Group action invariants
| Degree $n$ : | $35$ | |
| Transitive number $t$ : | $9$ | |
| Group : | $D_5\times C_7:C_3$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,35,14,3,32,11,5,34,13,2,31,15,4,33,12)(6,20,24,8,17,21,10,19,23,7,16,25,9,18,22)(26,30,29,28,27), (1,12,21,32,6,17,26,2,11,22,31,7,16,27)(3,15,23,35,8,20,28,5,13,25,33,10,18,30)(4,14,24,34,9,19,29) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 6: $C_6$ 10: $D_{5}$ 21: $C_7:C_3$ 30: $D_5\times C_3$ 42: $(C_7:C_3) \times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $D_{5}$
Degree 7: $C_7:C_3$
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, 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 $ | $7$ | $3$ | $( 6,21,11)( 7,22,12)( 8,23,13)( 9,24,14)(10,25,15)(16,26,31)(17,27,32) (18,28,33)(19,29,34)(20,30,35)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30) (28,29)(32,35)(33,34)$ |
| $ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ | $35$ | $6$ | $( 2, 5)( 3, 4)( 6,11,21)( 7,15,22,10,12,25)( 8,14,23, 9,13,24)(16,31,26) (17,35,27,20,32,30)(18,34,28,19,33,29)$ |
| $ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ | $35$ | $6$ | $( 2, 5)( 3, 4)( 6,21,11)( 7,25,12,10,22,15)( 8,24,13, 9,23,14)(16,26,31) (17,30,32,20,27,35)(18,29,33,19,28,34)$ |
| $ 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 $ | $3$ | $7$ | $( 1, 6,11,16,21,26,31)( 2, 7,12,17,22,27,32)( 3, 8,13,18,23,28,33) ( 4, 9,14,19,24,29,34)( 5,10,15,20,25,30,35)$ |
| $ 14, 14, 7 $ | $15$ | $14$ | $( 1, 6,11,16,21,26,31)( 2,10,12,20,22,30,32, 5, 7,15,17,25,27,35) ( 3, 9,13,19,23,29,33, 4, 8,14,18,24,28,34)$ |
| $ 35 $ | $6$ | $35$ | $( 1, 7,13,19,25,26,32, 3, 9,15,16,22,28,34, 5, 6,12,18,24,30,31, 2, 8,14,20, 21,27,33, 4,10,11,17,23,29,35)$ |
| $ 35 $ | $6$ | $35$ | $( 1, 8,15,17,24,26,33, 5, 7,14,16,23,30,32, 4, 6,13,20,22,29,31, 3,10,12,19, 21,28,35, 2, 9,11,18,25,27,34)$ |
| $ 7, 7, 7, 7, 7 $ | $3$ | $7$ | $( 1,16,31,11,26, 6,21)( 2,17,32,12,27, 7,22)( 3,18,33,13,28, 8,23) ( 4,19,34,14,29, 9,24)( 5,20,35,15,30,10,25)$ |
| $ 14, 14, 7 $ | $15$ | $14$ | $( 1,16,31,11,26, 6,21)( 2,20,32,15,27,10,22, 5,17,35,12,30, 7,25) ( 3,19,33,14,28, 9,23, 4,18,34,13,29, 8,24)$ |
| $ 35 $ | $6$ | $35$ | $( 1,17,33,14,30, 6,22, 3,19,35,11,27, 8,24, 5,16,32,13,29,10,21, 2,18,34,15, 26, 7,23, 4,20,31,12,28, 9,25)$ |
| $ 35 $ | $6$ | $35$ | $( 1,18,35,12,29, 6,23, 5,17,34,11,28,10,22, 4,16,33,15,27, 9,21, 3,20,32,14, 26, 8,25, 2,19,31,13,30, 7,24)$ |
Group invariants
| Order: | $210=2 \cdot 3 \cdot 5 \cdot 7$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [210, 2] |
| Character table: |
2 1 1 1 1 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 1 1 1
1a 3a 3b 2a 6a 6b 5a 15a 15b 5b 15c 15d 7a 14a 35a 35b 7b 14b 35c
2P 1a 3b 3a 1a 3b 3a 5b 15d 15c 5a 15b 15a 7a 7a 35b 35a 7b 7b 35d
3P 1a 1a 1a 2a 2a 2a 5b 5b 5b 5a 5a 5a 7b 14b 35d 35c 7a 14a 35b
5P 1a 3b 3a 2a 6b 6a 1a 3b 3a 1a 3b 3a 7b 14b 7b 7b 7a 14a 7a
7P 1a 3a 3b 2a 6a 6b 5b 15c 15d 5a 15a 15b 1a 2a 5b 5a 1a 2a 5b
11P 1a 3b 3a 2a 6b 6a 5a 15b 15a 5b 15d 15c 7a 14a 35a 35b 7b 14b 35c
13P 1a 3a 3b 2a 6a 6b 5b 15c 15d 5a 15a 15b 7b 14b 35d 35c 7a 14a 35b
17P 1a 3b 3a 2a 6b 6a 5b 15d 15c 5a 15b 15a 7b 14b 35d 35c 7a 14a 35b
19P 1a 3a 3b 2a 6a 6b 5a 15a 15b 5b 15c 15d 7b 14b 35c 35d 7a 14a 35a
23P 1a 3b 3a 2a 6b 6a 5b 15d 15c 5a 15b 15a 7a 14a 35b 35a 7b 14b 35d
29P 1a 3b 3a 2a 6b 6a 5a 15b 15a 5b 15d 15c 7a 14a 35a 35b 7b 14b 35c
31P 1a 3a 3b 2a 6a 6b 5a 15a 15b 5b 15c 15d 7b 14b 35c 35d 7a 14a 35a
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 A /A -1 -A -/A 1 A /A 1 A /A 1 -1 1 1 1 -1 1
X.4 1 /A A -1 -/A -A 1 /A A 1 /A A 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 2 B /B . . . C E /E *C F /F 2 . C *C 2 . C
X.8 2 /B B . . . C /E E *C /F F 2 . C *C 2 . C
X.9 2 B /B . . . *C F /F C E /E 2 . *C C 2 . *C
X.10 2 /B B . . . *C /F F C /E E 2 . *C C 2 . *C
X.11 2 2 2 . . . C C C *C *C *C 2 . C *C 2 . C
X.12 2 2 2 . . . *C *C *C C C C 2 . *C C 2 . *C
X.13 3 . . -3 . . 3 . . 3 . . G -G G G /G -/G /G
X.14 3 . . -3 . . 3 . . 3 . . /G -/G /G /G G -G G
X.15 3 . . 3 . . 3 . . 3 . . G G G G /G /G /G
X.16 3 . . 3 . . 3 . . 3 . . /G /G /G /G G G G
X.17 6 . . . . . D . . *D . . H . I J /H . /I
X.18 6 . . . . . D . . *D . . /H . /I /J H . I
X.19 6 . . . . . *D . . D . . H . J I /H . /J
X.20 6 . . . . . *D . . D . . /H . /J /I H . J
2 .
3 .
5 1
7 1
35d
2P 35c
3P 35a
5P 7a
7P 5a
11P 35d
13P 35a
17P 35a
19P 35b
23P 35c
29P 35d
31P 35b
X.1 1
X.2 1
X.3 1
X.4 1
X.5 1
X.6 1
X.7 *C
X.8 *C
X.9 C
X.10 C
X.11 *C
X.12 C
X.13 /G
X.14 G
X.15 /G
X.16 G
X.17 /J
X.18 J
X.19 /I
X.20 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)+3*E(5)^4
= (-3+3*Sqrt(5))/2 = 3b5
E = E(15)^2+E(15)^8
F = E(15)^11+E(15)^14
G = E(7)^3+E(7)^5+E(7)^6
= (-1-Sqrt(-7))/2 = -1-b7
H = 2*E(7)^3+2*E(7)^5+2*E(7)^6
= -1-Sqrt(-7) = -1-i7
I = E(35)^2+E(35)^8+E(35)^18+E(35)^22+E(35)^23+E(35)^32
J = E(35)+E(35)^4+E(35)^9+E(35)^11+E(35)^16+E(35)^29
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