Group action invariants
| Degree $n$ : | $28$ | |
| Transitive number $t$ : | $13$ | |
| Group : | $C_4\times C_7:C_3$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,7,17,15,21,3,2,8,18,16,22,4)(5,24,25,20,9,11,6,23,26,19,10,12)(13,27,14,28), (1,23,17,11,5,28,22,16,9,4,25,19,13,8,2,24,18,12,6,27,21,15,10,3,26,20,14,7) | |
| $|\Aut(F/K)|$: | $4$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 4: $C_4$ 6: $C_6$ 12: $C_{12}$ 21: $C_7:C_3$ 42: $(C_7:C_3) \times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $C_4$
Degree 7: $C_7:C_3$
Degree 14: $(C_7:C_3) \times C_2$
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, 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 $ | $7$ | $6$ | $( 1, 2)( 3,23,19, 4,24,20)( 5,17, 9, 6,18,10)( 7,12,27, 8,11,28) (13,22,26,14,21,25)(15,16)$ |
| $ 28 $ | $3$ | $28$ | $( 1, 3, 6, 8, 9,11,14,15,18,19,22,23,26,27, 2, 4, 5, 7,10,12,13,16,17,20,21, 24,25,28)$ |
| $ 12, 12, 4 $ | $7$ | $12$ | $( 1, 3,22,15,18, 7, 2, 4,21,16,17, 8)( 5,11,10,20,26,24, 6,12, 9,19,25,23) (13,27,14,28)$ |
| $ 12, 12, 4 $ | $7$ | $12$ | $( 1, 3,25,15,18,11, 2, 4,26,16,17,12)( 5,19, 6,20)( 7,14,23,21,27,10, 8,13,24, 22,28, 9)$ |
| $ 28 $ | $3$ | $28$ | $( 1, 4, 6, 7, 9,12,14,16,18,20,22,24,26,28, 2, 3, 5, 8,10,11,13,15,17,19,21, 23,25,27)$ |
| $ 12, 12, 4 $ | $7$ | $12$ | $( 1, 4,22,16,18, 8, 2, 3,21,15,17, 7)( 5,12,10,19,26,23, 6,11, 9,20,25,24) (13,28,14,27)$ |
| $ 12, 12, 4 $ | $7$ | $12$ | $( 1, 4,25,16,18,12, 2, 3,26,15,17,11)( 5,20, 6,19)( 7,13,23,22,27, 9, 8,14,24, 21,28,10)$ |
| $ 7, 7, 7, 7 $ | $3$ | $7$ | $( 1, 5, 9,13,18,21,26)( 2, 6,10,14,17,22,25)( 3, 7,11,16,19,24,27) ( 4, 8,12,15,20,23,28)$ |
| $ 14, 14 $ | $3$ | $14$ | $( 1, 6, 9,14,18,22,26, 2, 5,10,13,17,21,25)( 3, 8,11,15,19,23,27, 4, 7,12,16, 20,24,28)$ |
| $ 28 $ | $3$ | $28$ | $( 1, 7,14,20,26, 3,10,15,21,27, 6,12,18,24, 2, 8,13,19,25, 4, 9,16,22,28, 5, 11,17,23)$ |
| $ 28 $ | $3$ | $28$ | $( 1, 8,14,19,26, 4,10,16,21,28, 6,11,18,23, 2, 7,13,20,25, 3, 9,15,22,27, 5, 12,17,24)$ |
| $ 7, 7, 7, 7 $ | $3$ | $7$ | $( 1,13,26, 9,21, 5,18)( 2,14,25,10,22, 6,17)( 3,16,27,11,24, 7,19) ( 4,15,28,12,23, 8,20)$ |
| $ 14, 14 $ | $3$ | $14$ | $( 1,14,26,10,21, 6,18, 2,13,25, 9,22, 5,17)( 3,15,27,12,24, 8,19, 4,16,28,11, 23, 7,20)$ |
| $ 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,15, 2,16)( 3,18, 4,17)( 5,20, 6,19)( 7,21, 8,22)( 9,23,10,24)(11,26,12,25) (13,28,14,27)$ |
| $ 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,16, 2,15)( 3,17, 4,18)( 5,19, 6,20)( 7,22, 8,21)( 9,24,10,23)(11,25,12,26) (13,27,14,28)$ |
Group invariants
| Order: | $84=2^{2} \cdot 3 \cdot 7$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [84, 2] |
| Character table: |
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
3 1 1 1 1 1 1 . 1 1 . 1 1 . . . . . . 1
7 1 . . 1 . . 1 . . 1 . . 1 1 1 1 1 1 1
1a 3a 3b 2a 6a 6b 28a 12a 12b 28b 12c 12d 7a 14a 28c 28d 7b 14b 4a
2P 1a 3b 3a 1a 3b 3a 14a 6b 6a 14a 6b 6a 7a 7a 14b 14b 7b 7b 2a
3P 1a 1a 1a 2a 2a 2a 28d 4a 4a 28c 4b 4b 7b 14b 28b 28a 7a 14a 4b
5P 1a 3b 3a 2a 6b 6a 28c 12b 12a 28d 12d 12c 7b 14b 28a 28b 7a 14a 4a
7P 1a 3a 3b 2a 6a 6b 4a 12c 12d 4b 12a 12b 1a 2a 4a 4b 1a 2a 4b
11P 1a 3b 3a 2a 6b 6a 28b 12d 12c 28a 12b 12a 7a 14a 28d 28c 7b 14b 4b
13P 1a 3a 3b 2a 6a 6b 28c 12a 12b 28d 12c 12d 7b 14b 28a 28b 7a 14a 4a
17P 1a 3b 3a 2a 6b 6a 28c 12b 12a 28d 12d 12c 7b 14b 28a 28b 7a 14a 4a
19P 1a 3a 3b 2a 6a 6b 28d 12c 12d 28c 12a 12b 7b 14b 28b 28a 7a 14a 4b
23P 1a 3b 3a 2a 6b 6a 28b 12d 12c 28a 12b 12a 7a 14a 28d 28c 7b 14b 4b
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 B B B -B -B -B 1 -1 B -B 1 -1 -B
X.4 1 1 1 -1 -1 -1 -B -B -B B B B 1 -1 -B B 1 -1 B
X.5 1 A /A -1 -A -/A B E -/E -B -E /E 1 -1 B -B 1 -1 -B
X.6 1 A /A -1 -A -/A -B -E /E B E -/E 1 -1 -B B 1 -1 B
X.7 1 /A A -1 -/A -A B -/E E -B /E -E 1 -1 B -B 1 -1 -B
X.8 1 /A A -1 -/A -A -B /E -E B -/E E 1 -1 -B B 1 -1 B
X.9 1 A /A 1 A /A -1 -A -/A -1 -A -/A 1 1 -1 -1 1 1 -1
X.10 1 /A A 1 /A A -1 -/A -A -1 -/A -A 1 1 -1 -1 1 1 -1
X.11 1 A /A 1 A /A 1 A /A 1 A /A 1 1 1 1 1 1 1
X.12 1 /A A 1 /A A 1 /A A 1 /A A 1 1 1 1 1 1 1
X.13 3 . . -3 . . C . . -C . . -/D /D -/C /C -D D F
X.14 3 . . -3 . . -/C . . /C . . -D D C -C -/D /D F
X.15 3 . . -3 . . /C . . -/C . . -D D -C C -/D /D -F
X.16 3 . . -3 . . -C . . C . . -/D /D /C -/C -D D -F
X.17 3 . . 3 . . D . . D . . -D -D /D /D -/D -/D -3
X.18 3 . . 3 . . /D . . /D . . -/D -/D D D -D -D -3
X.19 3 . . 3 . . -/D . . -/D . . -/D -/D -D -D -D -D 3
X.20 3 . . 3 . . -D . . -D . . -D -D -/D -/D -/D -/D 3
2 2
3 1
7 1
4b
2P 2a
3P 4a
5P 4b
7P 4a
11P 4a
13P 4b
17P 4b
19P 4a
23P 4a
X.1 1
X.2 -1
X.3 B
X.4 -B
X.5 B
X.6 -B
X.7 B
X.8 -B
X.9 -1
X.10 -1
X.11 1
X.12 1
X.13 -F
X.14 -F
X.15 F
X.16 F
X.17 -3
X.18 -3
X.19 3
X.20 3
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
B = -E(4)
= -Sqrt(-1) = -i
C = -E(28)^3-E(28)^19-E(28)^27
D = -E(7)-E(7)^2-E(7)^4
= (1-Sqrt(-7))/2 = -b7
E = -E(12)^11
F = 3*E(4)
= 3*Sqrt(-1) = 3i
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