Group action invariants
| Degree $n$: | $35$ | |
| Transitive number $t$: | $4$ | |
| Group: | $D_{35}$ | |
| Parity: | $-1$ | |
| Primitive: | no | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $1$ | |
| Generators: | (2,5)(3,4)(6,31)(7,35)(8,34)(9,33)(10,32)(11,26)(12,30)(13,29)(14,28)(15,27)(16,21)(17,25)(18,24)(19,23)(20,22), (1,29)(2,28)(3,27)(4,26)(5,30)(6,24)(7,23)(8,22)(9,21)(10,25)(11,19)(12,18)(13,17)(14,16)(15,20)(31,34)(32,33) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $10$: $D_{5}$ $14$: $D_{7}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $D_{5}$
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 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $35$ | $2$ | $( 2, 5)( 3, 4)( 6,31)( 7,35)( 8,34)( 9,33)(10,32)(11,26)(12,30)(13,29)(14,28) (15,27)(16,21)(17,25)(18,24)(19,23)(20,22)$ |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$ |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)(26,28,30,27,29)(31,33,35,32,34)$ |
| $ 7, 7, 7, 7, 7 $ | $2$ | $7$ | $( 1, 6,11,16,21,26,31)( 2, 7,12,17,22,27,32)( 3, 8,13,18,23,28,33) ( 4, 9,14,19,24,29,34)( 5,10,15,20,25,30,35)$ |
| $ 35 $ | $2$ | $35$ | $( 1, 7,13,19,25,26,32, 3, 9,15,16,22,28,34, 5, 6,12,18,24,30,31, 2, 8,14,20, 21,27,33, 4,10,11,17,23,29,35)$ |
| $ 35 $ | $2$ | $35$ | $( 1, 8,15,17,24,26,33, 5, 7,14,16,23,30,32, 4, 6,13,20,22,29,31, 3,10,12,19, 21,28,35, 2, 9,11,18,25,27,34)$ |
| $ 35 $ | $2$ | $35$ | $( 1, 9,12,20,23,26,34, 2,10,13,16,24,27,35, 3, 6,14,17,25,28,31, 4, 7,15,18, 21,29,32, 5, 8,11,19,22,30,33)$ |
| $ 35 $ | $2$ | $35$ | $( 1,10,14,18,22,26,35, 4, 8,12,16,25,29,33, 2, 6,15,19,23,27,31, 5, 9,13,17, 21,30,34, 3, 7,11,20,24,28,32)$ |
| $ 7, 7, 7, 7, 7 $ | $2$ | $7$ | $( 1,11,21,31, 6,16,26)( 2,12,22,32, 7,17,27)( 3,13,23,33, 8,18,28) ( 4,14,24,34, 9,19,29)( 5,15,25,35,10,20,30)$ |
| $ 35 $ | $2$ | $35$ | $( 1,12,23,34,10,16,27, 3,14,25,31, 7,18,29, 5,11,22,33, 9,20,26, 2,13,24,35, 6,17,28, 4,15,21,32, 8,19,30)$ |
| $ 35 $ | $2$ | $35$ | $( 1,13,25,32, 9,16,28, 5,12,24,31, 8,20,27, 4,11,23,35, 7,19,26, 3,15,22,34, 6,18,30, 2,14,21,33,10,17,29)$ |
| $ 35 $ | $2$ | $35$ | $( 1,14,22,35, 8,16,29, 2,15,23,31, 9,17,30, 3,11,24,32,10,18,26, 4,12,25,33, 6,19,27, 5,13,21,34, 7,20,28)$ |
| $ 35 $ | $2$ | $35$ | $( 1,15,24,33, 7,16,30, 4,13,22,31,10,19,28, 2,11,25,34, 8,17,26, 5,14,23,32, 6,20,29, 3,12,21,35, 9,18,27)$ |
| $ 7, 7, 7, 7, 7 $ | $2$ | $7$ | $( 1,16,31,11,26, 6,21)( 2,17,32,12,27, 7,22)( 3,18,33,13,28, 8,23) ( 4,19,34,14,29, 9,24)( 5,20,35,15,30,10,25)$ |
| $ 35 $ | $2$ | $35$ | $( 1,17,33,14,30, 6,22, 3,19,35,11,27, 8,24, 5,16,32,13,29,10,21, 2,18,34,15, 26, 7,23, 4,20,31,12,28, 9,25)$ |
| $ 35 $ | $2$ | $35$ | $( 1,18,35,12,29, 6,23, 5,17,34,11,28,10,22, 4,16,33,15,27, 9,21, 3,20,32,14, 26, 8,25, 2,19,31,13,30, 7,24)$ |
| $ 35 $ | $2$ | $35$ | $( 1,19,32,15,28, 6,24, 2,20,33,11,29, 7,25, 3,16,34,12,30, 8,21, 4,17,35,13, 26, 9,22, 5,18,31,14,27,10,23)$ |
| $ 35 $ | $2$ | $35$ | $( 1,20,34,13,27, 6,25, 4,18,32,11,30, 9,23, 2,16,35,14,28, 7,21, 5,19,33,12, 26,10,24, 3,17,31,15,29, 8,22)$ |
Group invariants
| Order: | $70=2 \cdot 5 \cdot 7$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | yes | |
| GAP id: | [70, 3] |
| Character table: |
2 1 1 . . . . . . . . . . . . . . . . .
5 1 . 1 1 1 1 1 1 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 1 1 1
1a 2a 5a 5b 7a 35a 35b 35c 35d 7b 35e 35f 35g 35h 7c 35i 35j 35k 35l
2P 1a 1a 5b 5a 7b 35f 35h 35e 35g 7c 35k 35i 35l 35j 7a 35c 35a 35d 35b
3P 1a 2a 5b 5a 7c 35k 35i 35l 35j 7a 35b 35d 35a 35c 7b 35g 35e 35h 35f
5P 1a 2a 1a 1a 7b 7b 7b 7b 7b 7c 7c 7c 7c 7c 7a 7a 7a 7a 7a
7P 1a 2a 5b 5a 1a 5b 5a 5a 5b 1a 5b 5a 5a 5b 1a 5b 5a 5a 5b
11P 1a 2a 5a 5b 7c 35l 35k 35j 35i 7a 35a 35b 35c 35d 7b 35h 35g 35f 35e
13P 1a 2a 5b 5a 7a 35b 35d 35a 35c 7b 35f 35h 35e 35g 7c 35j 35l 35i 35k
17P 1a 2a 5b 5a 7c 35j 35l 35i 35k 7a 35c 35a 35d 35b 7b 35f 35h 35e 35g
19P 1a 2a 5a 5b 7b 35e 35f 35g 35h 7c 35l 35k 35j 35i 7a 35d 35c 35b 35a
23P 1a 2a 5b 5a 7b 35g 35e 35h 35f 7c 35j 35l 35i 35k 7a 35b 35d 35a 35c
29P 1a 2a 5a 5b 7a 35d 35c 35b 35a 7b 35h 35g 35f 35e 7c 35l 35k 35j 35i
31P 1a 2a 5a 5b 7c 35i 35j 35k 35l 7a 35d 35c 35b 35a 7b 35e 35f 35g 35h
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 2 . 2 2 B B B B B C C C C C D D D D D
X.4 2 . 2 2 C C C C C D D D D D B B B B B
X.5 2 . 2 2 D D D D D B B B B B C C C C C
X.6 2 . A *A 2 A *A *A A 2 A *A *A A 2 A *A *A A
X.7 2 . *A A 2 *A A A *A 2 *A A A *A 2 *A A A *A
X.8 2 . A *A D E L K F B J O P I C G N M H
X.9 2 . A *A D F K L E B I P O J C H M N G
X.10 2 . A *A C G N M H D F K L E B J O P I
X.11 2 . A *A C H M N G D E L K F B I P O J
X.12 2 . A *A B I P O J C G N M H D E L K F
X.13 2 . A *A B J O P I C H M N G D F K L E
X.14 2 . *A A D K E F L B P J I O C M G H N
X.15 2 . *A A D L F E K B O I J P C N H G M
X.16 2 . *A A C M G H N D L F E K B P J I O
X.17 2 . *A A C N H G M D K E F L B O I J P
X.18 2 . *A A B O I J P C M G H N D K E F L
X.19 2 . *A A B P J I O C N H G M D L F E K
A = E(5)^2+E(5)^3
= (-1-Sqrt(5))/2 = -1-b5
B = E(7)+E(7)^6
C = E(7)^2+E(7)^5
D = E(7)^3+E(7)^4
E = E(35)^6+E(35)^29
F = E(35)+E(35)^34
G = E(35)^11+E(35)^24
H = E(35)^4+E(35)^31
I = E(35)^16+E(35)^19
J = E(35)^9+E(35)^26
K = E(35)^13+E(35)^22
L = E(35)^8+E(35)^27
M = E(35)^17+E(35)^18
N = E(35)^3+E(35)^32
O = E(35)^12+E(35)^23
P = E(35)^2+E(35)^33
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