Show commands:
Magma
magma: G := TransitiveGroup(24, 45);
Group action invariants
Degree $n$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
Transitive number $t$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
Group: | $C_6:D_4$ | ||
Parity: | $1$ | magma: IsEven(G);
| |
Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
$\card{\Aut(F/K)}$: | $12$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
Generators: | (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24), (1,23)(2,24)(3,21)(4,22)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,13)(12,14), (1,9,17)(2,10,18)(3,24,19,15,11,7)(4,23,20,16,12,8)(5,14,22)(6,13,21) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_2^2$ x 7 $6$: $S_3$ $8$: $D_{4}$ x 2, $C_2^3$ $12$: $D_{6}$ x 3 $16$: $D_4\times C_2$ $24$: $S_3 \times C_2^2$, $(C_6\times C_2):C_2$ x 2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 3: $S_3$
Degree 4: $C_2^2$, $D_{4}$ x 2
Degree 8: $D_4\times C_2$
Degree 12: $D_6$, $(C_6\times C_2):C_2$ x 2
Low degree siblings
24T25 x 2, 24T45Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 3,15)( 4,16)( 7,19)( 8,20)(11,24)(12,23)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 2)( 3,16)( 4,15)( 5, 6)( 7,20)( 8,19)( 9,10)(11,23)(12,24)(13,14)(17,18) (21,22)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $6$ | $2$ | $( 1, 3)( 2, 4)( 5,24)( 6,23)( 7,22)( 8,21)( 9,19)(10,20)(11,17)(12,18)(13,16) (14,15)$ |
$ 4, 4, 4, 4, 4, 4 $ | $6$ | $4$ | $( 1, 3,14,15)( 2, 4,13,16)( 5,24,17,11)( 6,23,18,12)( 7, 9,19,22)( 8,10,20,21)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $6$ | $2$ | $( 1, 4)( 2, 3)( 5,23)( 6,24)( 7,21)( 8,22)( 9,20)(10,19)(11,18)(12,17)(13,15) (14,16)$ |
$ 4, 4, 4, 4, 4, 4 $ | $6$ | $4$ | $( 1, 4,14,16)( 2, 3,13,15)( 5,23,17,12)( 6,24,18,11)( 7,10,19,21)( 8, 9,20,22)$ |
$ 6, 6, 6, 6 $ | $2$ | $6$ | $( 1, 5, 9,14,17,22)( 2, 6,10,13,18,21)( 3, 7,11,15,19,24)( 4, 8,12,16,20,23)$ |
$ 6, 6, 3, 3, 3, 3 $ | $2$ | $6$ | $( 1, 5, 9,14,17,22)( 2, 6,10,13,18,21)( 3,19,11)( 4,20,12)( 7,24,15)( 8,23,16)$ |
$ 6, 6, 6, 6 $ | $2$ | $6$ | $( 1, 6, 9,13,17,21)( 2, 5,10,14,18,22)( 3, 8,11,16,19,23)( 4, 7,12,15,20,24)$ |
$ 6, 6, 6, 6 $ | $2$ | $6$ | $( 1, 6, 9,13,17,21)( 2, 5,10,14,18,22)( 3,20,11, 4,19,12)( 7,23,15, 8,24,16)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 9,17)( 2,10,18)( 3,11,19)( 4,12,20)( 5,14,22)( 6,13,21)( 7,15,24) ( 8,16,23)$ |
$ 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,10,17, 2, 9,18)( 3,12,19, 4,11,20)( 5,13,22, 6,14,21)( 7,16,24, 8,15,23)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,13)( 2,14)( 3,16)( 4,15)( 5,18)( 6,17)( 7,20)( 8,19)( 9,21)(10,22)(11,23) (12,24)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,14)( 2,13)( 3,15)( 4,16)( 5,17)( 6,18)( 7,19)( 8,20)( 9,22)(10,21)(11,24) (12,23)$ |
$ 6, 6, 3, 3, 3, 3 $ | $2$ | $6$ | $( 1,17, 9)( 2,18,10)( 3, 7,11,15,19,24)( 4, 8,12,16,20,23)( 5,22,14)( 6,21,13)$ |
$ 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,18, 9, 2,17,10)( 3, 8,11,16,19,23)( 4, 7,12,15,20,24)( 5,21,14, 6,22,13)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $48=2^{4} \cdot 3$ | magma: Order(G);
| |
Cyclic: | no | magma: IsCyclic(G);
| |
Abelian: | no | magma: IsAbelian(G);
| |
Solvable: | yes | magma: IsSolvable(G);
| |
Nilpotency class: | not nilpotent | ||
Label: | 48.43 | magma: IdentifyGroup(G);
|
Character table: |
2 4 3 4 3 3 3 3 3 3 3 3 3 3 3 4 4 3 3 3 1 1 1 1 . . . . 1 1 1 1 1 1 1 1 1 1 1a 2a 2b 2c 2d 4a 2e 4b 6a 6b 6c 6d 3a 6e 2f 2g 6f 6g 2P 1a 1a 1a 1a 1a 2g 1a 2g 3a 3a 3a 3a 3a 3a 1a 1a 3a 3a 3P 1a 2a 2b 2c 2d 4a 2e 4b 2g 2a 2f 2c 1a 2b 2f 2g 2a 2c 5P 1a 2a 2b 2c 2d 4a 2e 4b 6a 6f 6c 6g 3a 6e 2f 2g 6b 6d X.1 1 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 -1 1 X.3 1 -1 -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 -1 -1 X.5 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 1 -1 -1 X.6 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 1 -1 -1 1 1 -1 X.7 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 1 -1 X.8 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 X.9 2 -2 -2 2 . . . . -1 1 1 -1 -1 1 -2 2 1 -1 X.10 2 -2 2 -2 . . . . -1 1 -1 1 -1 -1 2 2 1 1 X.11 2 2 -2 -2 . . . . -1 -1 1 1 -1 1 -2 2 -1 1 X.12 2 2 2 2 . . . . -1 -1 -1 -1 -1 -1 2 2 -1 -1 X.13 2 . 2 . . . . . -2 . -2 . 2 2 -2 -2 . . X.14 2 . -2 . . . . . -2 . 2 . 2 -2 2 -2 . . X.15 2 . -2 . . . . . 1 A -1 -A -1 1 2 -2 -A A X.16 2 . -2 . . . . . 1 -A -1 A -1 1 2 -2 A -A X.17 2 . 2 . . . . . 1 A 1 A -1 -1 -2 -2 -A -A X.18 2 . 2 . . . . . 1 -A 1 -A -1 -1 -2 -2 A A A = -E(3)+E(3)^2 = -Sqrt(-3) = -i3 |
magma: CharacterTable(G);