Group action invariants
| Degree $n$: | $35$ | |
| Transitive number $t$: | $8$ | |
| Group: | $C_7:F_5$ | |
| Parity: | $1$ | |
| Primitive: | no | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $1$ | |
| Generators: | (1,14,21,34,6,19,26,4,11,24,31,9,16,29)(2,13,22,33,7,18,27,3,12,23,32,8,17,28)(5,15,25,35,10,20,30), (1,33,2,35)(3,32,5,31)(4,34)(6,28,7,30)(8,27,10,26)(9,29)(11,23,12,25)(13,22,15,21)(14,24)(16,18,17,20) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $14$: $D_{7}$ $20$: $F_5$ $28$: 28T3 Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $F_5$
Degree 7: $D_{7}$
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, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ | $35$ | $4$ | $( 2, 3, 5, 4)( 6,31)( 7,33,10,34)( 8,35, 9,32)(11,26)(12,28,15,29) (13,30,14,27)(16,21)(17,23,20,24)(18,25,19,22)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ | $35$ | $4$ | $( 2, 4, 5, 3)( 6,31)( 7,34,10,33)( 8,32, 9,35)(11,26)(12,29,15,28) (13,27,14,30)(16,21)(17,24,20,23)(18,22,19,25)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30) (28,29)(32,35)(33,34)$ |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$ |
| $ 7, 7, 7, 7, 7 $ | $2$ | $7$ | $( 1, 6,11,16,21,26,31)( 2, 7,12,17,22,27,32)( 3, 8,13,18,23,28,33) ( 4, 9,14,19,24,29,34)( 5,10,15,20,25,30,35)$ |
| $ 14, 14, 7 $ | $10$ | $14$ | $( 1, 6,11,16,21,26,31)( 2,10,12,20,22,30,32, 5, 7,15,17,25,27,35) ( 3, 9,13,19,23,29,33, 4, 8,14,18,24,28,34)$ |
| $ 35 $ | $4$ | $35$ | $( 1, 7,13,19,25,26,32, 3, 9,15,16,22,28,34, 5, 6,12,18,24,30,31, 2, 8,14,20, 21,27,33, 4,10,11,17,23,29,35)$ |
| $ 35 $ | $4$ | $35$ | $( 1, 8,15,17,24,26,33, 5, 7,14,16,23,30,32, 4, 6,13,20,22,29,31, 3,10,12,19, 21,28,35, 2, 9,11,18,25,27,34)$ |
| $ 7, 7, 7, 7, 7 $ | $2$ | $7$ | $( 1,11,21,31, 6,16,26)( 2,12,22,32, 7,17,27)( 3,13,23,33, 8,18,28) ( 4,14,24,34, 9,19,29)( 5,15,25,35,10,20,30)$ |
| $ 14, 14, 7 $ | $10$ | $14$ | $( 1,11,21,31, 6,16,26)( 2,15,22,35, 7,20,27, 5,12,25,32,10,17,30) ( 3,14,23,34, 8,19,28, 4,13,24,33, 9,18,29)$ |
| $ 35 $ | $4$ | $35$ | $( 1,12,23,34,10,16,27, 3,14,25,31, 7,18,29, 5,11,22,33, 9,20,26, 2,13,24,35, 6,17,28, 4,15,21,32, 8,19,30)$ |
| $ 35 $ | $4$ | $35$ | $( 1,13,25,32, 9,16,28, 5,12,24,31, 8,20,27, 4,11,23,35, 7,19,26, 3,15,22,34, 6,18,30, 2,14,21,33,10,17,29)$ |
| $ 7, 7, 7, 7, 7 $ | $2$ | $7$ | $( 1,16,31,11,26, 6,21)( 2,17,32,12,27, 7,22)( 3,18,33,13,28, 8,23) ( 4,19,34,14,29, 9,24)( 5,20,35,15,30,10,25)$ |
| $ 14, 14, 7 $ | $10$ | $14$ | $( 1,16,31,11,26, 6,21)( 2,20,32,15,27,10,22, 5,17,35,12,30, 7,25) ( 3,19,33,14,28, 9,23, 4,18,34,13,29, 8,24)$ |
| $ 35 $ | $4$ | $35$ | $( 1,17,33,14,30, 6,22, 3,19,35,11,27, 8,24, 5,16,32,13,29,10,21, 2,18,34,15, 26, 7,23, 4,20,31,12,28, 9,25)$ |
| $ 35 $ | $4$ | $35$ | $( 1,18,35,12,29, 6,23, 5,17,34,11,28,10,22, 4,16,33,15,27, 9,21, 3,20,32,14, 26, 8,25, 2,19,31,13,30, 7,24)$ |
Group invariants
| Order: | $140=2^{2} \cdot 5 \cdot 7$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | yes | |
| GAP id: | [140, 6] |
| Character table: |
2 2 2 2 2 . 1 1 . . 1 1 . . 1 1 . .
5 1 . . . 1 1 . 1 1 1 . 1 1 1 . 1 1
7 1 . . 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1a 4a 4b 2a 5a 7a 14a 35a 35b 7b 14b 35c 35d 7c 14c 35e 35f
2P 1a 2a 2a 1a 5a 7b 7b 35d 35c 7c 7c 35e 35f 7a 7a 35a 35b
3P 1a 4b 4a 2a 5a 7c 14c 35f 35e 7a 14a 35a 35b 7b 14b 35d 35c
5P 1a 4a 4b 2a 1a 7b 14b 7b 7b 7c 14c 7c 7c 7a 14a 7a 7a
7P 1a 4b 4a 2a 5a 1a 2a 5a 5a 1a 2a 5a 5a 1a 2a 5a 5a
11P 1a 4b 4a 2a 5a 7c 14c 35f 35e 7a 14a 35a 35b 7b 14b 35d 35c
13P 1a 4a 4b 2a 5a 7a 14a 35a 35b 7b 14b 35c 35d 7c 14c 35e 35f
17P 1a 4a 4b 2a 5a 7c 14c 35f 35e 7a 14a 35a 35b 7b 14b 35d 35c
19P 1a 4b 4a 2a 5a 7b 14b 35d 35c 7c 14c 35e 35f 7a 14a 35a 35b
23P 1a 4b 4a 2a 5a 7b 14b 35d 35c 7c 14c 35e 35f 7a 14a 35a 35b
29P 1a 4a 4b 2a 5a 7a 14a 35a 35b 7b 14b 35c 35d 7c 14c 35e 35f
31P 1a 4b 4a 2a 5a 7c 14c 35e 35f 7a 14a 35b 35a 7b 14b 35c 35d
X.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
X.3 1 A -A -1 1 1 -1 1 1 1 -1 1 1 1 -1 1 1
X.4 1 -A A -1 1 1 -1 1 1 1 -1 1 1 1 -1 1 1
X.5 2 . . -2 2 B -B B B D -D D D C -C C C
X.6 2 . . -2 2 C -C C C B -B B B D -D D D
X.7 2 . . -2 2 D -D D D C -C C C B -B B B
X.8 2 . . 2 2 B B B B D D D D C C C C
X.9 2 . . 2 2 C C C C B B B B D D D D
X.10 2 . . 2 2 D D D D C C C C B B B B
X.11 4 . . . -1 4 . -1 -1 4 . -1 -1 4 . -1 -1
X.12 4 . . . -1 E . H /H G . /J J F . /I I
X.13 4 . . . -1 E . /H H G . J /J F . I /I
X.14 4 . . . -1 F . I /I E . H /H G . J /J
X.15 4 . . . -1 F . /I I E . /H H G . /J J
X.16 4 . . . -1 G . J /J F . /I I E . H /H
X.17 4 . . . -1 G . /J J F . I /I E . /H H
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
E = 2*E(7)^3+2*E(7)^4
F = 2*E(7)^2+2*E(7)^5
G = 2*E(7)+2*E(7)^6
H = E(35)^6+E(35)^8+E(35)^22+E(35)^34
I = E(35)^18+E(35)^24+E(35)^31+E(35)^32
J = E(35)^9+E(35)^12+E(35)^16+E(35)^33
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