Group action invariants
| Degree $n$ : | $24$ | |
| Transitive number $t$ : | $4$ | |
| Group : | $C_3\times Q_8$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $2$ | |
| Generators: | (1,4,18,19,9,12,2,3,17,20,10,11)(5,8,21,24,14,16,6,7,22,23,13,15), (1,6,10,14,17,21,2,5,9,13,18,22)(3,7,12,16,19,24,4,8,11,15,20,23) | |
| $|\Aut(F/K)|$: | $24$ |
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 8: $Q_8$ 12: $C_6\times C_2$ 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: $Q_8$
Degree 12: $C_6\times C_2$
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, 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 $ | $2$ | $12$ | $( 1, 3,18,20, 9,11, 2, 4,17,19,10,12)( 5, 7,21,23,14,15, 6, 8,22,24,13,16)$ |
| $ 12, 12 $ | $2$ | $12$ | $( 1, 5,10,13,17,22, 2, 6, 9,14,18,21)( 3, 8,12,15,19,23, 4, 7,11,16,20,24)$ |
| $ 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 7, 2, 8)( 3,22, 4,21)( 5,12, 6,11)( 9,15,10,16)(13,19,14,20)(17,24,18,23)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 9,17)( 2,10,18)( 3,11,19)( 4,12,20)( 5,14,22)( 6,13,21)( 7,15,24) ( 8,16,23)$ |
| $ 6, 6, 6, 6 $ | $1$ | $6$ | $( 1,10,17, 2, 9,18)( 3,12,19, 4,11,20)( 5,13,22, 6,14,21)( 7,16,24, 8,15,23)$ |
| $ 12, 12 $ | $2$ | $12$ | $( 1,11,10,20,17, 3, 2,12, 9,19,18, 4)( 5,15,13,23,22, 7, 6,16,14,24,21, 8)$ |
| $ 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,13, 2,14)( 3,15, 4,16)( 5,17, 6,18)( 7,20, 8,19)( 9,21,10,22)(11,24,12,23)$ |
| $ 12, 12 $ | $2$ | $12$ | $( 1,15,18, 8, 9,24, 2,16,17, 7,10,23)( 3, 5,20,21,11,14, 4, 6,19,22,12,13)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,17, 9)( 2,18,10)( 3,19,11)( 4,20,12)( 5,22,14)( 6,21,13)( 7,24,15) ( 8,23,16)$ |
| $ 6, 6, 6, 6 $ | $1$ | $6$ | $( 1,18, 9, 2,17,10)( 3,20,11, 4,19,12)( 5,21,14, 6,22,13)( 7,23,15, 8,24,16)$ |
| $ 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,19, 2,20)( 3,10, 4, 9)( 5,24, 6,23)( 7,13, 8,14)(11,18,12,17)(15,21,16,22)$ |
| $ 12, 12 $ | $2$ | $12$ | $( 1,21,18,14, 9, 6, 2,22,17,13,10, 5)( 3,24,20,16,11, 7, 4,23,19,15,12, 8)$ |
| $ 12, 12 $ | $2$ | $12$ | $( 1,23,10, 7,17,16, 2,24, 9, 8,18,15)( 3,13,12,22,19, 6, 4,14,11,21,20, 5)$ |
Group invariants
| Order: | $24=2^{3} \cdot 3$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [24, 11] |
| Character table: |
2 3 3 2 2 2 3 3 2 2 2 3 3 2 2 2
3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1a 2a 12a 12b 4a 3a 6a 12c 4b 12d 3b 6b 4c 12e 12f
2P 1a 1a 6b 6a 2a 3b 3b 6a 2a 6b 3a 3a 2a 6b 6a
3P 1a 2a 4c 4b 4a 1a 2a 4c 4b 4a 1a 2a 4c 4b 4a
5P 1a 2a 12c 12e 4a 3b 6b 12a 4b 12f 3a 6a 4c 12b 12d
7P 1a 2a 12a 12b 4a 3a 6a 12c 4b 12d 3b 6b 4c 12e 12f
11P 1a 2a 12c 12e 4a 3b 6b 12a 4b 12f 3a 6a 4c 12b 12d
X.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
X.3 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
X.5 1 1 A /A 1 -A -A /A -1 -A -/A -/A -1 A -/A
X.6 1 1 /A A 1 -/A -/A A -1 -/A -A -A -1 /A -A
X.7 1 1 A -/A -1 -A -A /A 1 A -/A -/A -1 -A /A
X.8 1 1 /A -A -1 -/A -/A A 1 /A -A -A -1 -/A A
X.9 1 1 -/A A -1 -/A -/A -A -1 /A -A -A 1 /A A
X.10 1 1 -A /A -1 -A -A -/A -1 A -/A -/A 1 A /A
X.11 1 1 -/A -A 1 -/A -/A -A 1 -/A -A -A 1 -/A -A
X.12 1 1 -A -/A 1 -A -A -/A 1 -A -/A -/A 1 -A -/A
X.13 2 -2 . . . 2 -2 . . . 2 -2 . . .
X.14 2 -2 . . . B -B . . . /B -/B . . .
X.15 2 -2 . . . /B -/B . . . B -B . . .
A = -E(3)
= (1-Sqrt(-3))/2 = -b3
B = 2*E(3)^2
= -1-Sqrt(-3) = -1-i3
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