Group action invariants
| Degree $n$ : | $28$ | |
| Transitive number $t$ : | $12$ | |
| Group : | $C_7:C_{12}$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,8,25,24,18,28,2,7,26,23,17,27)(3,13,15,22,11,9,4,14,16,21,12,10)(5,20,6,19), (1,9,13)(2,10,14)(5,26,21)(6,25,22)(7,19,11)(8,20,12)(15,23,28)(16,24,27) | |
| $|\Aut(F/K)|$: | $4$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 4: $C_4$ 6: $C_6$ 12: $C_{12}$ 42: $F_7$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $C_4$
Degree 7: $F_7$
Degree 14: $F_7$
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, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ | $7$ | $3$ | $( 3,19,24)( 4,20,23)( 5, 9,18)( 6,10,17)( 7,27,11)( 8,28,12)(13,26,21) (14,25,22)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ | $7$ | $3$ | $( 3,24,19)( 4,23,20)( 5,18, 9)( 6,17,10)( 7,11,27)( 8,12,28)(13,21,26) (14,22,25)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)$ |
| $ 6, 6, 6, 6, 2, 2 $ | $7$ | $6$ | $( 1, 2)( 3,20,24, 4,19,23)( 5,10,18, 6, 9,17)( 7,28,11, 8,27,12) (13,25,21,14,26,22)(15,16)$ |
| $ 6, 6, 6, 6, 2, 2 $ | $7$ | $6$ | $( 1, 2)( 3,23,19, 4,24,20)( 5,17, 9, 6,18,10)( 7,12,27, 8,11,28) (13,22,26,14,21,25)(15,16)$ |
| $ 4, 4, 4, 4, 4, 4, 4 $ | $7$ | $4$ | $( 1, 3, 2, 4)( 5,27, 6,28)( 7,25, 8,26)( 9,24,10,23)(11,22,12,21)(13,19,14,20) (15,18,16,17)$ |
| $ 12, 12, 4 $ | $7$ | $12$ | $( 1, 3,10,28,26,19, 2, 4, 9,27,25,20)( 5,16,17,23,13,11, 6,15,18,24,14,12) ( 7,22, 8,21)$ |
| $ 12, 12, 4 $ | $7$ | $12$ | $( 1, 3,14, 8, 5,24, 2, 4,13, 7, 6,23)( 9,16,17,28,21,19,10,15,18,27,22,20) (11,25,12,26)$ |
| $ 4, 4, 4, 4, 4, 4, 4 $ | $7$ | $4$ | $( 1, 4, 2, 3)( 5,28, 6,27)( 7,26, 8,25)( 9,23,10,24)(11,21,12,22)(13,20,14,19) (15,17,16,18)$ |
| $ 12, 12, 4 $ | $7$ | $12$ | $( 1, 4,10,27,26,20, 2, 3, 9,28,25,19)( 5,15,17,24,13,12, 6,16,18,23,14,11) ( 7,21, 8,22)$ |
| $ 12, 12, 4 $ | $7$ | $12$ | $( 1, 4,14, 7, 5,23, 2, 3,13, 8, 6,24)( 9,15,17,27,21,20,10,16,18,28,22,19) (11,26,12,25)$ |
| $ 7, 7, 7, 7 $ | $6$ | $7$ | $( 1, 5, 9,13,18,21,26)( 2, 6,10,14,17,22,25)( 3, 7,11,16,19,24,27) ( 4, 8,12,15,20,23,28)$ |
| $ 14, 14 $ | $6$ | $14$ | $( 1, 6, 9,14,18,22,26, 2, 5,10,13,17,21,25)( 3, 8,11,15,19,23,27, 4, 7,12,16, 20,24,28)$ |
Group invariants
| Order: | $84=2^{2} \cdot 3 \cdot 7$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [84, 1] |
| Character table: |
2 2 2 2 2 2 2 2 2 2 2 2 2 1 1
3 1 1 1 1 1 1 1 1 1 1 1 1 . .
7 1 . . 1 . . . . . . . . 1 1
1a 3a 3b 2a 6a 6b 4a 12a 12b 4b 12c 12d 7a 14a
2P 1a 3b 3a 1a 3b 3a 2a 6a 6b 2a 6a 6b 7a 7a
3P 1a 1a 1a 2a 2a 2a 4b 4b 4b 4a 4a 4a 7a 14a
5P 1a 3b 3a 2a 6b 6a 4a 12b 12a 4b 12d 12c 7a 14a
7P 1a 3a 3b 2a 6a 6b 4b 12c 12d 4a 12a 12b 1a 2a
11P 1a 3b 3a 2a 6b 6a 4b 12d 12c 4a 12b 12a 7a 14a
13P 1a 3a 3b 2a 6a 6b 4a 12a 12b 4b 12c 12d 7a 14a
X.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
X.3 1 1 1 -1 -1 -1 B B B -B -B -B 1 -1
X.4 1 1 1 -1 -1 -1 -B -B -B B B B 1 -1
X.5 1 A /A -1 -A -/A B C -/C -B -C /C 1 -1
X.6 1 A /A -1 -A -/A -B -C /C B C -/C 1 -1
X.7 1 /A A -1 -/A -A B -/C C -B /C -C 1 -1
X.8 1 /A A -1 -/A -A -B /C -C B -/C C 1 -1
X.9 1 A /A 1 A /A -1 -/A -A -1 -/A -A 1 1
X.10 1 /A A 1 /A A -1 -A -/A -1 -A -/A 1 1
X.11 1 A /A 1 A /A 1 /A A 1 /A A 1 1
X.12 1 /A A 1 /A A 1 A /A 1 A /A 1 1
X.13 6 . . -6 . . . . . . . . -1 1
X.14 6 . . 6 . . . . . . . . -1 -1
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
B = -E(4)
= -Sqrt(-1) = -i
C = -E(12)^7
|