Group action invariants
| Degree $n$ : | $28$ | |
| Transitive number $t$ : | $23$ | |
| Group : | $D_{28}:C_3$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,4,22,16,18,8,2,3,21,15,17,7)(5,12,10,19,26,23,6,11,9,20,25,24)(13,28,14,27), (1,13,21,18,5,26)(2,14,22,17,6,25)(3,20,11,15,27,8)(4,19,12,16,28,7)(23,24) | |
| $|\Aut(F/K)|$: | $2$ |
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: $D_{4}$ 12: $C_6\times C_2$ 24: $D_4 \times C_3$ 42: $F_7$ 84: $F_7 \times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $D_{4}$
Degree 7: $F_7$
Degree 14: $F_7 \times C_2$
Low degree siblings
28T23Siblings 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 $ | $1$ | $1$ | $()$ |
| $ 6, 6, 6, 6, 2, 1, 1 $ | $14$ | $6$ | $( 3, 8,19,28,24,12)( 4, 7,20,27,23,11)( 5,13, 9,26,18,21)( 6,14,10,25,17,22) (15,16)$ |
| $ 6, 6, 6, 6, 2, 1, 1 $ | $14$ | $6$ | $( 3,12,24,28,19, 8)( 4,11,23,27,20, 7)( 5,21,18,26, 9,13)( 6,22,17,25,10,14) (15,16)$ |
| $ 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, 1, 1 $ | $14$ | $2$ | $( 3,28)( 4,27)( 5,26)( 6,25)( 7,23)( 8,24)( 9,21)(10,22)(11,20)(12,19)(13,18) (14,17)(15,16)$ |
| $ 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 $ | $7$ | $6$ | $( 1, 2)( 3,23,19, 4,24,20)( 5,17, 9, 6,18,10)( 7,12,27, 8,11,28) (13,22,26,14,21,25)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $14$ | $2$ | $( 1, 3)( 2, 4)( 5,27)( 6,28)( 7,26)( 8,25)( 9,24)(10,23)(11,21)(12,22)(13,19) (14,20)(15,17)(16,18)$ |
| $ 28 $ | $6$ | $28$ | $( 1, 3, 6, 8, 9,11,14,15,18,19,22,23,26,27, 2, 4, 5, 7,10,12,13,16,17,20,21, 24,25,28)$ |
| $ 6, 6, 6, 6, 2, 2 $ | $14$ | $6$ | $( 1, 3, 9,27,26,19)( 2, 4,10,28,25,20)( 5,16,18,24,13,11)( 6,15,17,23,14,12) ( 7,21)( 8,22)$ |
| $ 6, 6, 6, 6, 2, 2 $ | $14$ | $6$ | $( 1, 3,13, 7, 5,24)( 2, 4,14, 8, 6,23)( 9,16,18,27,21,19)(10,15,17,28,22,20) (11,26)(12,25)$ |
| $ 12, 12, 4 $ | $14$ | $12$ | $( 1, 3,22,15,18, 7, 2, 4,21,16,17, 8)( 5,11,10,20,26,24, 6,12, 9,19,25,23) (13,27,14,28)$ |
| $ 12, 12, 4 $ | $14$ | $12$ | $( 1, 3,25,15,18,11, 2, 4,26,16,17,12)( 5,19, 6,20)( 7,14,23,21,27,10, 8,13,24, 22,28, 9)$ |
| $ 28 $ | $6$ | $28$ | $( 1, 4, 6, 7, 9,12,14,16,18,20,22,24,26,28, 2, 3, 5, 8,10,11,13,15,17,19,21, 23,25,27)$ |
| $ 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 $ | $6$ | $14$ | $( 1, 6, 9,14,18,22,26, 2, 5,10,13,17,21,25)( 3, 8,11,15,19,23,27, 4, 7,12,16, 20,24,28)$ |
| $ 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,15, 2,16)( 3,18, 4,17)( 5,20, 6,19)( 7,21, 8,22)( 9,23,10,24)(11,26,12,25) (13,28,14,27)$ |
Group invariants
| Order: | $168=2^{3} \cdot 3 \cdot 7$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [168, 9] |
| Character table: |
2 3 2 2 3 3 2 3 3 3 2 2 2 2 2 2 2 2 2 2
3 1 1 1 1 1 1 1 1 1 1 . 1 1 1 1 . . . 1
7 1 . . . . . 1 . . . 1 . . . . 1 1 1 1
1a 6a 6b 3a 3b 2a 2b 6c 6d 2c 28a 6e 6f 12a 12b 28b 7a 14a 4a
2P 1a 3a 3b 3b 3a 1a 1a 3b 3a 1a 14a 3a 3b 6d 6c 14a 7a 7a 2b
3P 1a 2a 2a 1a 1a 2a 2b 2b 2b 2c 28a 2c 2c 4a 4a 28b 7a 14a 4a
5P 1a 6b 6a 3b 3a 2a 2b 6d 6c 2c 28b 6f 6e 12b 12a 28a 7a 14a 4a
7P 1a 6a 6b 3a 3b 2a 2b 6c 6d 2c 4a 6e 6f 12a 12b 4a 1a 2b 4a
11P 1a 6b 6a 3b 3a 2a 2b 6d 6c 2c 28b 6f 6e 12b 12a 28a 7a 14a 4a
13P 1a 6a 6b 3a 3b 2a 2b 6c 6d 2c 28b 6e 6f 12a 12b 28a 7a 14a 4a
17P 1a 6b 6a 3b 3a 2a 2b 6d 6c 2c 28b 6f 6e 12b 12a 28a 7a 14a 4a
19P 1a 6a 6b 3a 3b 2a 2b 6c 6d 2c 28a 6e 6f 12a 12b 28b 7a 14a 4a
23P 1a 6b 6a 3b 3a 2a 2b 6d 6c 2c 28b 6f 6e 12b 12a 28a 7a 14a 4a
X.1 1 1 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 1 1 1
X.3 1 -1 -1 1 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 -1 1 1 -1
X.5 1 A /A -/A -A -1 1 -/A -A -1 1 A /A -/A -A 1 1 1 1
X.6 1 /A A -A -/A -1 1 -A -/A -1 1 /A A -A -/A 1 1 1 1
X.7 1 A /A -/A -A -1 1 -/A -A 1 -1 -A -/A /A A -1 1 1 -1
X.8 1 /A A -A -/A -1 1 -A -/A 1 -1 -/A -A A /A -1 1 1 -1
X.9 1 -/A -A -A -/A 1 1 -A -/A -1 -1 /A A A /A -1 1 1 -1
X.10 1 -A -/A -/A -A 1 1 -/A -A -1 -1 A /A /A A -1 1 1 -1
X.11 1 -/A -A -A -/A 1 1 -A -/A 1 1 -/A -A -A -/A 1 1 1 1
X.12 1 -A -/A -/A -A 1 1 -/A -A 1 1 -A -/A -/A -A 1 1 1 1
X.13 2 . . 2 2 . -2 -2 -2 . . . . . . . 2 -2 .
X.14 2 . . B /B . -2 -B -/B . . . . . . . 2 -2 .
X.15 2 . . /B B . -2 -/B -B . . . . . . . 2 -2 .
X.16 6 . . . . . 6 . . . -1 . . . . -1 -1 -1 6
X.17 6 . . . . . 6 . . . 1 . . . . 1 -1 -1 -6
X.18 6 . . . . . -6 . . . C . . . . -C -1 1 .
X.19 6 . . . . . -6 . . . -C . . . . C -1 1 .
A = -E(3)
= (1-Sqrt(-3))/2 = -b3
B = 2*E(3)^2
= -1-Sqrt(-3) = -1-i3
C = -E(28)^3+E(28)^11+E(28)^15-E(28)^19+E(28)^23-E(28)^27
= -Sqrt(7) = -r7
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