Group action invariants
Degree $n$: | $28$ | |
Transitive number $t$: | $44$ | |
Group: | $C_2\times F_8:C_3$ | |
Parity: | $1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $4$ | |
Generators: | (1,10,18,26,20,13,22,15,24,4,12,6,27,8)(2,9,17,25,19,14,21,16,23,3,11,5,28,7), (1,13,9,15,27,23)(2,14,10,16,28,24)(3,17)(4,18)(5,8,12,19,22,26)(6,7,11,20,21,25) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ $21$: $C_7:C_3$ $42$: $(C_7:C_3) \times C_2$ $168$: $C_2^3:(C_7: C_3)$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: None
Degree 7: $C_7:C_3$
Degree 14: $(C_7:C_3) \times C_2$, 14T11, 14T18
Low degree siblings
14T18, 16T712, 42T67Siblings 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, 1, 1 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $7$ | $2$ | $( 5,20)( 6,19)( 9,24)(10,23)(11,26)(12,25)(13,28)(14,27)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ | $28$ | $3$ | $( 3, 5, 9)( 4, 6,10)( 7,14,25)( 8,13,26)(11,21,28)(12,22,27)(17,19,23) (18,20,24)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ | $28$ | $3$ | $( 3, 9, 5)( 4,10, 6)( 7,25,14)( 8,26,13)(11,28,21)(12,27,22)(17,23,19) (18,24,20)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $7$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7,21)( 8,22)( 9,23)(10,24)(11,12)(13,27)(14,28)(15,16) (17,18)(19,20)(25,26)$ |
$ 6, 6, 6, 6, 2, 2 $ | $28$ | $6$ | $( 1, 2)( 3, 6, 9,17,20,23)( 4, 5,10,18,19,24)( 7,28,25, 8,27,26) (11,22,13,12,21,14)(15,16)$ |
$ 6, 6, 6, 6, 2, 2 $ | $28$ | $6$ | $( 1, 2)( 3,10,20,17,24, 6)( 4, 9,19,18,23, 5)( 7,11,27, 8,12,28) (13,22,26,14,21,25)(15,16)$ |
$ 7, 7, 7, 7 $ | $24$ | $7$ | $( 1, 3, 5, 7,24,12,14)( 2, 4, 6, 8,23,11,13)( 9,25,27,16,18,20,22) (10,26,28,15,17,19,21)$ |
$ 6, 6, 3, 3, 3, 3, 2, 2 $ | $28$ | $6$ | $( 1, 3, 7)( 2, 4, 8)( 5,25,24,20,12, 9)( 6,26,23,19,11,10)(13,28)(14,27) (15,17,21)(16,18,22)$ |
$ 6, 6, 3, 3, 3, 3, 2, 2 $ | $28$ | $6$ | $( 1, 3,12,16,18,25)( 2, 4,11,15,17,26)( 5,20)( 6,19)( 7,14, 9)( 8,13,10) (21,28,23)(22,27,24)$ |
$ 14, 14 $ | $24$ | $14$ | $( 1, 4, 5, 8, 9,11,14,15,18,19,22,23,25,28)( 2, 3, 6, 7,10,12,13,16,17,20,21, 24,26,27)$ |
$ 6, 6, 6, 6, 2, 2 $ | $28$ | $6$ | $( 1, 4, 7,15,18,21)( 2, 3, 8,16,17,22)( 5,11,24,19,25,10)( 6,12,23,20,26, 9) (13,27)(14,28)$ |
$ 6, 6, 6, 6, 2, 2 $ | $28$ | $6$ | $( 1, 4,12,15,18,26)( 2, 3,11,16,17,25)( 5,19)( 6,20)( 7,28, 9,21,14,23) ( 8,27,10,22,13,24)$ |
$ 7, 7, 7, 7 $ | $24$ | $7$ | $( 1, 7,27,20,25, 3, 9)( 2, 8,28,19,26, 4,10)( 5,12,18,24,16,22,14) ( 6,11,17,23,15,21,13)$ |
$ 14, 14 $ | $24$ | $14$ | $( 1, 8,14,19,25, 4, 9,15,22,28, 5,11,18,23)( 2, 7,13,20,26, 3,10,16,21,27, 6, 12,17,24)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,15)( 2,16)( 3,17)( 4,18)( 5,19)( 6,20)( 7,21)( 8,22)( 9,23)(10,24)(11,25) (12,26)(13,27)(14,28)$ |
Group invariants
Order: | $336=2^{4} \cdot 3 \cdot 7$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [336, 210] |
Character table: |
2 4 4 2 2 4 2 2 1 2 2 1 2 2 1 1 4 3 1 1 1 1 1 1 1 . 1 1 . 1 1 . . 1 7 1 . . . . . . 1 . . 1 . . 1 1 1 1a 2a 3a 3b 2b 6a 6b 7a 6c 6d 14a 6e 6f 7b 14b 2c 2P 1a 1a 3b 3a 1a 3b 3a 7a 3b 3a 7a 3b 3a 7b 7b 1a 3P 1a 2a 1a 1a 2b 2b 2b 7b 2a 2a 14b 2c 2c 7a 14a 2c 5P 1a 2a 3b 3a 2b 6b 6a 7b 6d 6c 14b 6f 6e 7a 14a 2c 7P 1a 2a 3a 3b 2b 6a 6b 1a 6c 6d 2c 6e 6f 1a 2c 2c 11P 1a 2a 3b 3a 2b 6b 6a 7a 6d 6c 14a 6f 6e 7b 14b 2c 13P 1a 2a 3a 3b 2b 6a 6b 7b 6c 6d 14b 6e 6f 7a 14a 2c 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 A /A -1 -A -/A 1 A /A -1 -A -/A 1 -1 -1 X.4 1 1 /A A -1 -/A -A 1 /A A -1 -/A -A 1 -1 -1 X.5 1 1 A /A 1 A /A 1 A /A 1 A /A 1 1 1 X.6 1 1 /A A 1 /A A 1 /A A 1 /A A 1 1 1 X.7 3 3 . . -3 . . B . . -B . . /B -/B -3 X.8 3 3 . . -3 . . /B . . -/B . . B -B -3 X.9 3 3 . . 3 . . B . . B . . /B /B 3 X.10 3 3 . . 3 . . /B . . /B . . B B 3 X.11 7 -1 1 1 -1 -1 -1 . -1 -1 . 1 1 . . 7 X.12 7 -1 1 1 1 1 1 . -1 -1 . -1 -1 . . -7 X.13 7 -1 A /A -1 -A -/A . -A -/A . A /A . . 7 X.14 7 -1 /A A -1 -/A -A . -/A -A . /A A . . 7 X.15 7 -1 A /A 1 A /A . -A -/A . -A -/A . . -7 X.16 7 -1 /A A 1 /A A . -/A -A . -/A -A . . -7 A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = E(7)^3+E(7)^5+E(7)^6 = (-1-Sqrt(-7))/2 = -1-b7 |