Group action invariants
Degree $n$: | $24$ | |
Transitive number $t$: | $50$ | |
Group: | $C_2^2\times A_4$ | |
Parity: | $1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $8$ | |
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) |
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
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 $ | $1$ | $2$ | $( 1,23)( 2,24)( 3, 6)( 4, 5)( 7, 9)( 8,10)(11,13)(12,14)(15,18)(16,17)(19,21) (20,22)$ |
Group invariants
Order: | $48=2^{4} \cdot 3$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [48, 49] |
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 2d 2g 1a 2f 2d 2g 2f 1a 2c 2d 2e 2f 2g 5P 1a 2a 2b 6d 6e 3b 6f 6a 6b 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 3 -1 3 3 X.14 3 -1 -1 . . . . . . . . 1 -3 1 3 -3 X.15 3 -1 1 . . . . . . . . -1 3 1 -3 -3 X.16 3 -1 1 . . . . . . . . 1 -3 -1 -3 3 A = -E(3) = (1-Sqrt(-3))/2 = -b3 |