Group action invariants
| Degree $n$: | $28$ | |
| Transitive number $t$: | $14$ | |
| Group: | $C_2^2\times C_7:C_3$ | |
| Parity: | $1$ | |
| Primitive: | no | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $4$ | |
| Generators: | (1,15)(2,16)(3,6,24,17,19,10)(4,5,23,18,20,9)(7,14,11,22,27,25)(8,13,12,21,28,26), (1,25,21,17,13,10,5,2,26,22,18,14,9,6)(3,28,24,20,16,12,7,4,27,23,19,15,11,8) |
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 $12$: $C_6\times C_2$ $21$: $C_7:C_3$ $42$: $(C_7:C_3) \times C_2$ x 3 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_2^2$
Degree 7: $C_7:C_3$
Degree 14: $(C_7:C_3) \times C_2$ x 3
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 $ | $7$ | $6$ | $( 1, 2)( 3,23,19, 4,24,20)( 5,17, 9, 6,18,10)( 7,12,27, 8,11,28) (13,22,26,14,21,25)(15,16)$ |
| $ 14, 14 $ | $3$ | $14$ | $( 1, 3, 5, 7, 9,11,13,16,18,19,21,24,26,27)( 2, 4, 6, 8,10,12,14,15,17,20,22, 23,25,28)$ |
| $ 6, 6, 6, 6, 2, 2 $ | $7$ | $6$ | $( 1, 3,21,16,18, 7)( 2, 4,22,15,17, 8)( 5,11, 9,19,26,24)( 6,12,10,20,25,23) (13,27)(14,28)$ |
| $ 6, 6, 6, 6, 2, 2 $ | $7$ | $6$ | $( 1, 3,26,16,18,11)( 2, 4,25,15,17,12)( 5,19)( 6,20)( 7,13,24,21,27, 9) ( 8,14,23,22,28,10)$ |
| $ 14, 14 $ | $3$ | $14$ | $( 1, 4, 5, 8, 9,12,13,15,18,20,21,23,26,28)( 2, 3, 6, 7,10,11,14,16,17,19,22, 24,25,27)$ |
| $ 6, 6, 6, 6, 2, 2 $ | $7$ | $6$ | $( 1, 4,21,15,18, 8)( 2, 3,22,16,17, 7)( 5,12, 9,20,26,23)( 6,11,10,19,25,24) (13,28)(14,27)$ |
| $ 6, 6, 6, 6, 2, 2 $ | $7$ | $6$ | $( 1, 4,26,15,18,12)( 2, 3,25,16,17,11)( 5,20)( 6,19)( 7,14,24,22,27,10) ( 8,13,23,21,28, 9)$ |
| $ 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 $ | $3$ | $14$ | $( 1, 6, 9,14,18,22,26, 2, 5,10,13,17,21,25)( 3, 8,11,15,19,23,27, 4, 7,12,16, 20,24,28)$ |
| $ 14, 14 $ | $3$ | $14$ | $( 1, 7,13,19,26, 3, 9,16,21,27, 5,11,18,24)( 2, 8,14,20,25, 4,10,15,22,28, 6, 12,17,23)$ |
| $ 14, 14 $ | $3$ | $14$ | $( 1, 8,13,20,26, 4, 9,15,21,28, 5,12,18,23)( 2, 7,14,19,25, 3,10,16,22,27, 6, 11,17,24)$ |
| $ 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 $ | $3$ | $14$ | $( 1,14,26,10,21, 6,18, 2,13,25, 9,22, 5,17)( 3,15,27,12,24, 8,19, 4,16,28,11, 23, 7,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,15)( 2,16)( 3,17)( 4,18)( 5,20)( 6,19)( 7,22)( 8,21)( 9,23)(10,24)(11,25) (12,26)(13,28)(14,27)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,16)( 2,15)( 3,18)( 4,17)( 5,19)( 6,20)( 7,21)( 8,22)( 9,24)(10,23)(11,26) (12,25)(13,27)(14,28)$ |
Group invariants
| Order: | $84=2^{2} \cdot 3 \cdot 7$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | yes | |
| GAP id: | [84, 9] |
| 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 14a 6c 6d 14b 6e 6f 7a 14c 14d 14e 7b 14f 2b
2P 1a 3b 3a 1a 3b 3a 7a 3b 3a 7a 3b 3a 7a 7a 7b 7b 7b 7b 1a
3P 1a 1a 1a 2a 2a 2a 14d 2c 2c 14e 2b 2b 7b 14f 14a 14b 7a 14c 2b
5P 1a 3b 3a 2a 6b 6a 14d 6d 6c 14e 6f 6e 7b 14f 14a 14b 7a 14c 2b
7P 1a 3a 3b 2a 6a 6b 2c 6c 6d 2b 6e 6f 1a 2a 2c 2b 1a 2a 2b
11P 1a 3b 3a 2a 6b 6a 14a 6d 6c 14b 6f 6e 7a 14c 14d 14e 7b 14f 2b
13P 1a 3a 3b 2a 6a 6b 14d 6c 6d 14e 6e 6f 7b 14f 14a 14b 7a 14c 2b
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 -1 -A -/A -1 -A -/A 1 A /A 1 -1 -1 1 1 -1 1
X.6 1 /A A -1 -/A -A -1 -/A -A 1 /A A 1 -1 -1 1 1 -1 1
X.7 1 A /A -1 -A -/A 1 A /A -1 -A -/A 1 -1 1 -1 1 -1 -1
X.8 1 /A A -1 -/A -A 1 /A A -1 -/A -A 1 -1 1 -1 1 -1 -1
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 . . B . . -B . . -B B /B -/B -/B /B 3
X.14 3 . . -3 . . /B . . -/B . . -/B /B B -B -B B 3
X.15 3 . . -3 . . -/B . . /B . . -/B /B -B B -B B -3
X.16 3 . . -3 . . -B . . B . . -B B -/B /B -/B /B -3
X.17 3 . . 3 . . B . . B . . -B -B /B /B -/B -/B -3
X.18 3 . . 3 . . /B . . /B . . -/B -/B B B -B -B -3
X.19 3 . . 3 . . -/B . . -/B . . -/B -/B -B -B -B -B 3
X.20 3 . . 3 . . -B . . -B . . -B -B -/B -/B -/B -/B 3
2 2
3 1
7 1
2c
2P 1a
3P 2c
5P 2c
7P 2c
11P 2c
13P 2c
X.1 1
X.2 -1
X.3 1
X.4 -1
X.5 -1
X.6 -1
X.7 1
X.8 1
X.9 -1
X.10 -1
X.11 1
X.12 1
X.13 -3
X.14 -3
X.15 3
X.16 3
X.17 -3
X.18 -3
X.19 3
X.20 3
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
B = -E(7)-E(7)^2-E(7)^4
= (1-Sqrt(-7))/2 = -b7
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