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