Group action invariants
| Degree $n$: | $24$ | |
| Transitive number $t$: | $36$ | |
| Group: | $Q_8:S_3$ | |
| Parity: | $-1$ | |
| Primitive: | no | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $2$ | |
| Generators: | (1,3,18,20,9,11,2,4,17,19,10,12)(5,8,21,24,14,16,6,7,22,23,13,15), (3,23)(4,24)(5,21)(6,22)(7,20)(8,19)(9,17)(10,18)(11,16)(12,15)(13,14) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ $8$: $D_{4}$ $12$: $D_{6}$ $16$: $QD_{16}$ $24$: $(C_6\times C_2):C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 4: $D_{4}$
Degree 6: $D_{6}$
Degree 8: $QD_{16}$
Degree 12: $(C_6\times C_2):C_2$
Low degree siblings
There are no siblings 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, 2, 2, 2, 1, 1 $ | $12$ | $2$ | $( 3,23)( 4,24)( 5,21)( 6,22)( 7,20)( 8,19)( 9,17)(10,18)(11,16)(12,15)(13,14)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)$ |
| $ 8, 8, 8 $ | $6$ | $8$ | $( 1, 3,13,16, 2, 4,14,15)( 5,24,17,11, 6,23,18,12)( 7, 9,19,21, 8,10,20,22)$ |
| $ 12, 12 $ | $4$ | $12$ | $( 1, 3,18,20, 9,11, 2, 4,17,19,10,12)( 5, 8,21,24,14,16, 6, 7,22,23,13,15)$ |
| $ 8, 8, 8 $ | $6$ | $8$ | $( 1, 4,13,15, 2, 3,14,16)( 5,23,17,12, 6,24,18,11)( 7,10,19,22, 8, 9,20,21)$ |
| $ 12, 12 $ | $4$ | $12$ | $( 1, 5,10,13,17,22, 2, 6, 9,14,18,21)( 3, 7,12,16,19,24, 4, 8,11,15,20,23)$ |
| $ 4, 4, 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 7, 2, 8)( 3,21, 4,22)( 5,11, 6,12)( 9,15,10,16)(13,20,14,19)(17,24,18,23)$ |
| $ 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 $ | $2$ | $6$ | $( 1,10,17, 2, 9,18)( 3,12,19, 4,11,20)( 5,13,22, 6,14,21)( 7,16,24, 8,15,23)$ |
| $ 12, 12 $ | $4$ | $12$ | $( 1,11,10,20,17, 3, 2,12, 9,19,18, 4)( 5,16,13,24,22, 8, 6,15,14,23,21, 7)$ |
| $ 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,13, 2,14)( 3,16, 4,15)( 5,17, 6,18)( 7,19, 8,20)( 9,21,10,22)(11,23,12,24)$ |
Group invariants
| Order: | $48=2^{4} \cdot 3$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | yes | |
| GAP id: | [48, 17] |
| Character table: |
2 4 2 4 3 2 3 2 2 3 3 2 3
3 1 . 1 . 1 . 1 1 1 1 1 1
1a 2a 2b 8a 12a 8b 12b 4a 3a 6a 12c 4b
2P 1a 1a 1a 4b 6a 4b 6a 2b 3a 3a 6a 2b
3P 1a 2a 2b 8a 4a 8b 4b 4a 1a 2b 4a 4b
5P 1a 2a 2b 8b 12c 8a 12b 4a 3a 6a 12a 4b
7P 1a 2a 2b 8b 12a 8a 12b 4a 3a 6a 12c 4b
11P 1a 2a 2b 8a 12c 8b 12b 4a 3a 6a 12a 4b
X.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
X.3 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
X.5 2 . 2 . . . -2 . 2 2 . -2
X.6 2 . 2 . -1 . -1 2 -1 -1 -1 2
X.7 2 . 2 . 1 . -1 -2 -1 -1 1 2
X.8 2 . -2 A . -A . . 2 -2 . .
X.9 2 . -2 -A . A . . 2 -2 . .
X.10 2 . 2 . B . 1 . -1 -1 -B -2
X.11 2 . 2 . -B . 1 . -1 -1 B -2
X.12 4 . -4 . . . . . -2 2 . .
A = -E(8)-E(8)^3
= -Sqrt(-2) = -i2
B = -E(3)+E(3)^2
= -Sqrt(-3) = -i3
|