Group action invariants
| Degree $n$ : | $35$ | |
| Transitive number $t$ : | $11$ | |
| Group : | $D_{35}:C_3$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,7,21,32,26,12)(2,6,22,31,27,11)(3,10,23,35,28,15)(4,9,24,34,29,14)(5,8,25,33,30,13)(16,17)(18,20), (1,16)(2,20)(3,19)(4,18)(5,17)(6,11)(7,15)(8,14)(9,13)(10,12)(21,31)(22,35)(23,34)(24,33)(25,32)(27,30)(28,29) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 6: $C_6$ 10: $D_{5}$ 30: $D_5\times C_3$ 42: $F_7$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $D_{5}$
Degree 7: $F_7$
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 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $7$ | $3$ | $( 6,11,21)( 7,12,22)( 8,13,23)( 9,14,24)(10,15,25)(16,31,26)(17,32,27) (18,33,28)(19,34,29)(20,35,30)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $7$ | $3$ | $( 6,21,11)( 7,22,12)( 8,23,13)( 9,24,14)(10,25,15)(16,26,31)(17,27,32) (18,28,33)(19,29,34)(20,30,35)$ |
| $ 6, 6, 6, 6, 6, 2, 2, 1 $ | $35$ | $6$ | $( 2, 5)( 3, 4)( 6,16,11,31,21,26)( 7,20,12,35,22,30)( 8,19,13,34,23,29) ( 9,18,14,33,24,28)(10,17,15,32,25,27)$ |
| $ 6, 6, 6, 6, 6, 2, 2, 1 $ | $35$ | $6$ | $( 2, 5)( 3, 4)( 6,26,21,31,11,16)( 7,30,22,35,12,20)( 8,29,23,34,13,19) ( 9,28,24,33,14,18)(10,27,25,32,15,17)$ |
| $ 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 $ | $2$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$ |
| $ 15, 15, 5 $ | $14$ | $15$ | $( 1, 2, 3, 4, 5)( 6,12,23, 9,15,21, 7,13,24,10,11,22, 8,14,25)(16,32,28,19,35, 26,17,33,29,20,31,27,18,34,30)$ |
| $ 15, 15, 5 $ | $14$ | $15$ | $( 1, 2, 3, 4, 5)( 6,22,13, 9,25,11, 7,23,14,10,21,12, 8,24,15)(16,27,33,19,30, 31,17,28,34,20,26,32,18,29,35)$ |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)(26,28,30,27,29)(31,33,35,32,34)$ |
| $ 15, 15, 5 $ | $14$ | $15$ | $( 1, 3, 5, 2, 4)( 6,13,25, 7,14,21, 8,15,22, 9,11,23,10,12,24)(16,33,30,17,34, 26,18,35,27,19,31,28,20,32,29)$ |
| $ 15, 15, 5 $ | $14$ | $15$ | $( 1, 3, 5, 2, 4)( 6,23,15, 7,24,11, 8,25,12, 9,21,13,10,22,14)(16,28,35,17,29, 31,18,30,32,19,26,33,20,27,34)$ |
| $ 7, 7, 7, 7, 7 $ | $6$ | $7$ | $( 1, 6,11,16,21,26,31)( 2, 7,12,17,22,27,32)( 3, 8,13,18,23,28,33) ( 4, 9,14,19,24,29,34)( 5,10,15,20,25,30,35)$ |
| $ 35 $ | $6$ | $35$ | $( 1, 7,13,19,25,26,32, 3, 9,15,16,22,28,34, 5, 6,12,18,24,30,31, 2, 8,14,20, 21,27,33, 4,10,11,17,23,29,35)$ |
| $ 35 $ | $6$ | $35$ | $( 1, 8,15,17,24,26,33, 5, 7,14,16,23,30,32, 4, 6,13,20,22,29,31, 3,10,12,19, 21,28,35, 2, 9,11,18,25,27,34)$ |
| $ 35 $ | $6$ | $35$ | $( 1, 9,12,20,23,26,34, 2,10,13,16,24,27,35, 3, 6,14,17,25,28,31, 4, 7,15,18, 21,29,32, 5, 8,11,19,22,30,33)$ |
| $ 35 $ | $6$ | $35$ | $( 1,10,14,18,22,26,35, 4, 8,12,16,25,29,33, 2, 6,15,19,23,27,31, 5, 9,13,17, 21,30,34, 3, 7,11,20,24,28,32)$ |
Group invariants
| Order: | $210=2 \cdot 3 \cdot 5 \cdot 7$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [210, 3] |
| Character table: |
2 1 1 1 1 1 1 . . . . . . . . . . .
3 1 1 1 1 1 1 1 1 1 1 1 1 . . . . .
5 1 1 1 . . . 1 1 1 1 1 1 1 1 1 1 1
7 1 . . . . . 1 . . 1 . . 1 1 1 1 1
1a 3a 3b 6a 6b 2a 5a 15a 15b 5b 15c 15d 7a 35a 35b 35c 35d
2P 1a 3b 3a 3a 3b 1a 5b 15d 15c 5a 15b 15a 7a 35b 35d 35a 35c
3P 1a 1a 1a 2a 2a 2a 5b 5b 5b 5a 5a 5a 7a 35b 35d 35a 35c
5P 1a 3b 3a 6b 6a 2a 1a 3b 3a 1a 3b 3a 7a 7a 7a 7a 7a
7P 1a 3a 3b 6a 6b 2a 5b 15c 15d 5a 15a 15b 1a 5b 5a 5a 5b
11P 1a 3b 3a 6b 6a 2a 5a 15b 15a 5b 15d 15c 7a 35a 35b 35c 35d
13P 1a 3a 3b 6a 6b 2a 5b 15c 15d 5a 15a 15b 7a 35b 35d 35a 35c
17P 1a 3b 3a 6b 6a 2a 5b 15d 15c 5a 15b 15a 7a 35c 35a 35d 35b
19P 1a 3a 3b 6a 6b 2a 5a 15a 15b 5b 15c 15d 7a 35a 35b 35c 35d
23P 1a 3b 3a 6b 6a 2a 5b 15d 15c 5a 15b 15a 7a 35c 35a 35d 35b
29P 1a 3b 3a 6b 6a 2a 5a 15b 15a 5b 15d 15c 7a 35d 35c 35b 35a
31P 1a 3a 3b 6a 6b 2a 5a 15a 15b 5b 15c 15d 7a 35d 35c 35b 35a
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 -/A -A -1 1 A /A 1 A /A 1 1 1 1 1
X.4 1 /A A -A -/A -1 1 /A A 1 /A A 1 1 1 1 1
X.5 1 A /A /A A 1 1 A /A 1 A /A 1 1 1 1 1
X.6 1 /A A A /A 1 1 /A A 1 /A A 1 1 1 1 1
X.7 2 B /B . . . C E /E *C F /F 2 C *C *C C
X.8 2 /B B . . . C /E E *C /F F 2 C *C *C C
X.9 2 B /B . . . *C F /F C E /E 2 *C C C *C
X.10 2 /B B . . . *C /F F C /E E 2 *C C C *C
X.11 2 2 2 . . . C C C *C *C *C 2 C *C *C C
X.12 2 2 2 . . . *C *C *C C C C 2 *C C C *C
X.13 6 . . . . . 6 . . 6 . . -1 -1 -1 -1 -1
X.14 6 . . . . . D . . *D . . -1 G I J H
X.15 6 . . . . . D . . *D . . -1 H J I G
X.16 6 . . . . . *D . . D . . -1 I H G J
X.17 6 . . . . . *D . . D . . -1 J G H I
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
B = 2*E(3)
= -1+Sqrt(-3) = 2b3
C = E(5)+E(5)^4
= (-1+Sqrt(5))/2 = b5
D = 3*E(5)^2+3*E(5)^3
= (-3-3*Sqrt(5))/2 = -3-3b5
E = E(15)^2+E(15)^8
F = E(15)^11+E(15)^14
G = E(35)^4+E(35)^6+E(35)^9+E(35)^26+E(35)^29+E(35)^31
H = E(35)+E(35)^11+E(35)^16+E(35)^19+E(35)^24+E(35)^34
I = E(35)^8+E(35)^12+E(35)^17+E(35)^18+E(35)^23+E(35)^27
J = E(35)^2+E(35)^3+E(35)^13+E(35)^22+E(35)^32+E(35)^33
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