $( 1, 4,10,11, 6, 8, 2, 3, 9,12, 5, 7)(13,16,21,24,18,19,14,15,22,23,17,20)$ | |
$ 6, 6, 6, 6 $ | $1$ | $6$ | $( 1, 5, 9, 2, 6,10)( 3, 8,11, 4, 7,12)(13,17,22,14,18,21)(15,19,24,16,20,23)$ | |
$ 6, 6, 3, 3, 3, 3 $ | $2$ | $6$ | $( 1, 5, 9, 2, 6,10)( 3, 8,11, 4, 7,12)(13,18,22)(14,17,21)(15,20,24)(16,19,23)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 6, 9)( 2, 5,10)( 3, 7,11)( 4, 8,12)(13,18,22)(14,17,21)(15,20,24) (16,19,23)$ | |
$ 12, 12 $ | $2$ | $12$ | $( 1, 7, 5,12, 9, 3, 2, 8, 6,11,10, 4)(13,19,17,24,22,16,14,20,18,23,21,15)$ | |
$ 12, 12 $ | $1$ | $12$ | $( 1, 7, 5,12, 9, 3, 2, 8, 6,11,10, 4)(13,20,17,23,22,15,14,19,18,24,21,16)$ | |
$ 12, 12 $ | $1$ | $12$ | $( 1, 8, 5,11, 9, 4, 2, 7, 6,12,10, 3)(13,19,17,24,22,16,14,20,18,23,21,15)$ | |
$ 6, 6, 3, 3, 3, 3 $ | $2$ | $6$ | $( 1, 9, 6)( 2,10, 5)( 3,11, 7)( 4,12, 8)(13,21,18,14,22,17)(15,23,20,16,24,19)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 9, 6)( 2,10, 5)( 3,11, 7)( 4,12, 8)(13,22,18)(14,21,17)(15,24,20) (16,23,19)$ | |
$ 6, 6, 6, 6 $ | $1$ | $6$ | $( 1,10, 6, 2, 9, 5)( 3,12, 7, 4,11, 8)(13,21,18,14,22,17)(15,23,20,16,24,19)$ | |
$ 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,11, 2,12)( 3, 5, 4, 6)( 7,10, 8, 9)(13,23,14,24)(15,18,16,17)(19,21,20,22)$ | |
$ 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,11, 2,12)( 3, 5, 4, 6)( 7,10, 8, 9)(13,24,14,23)(15,17,16,18)(19,22,20,21)$ | |
$ 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,12, 2,11)( 3, 6, 4, 5)( 7, 9, 8,10)(13,23,14,24)(15,18,16,17)(19,21,20,22)$ | |
$ 12, 12 $ | $2$ | $12$ | $( 1,13, 5,17, 9,22, 2,14, 6,18,10,21)( 3,15, 8,19,11,24, 4,16, 7,20,12,23)$ | |
$ 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,13, 6,18, 9,22)( 2,14, 5,17,10,21)( 3,15, 7,20,11,24)( 4,16, 8,19,12,23)$ | |
$ 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,15, 2,16)( 3,21, 4,22)( 5,19, 6,20)( 7,14, 8,13)( 9,24,10,23)(11,17,12,18)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1,15)( 2,16)( 3,21)( 4,22)( 5,19)( 6,20)( 7,14)( 8,13)( 9,24)(10,23)(11,17) (12,18)$ | |
$ 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,17, 2,18)( 3,19, 4,20)( 5,22, 6,21)( 7,23, 8,24)( 9,14,10,13)(11,16,12,15)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1,17)( 2,18)( 3,19)( 4,20)( 5,22)( 6,21)( 7,23)( 8,24)( 9,14)(10,13)(11,16) (12,15)$ | |
$ 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,19, 9,16, 6,23)( 2,20,10,15, 5,24)( 3,13,11,22, 7,18)( 4,14,12,21, 8,17)$ | |
$ 12, 12 $ | $2$ | $12$ | $( 1,19,10,15, 6,23, 2,20, 9,16, 5,24)( 3,13,12,21, 7,18, 4,14,11,22, 8,17)$ | |
$ 12, 12 $ | $2$ | $12$ | $( 1,21,10,18, 6,14, 2,22, 9,17, 5,13)( 3,23,12,20, 7,16, 4,24,11,19, 8,15)$ | |
$ 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,21, 9,17, 6,14)( 2,22,10,18, 5,13)( 3,23,11,19, 7,16)( 4,24,12,20, 8,15)$ | |
$ 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,23, 6,16, 9,19)( 2,24, 5,15,10,20)( 3,18, 7,22,11,13)( 4,17, 8,21,12,14)$ | |
$ 12, 12 $ | $2$ | $12$ | $( 1,23, 5,15, 9,19, 2,24, 6,16,10,20)( 3,18, 8,21,11,13, 4,17, 7,22,12,14)$ |
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: | $2$ | ||
Label: | 48.47 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 2D | 3A1 | 3A-1 | 4A1 | 4A-1 | 4B | 4C | 4D | 6A1 | 6A-1 | 6B1 | 6B-1 | 6C1 | 6C-1 | 6D1 | 6D-1 | 12A1 | 12A-1 | 12A5 | 12A-5 | 12B1 | 12B-1 | 12C1 | 12C-1 | 12D1 | 12D-1 | ||
Size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | |
2 P | 1A | 1A | 1A | 1A | 1A | 3A-1 | 3A1 | 2A | 2A | 2A | 2A | 2A | 3A1 | 3A-1 | 3A1 | 3A1 | 3A-1 | 3A1 | 3A-1 | 3A-1 | 6A1 | 6A1 | 6A-1 | 6A-1 | 6A-1 | 6A1 | 6A-1 | 6A1 | 6A-1 | 6A1 | |
3 P | 1A | 2A | 2B | 2C | 2D | 1A | 1A | 4A-1 | 4A1 | 4D | 4B | 4C | 2A | 2A | 2C | 2D | 2C | 2B | 2D | 2B | 4A-1 | 4A1 | 4A1 | 4A-1 | 4B | 4B | 4C | 4C | 4D | 4D | |
Type | |||||||||||||||||||||||||||||||
48.47.1a | R | ||||||||||||||||||||||||||||||
48.47.1b | R | ||||||||||||||||||||||||||||||
48.47.1c | R | ||||||||||||||||||||||||||||||
48.47.1d | R | ||||||||||||||||||||||||||||||
48.47.1e | R | ||||||||||||||||||||||||||||||
48.47.1f | R | ||||||||||||||||||||||||||||||
48.47.1g | R | ||||||||||||||||||||||||||||||
48.47.1h | R | ||||||||||||||||||||||||||||||
48.47.1i1 | C | ||||||||||||||||||||||||||||||
48.47.1i2 | C | ||||||||||||||||||||||||||||||
48.47.1j1 | C | ||||||||||||||||||||||||||||||
48.47.1j2 | C | ||||||||||||||||||||||||||||||
48.47.1k1 | C | ||||||||||||||||||||||||||||||
48.47.1k2 | C | ||||||||||||||||||||||||||||||
48.47.1l1 | C | ||||||||||||||||||||||||||||||
48.47.1l2 | C | ||||||||||||||||||||||||||||||
48.47.1m1 | C | ||||||||||||||||||||||||||||||
48.47.1m2 | C | ||||||||||||||||||||||||||||||
48.47.1n1 | C | ||||||||||||||||||||||||||||||
48.47.1n2 | C | ||||||||||||||||||||||||||||||
48.47.1o1 | C | ||||||||||||||||||||||||||||||
48.47.1o2 | C | ||||||||||||||||||||||||||||||
48.47.1p1 | C | ||||||||||||||||||||||||||||||
48.47.1p2 | C | ||||||||||||||||||||||||||||||
48.47.2a1 | C | ||||||||||||||||||||||||||||||
48.47.2a2 | C | ||||||||||||||||||||||||||||||
48.47.2b1 | C | ||||||||||||||||||||||||||||||
48.47.2b2 | C | ||||||||||||||||||||||||||||||
48.47.2b3 | C | ||||||||||||||||||||||||||||||
48.47.2b4 | C |
magma: CharacterTable(G);