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Group invariants
| Abstract group: | $C_7\times D_5$ |
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| Order: | $70=2 \cdot 5 \cdot 7$ |
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| Cyclic: | no |
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| Abelian: | no |
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| Solvable: | yes |
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| Nilpotency class: | not nilpotent |
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Group action invariants
| Degree $n$: | $35$ |
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| Transitive number $t$: | $2$ |
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| Parity: | $1$ |
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| Primitive: | no |
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| $\card{\Aut(F/K)}$: | $7$ |
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| Generators: | $(1,10,14,18,22,26,35,4,8,12,16,25,29,33,2,6,15,19,23,27,31,5,9,13,17,21,30,34,3,7,11,20,24,28,32)$, $(1,11,21,31,6,16,26)(2,15,22,35,7,20,27,5,12,25,32,10,17,30)(3,14,23,34,8,19,28,4,13,24,33,9,18,29)$ |
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ $7$: $C_7$ $10$: $D_{5}$ $14$: $C_{14}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $D_{5}$
Degree 7: $C_7$
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
| Label | Cycle Type | Size | Order | Index | Representative |
| 1A | $1^{35}$ | $1$ | $1$ | $0$ | $()$ |
| 2A | $2^{14},1^{7}$ | $5$ | $2$ | $14$ | $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)$ |
| 5A1 | $5^{7}$ | $2$ | $5$ | $28$ | $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18)(21,24,22,25,23)(26,29,27,30,28)(31,34,32,35,33)$ |
| 5A2 | $5^{7}$ | $2$ | $5$ | $28$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$ |
| 7A1 | $7^{5}$ | $1$ | $7$ | $30$ | $( 1,16,31,11,26, 6,21)( 2,17,32,12,27, 7,22)( 3,18,33,13,28, 8,23)( 4,19,34,14,29, 9,24)( 5,20,35,15,30,10,25)$ |
| 7A-1 | $7^{5}$ | $1$ | $7$ | $30$ | $( 1,21, 6,26,11,31,16)( 2,22, 7,27,12,32,17)( 3,23, 8,28,13,33,18)( 4,24, 9,29,14,34,19)( 5,25,10,30,15,35,20)$ |
| 7A2 | $7^{5}$ | $1$ | $7$ | $30$ | $( 1,31,26,21,16,11, 6)( 2,32,27,22,17,12, 7)( 3,33,28,23,18,13, 8)( 4,34,29,24,19,14, 9)( 5,35,30,25,20,15,10)$ |
| 7A-2 | $7^{5}$ | $1$ | $7$ | $30$ | $( 1, 6,11,16,21,26,31)( 2, 7,12,17,22,27,32)( 3, 8,13,18,23,28,33)( 4, 9,14,19,24,29,34)( 5,10,15,20,25,30,35)$ |
| 7A3 | $7^{5}$ | $1$ | $7$ | $30$ | $( 1,11,21,31, 6,16,26)( 2,12,22,32, 7,17,27)( 3,13,23,33, 8,18,28)( 4,14,24,34, 9,19,29)( 5,15,25,35,10,20,30)$ |
| 7A-3 | $7^{5}$ | $1$ | $7$ | $30$ | $( 1,26,16, 6,31,21,11)( 2,27,17, 7,32,22,12)( 3,28,18, 8,33,23,13)( 4,29,19, 9,34,24,14)( 5,30,20,10,35,25,15)$ |
| 14A1 | $14^{2},7$ | $5$ | $14$ | $32$ | $( 1,26,16, 6,31,21,11)( 2,30,17,10,32,25,12, 5,27,20, 7,35,22,15)( 3,29,18, 9,33,24,13, 4,28,19, 8,34,23,14)$ |
| 14A-1 | $14^{2},7$ | $5$ | $14$ | $32$ | $( 1,14,21,34, 6,19,26, 4,11,24,31, 9,16,29)( 2,13,22,33, 7,18,27, 3,12,23,32, 8,17,28)( 5,15,25,35,10,20,30)$ |
| 14A3 | $14^{2},7$ | $5$ | $14$ | $32$ | $( 1,10,11,20,21,30,31, 5, 6,15,16,25,26,35)( 2, 9,12,19,22,29,32, 4, 7,14,17,24,27,34)( 3, 8,13,18,23,28,33)$ |
| 14A-3 | $14^{2},7$ | $5$ | $14$ | $32$ | $( 1,31,26,21,16,11, 6)( 2,35,27,25,17,15, 7, 5,32,30,22,20,12,10)( 3,34,28,24,18,14, 8, 4,33,29,23,19,13, 9)$ |
| 14A5 | $14^{2},7$ | $5$ | $14$ | $32$ | $( 1,22, 6,27,11,32,16, 2,21, 7,26,12,31,17)( 3,25, 8,30,13,35,18, 5,23,10,28,15,33,20)( 4,24, 9,29,14,34,19)$ |
| 14A-5 | $14^{2},7$ | $5$ | $14$ | $32$ | $( 1,16,31,11,26, 6,21)( 2,20,32,15,27,10,22, 5,17,35,12,30, 7,25)( 3,19,33,14,28, 9,23, 4,18,34,13,29, 8,24)$ |
| 35A1 | $35$ | $2$ | $35$ | $34$ | $( 1,10,14,18,22,26,35, 4, 8,12,16,25,29,33, 2, 6,15,19,23,27,31, 5, 9,13,17,21,30,34, 3, 7,11,20,24,28,32)$ |
| 35A-1 | $35$ | $2$ | $35$ | $34$ | $( 1,35,29,23,17,11,10, 4,33,27,21,20,14, 8, 2,31,30,24,18,12, 6, 5,34,28,22,16,15, 9, 3,32,26,25,19,13, 7)$ |
| 35A2 | $35$ | $2$ | $35$ | $34$ | $( 1,14,22,35, 8,16,29, 2,15,23,31, 9,17,30, 3,11,24,32,10,18,26, 4,12,25,33, 6,19,27, 5,13,21,34, 7,20,28)$ |
| 35A-2 | $35$ | $2$ | $35$ | $34$ | $( 1,29,17,10,33,21,14, 2,30,18, 6,34,22,15, 3,26,19, 7,35,23,11, 4,27,20, 8,31,24,12, 5,28,16, 9,32,25,13)$ |
| 35A3 | $35$ | $2$ | $35$ | $34$ | $( 1,18,35,12,29, 6,23, 5,17,34,11,28,10,22, 4,16,33,15,27, 9,21, 3,20,32,14,26, 8,25, 2,19,31,13,30, 7,24)$ |
| 35A-3 | $35$ | $2$ | $35$ | $34$ | $( 1,23,10,27,14,31,18, 5,22, 9,26,13,35,17, 4,21, 8,30,12,34,16, 3,25, 7,29,11,33,20, 2,24, 6,28,15,32,19)$ |
| 35A4 | $35$ | $2$ | $35$ | $34$ | $( 1,22, 8,29,15,31,17, 3,24,10,26,12,33,19, 5,21, 7,28,14,35,16, 2,23, 9,30,11,32,18, 4,25, 6,27,13,34,20)$ |
| 35A-4 | $35$ | $2$ | $35$ | $34$ | $( 1,20,34,13,27, 6,25, 4,18,32,11,30, 9,23, 2,16,35,14,28, 7,21, 5,19,33,12,26,10,24, 3,17,31,15,29, 8,22)$ |
| 35A8 | $35$ | $2$ | $35$ | $34$ | $( 1, 8,15,17,24,26,33, 5, 7,14,16,23,30,32, 4, 6,13,20,22,29,31, 3,10,12,19,21,28,35, 2, 9,11,18,25,27,34)$ |
| 35A-8 | $35$ | $2$ | $35$ | $34$ | $( 1,33,30,22,19,11, 8, 5,32,29,21,18,15, 7, 4,31,28,25,17,14, 6, 3,35,27,24,16,13,10, 2,34,26,23,20,12, 9)$ |
| 35A9 | $35$ | $2$ | $35$ | $34$ | $( 1,15,24,33, 7,16,30, 4,13,22,31,10,19,28, 2,11,25,34, 8,17,26, 5,14,23,32, 6,20,29, 3,12,21,35, 9,18,27)$ |
| 35A-9 | $35$ | $2$ | $35$ | $34$ | $( 1,30,19, 8,32,21,15, 4,28,17, 6,35,24,13, 2,26,20, 9,33,22,11, 5,29,18, 7,31,25,14, 3,27,16,10,34,23,12)$ |
Malle's constant $a(G)$: $1/14$
Character table
| 1A | 2A | 5A1 | 5A2 | 7A1 | 7A-1 | 7A2 | 7A-2 | 7A3 | 7A-3 | 14A1 | 14A-1 | 14A3 | 14A-3 | 14A5 | 14A-5 | 35A1 | 35A-1 | 35A2 | 35A-2 | 35A3 | 35A-3 | 35A4 | 35A-4 | 35A8 | 35A-8 | 35A9 | 35A-9 | ||
| Size | 1 | 5 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 5 | 5 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| 2 P | 1A | 1A | 5A2 | 5A1 | 7A2 | 7A-2 | 7A-3 | 7A3 | 7A-1 | 7A1 | 7A1 | 7A-1 | 7A3 | 7A-3 | 7A-2 | 7A2 | 35A2 | 35A-2 | 35A4 | 35A-4 | 35A-1 | 35A1 | 35A8 | 35A-8 | 35A9 | 35A-9 | 35A-3 | 35A3 | |
| 5 P | 1A | 2A | 5A2 | 5A1 | 7A3 | 7A-3 | 7A-1 | 7A1 | 7A2 | 7A-2 | 14A3 | 14A-3 | 14A-5 | 14A5 | 14A1 | 14A-1 | 35A3 | 35A-3 | 35A-1 | 35A1 | 35A9 | 35A-9 | 35A-2 | 35A2 | 35A-4 | 35A4 | 35A-8 | 35A8 | |
| 7 P | 1A | 2A | 1A | 1A | 7A-2 | 7A2 | 7A3 | 7A-3 | 7A1 | 7A-1 | 14A5 | 14A-5 | 14A1 | 14A-1 | 14A-3 | 14A3 | 7A-3 | 7A3 | 7A1 | 7A-1 | 7A-2 | 7A2 | 7A2 | 7A-2 | 7A-3 | 7A3 | 7A1 | 7A-1 | |
| Type | |||||||||||||||||||||||||||||
| 70.1.1a | R | ||||||||||||||||||||||||||||
| 70.1.1b | R | ||||||||||||||||||||||||||||
| 70.1.1c1 | C | ||||||||||||||||||||||||||||
| 70.1.1c2 | C | ||||||||||||||||||||||||||||
| 70.1.1c3 | C | ||||||||||||||||||||||||||||
| 70.1.1c4 | C | ||||||||||||||||||||||||||||
| 70.1.1c5 | C | ||||||||||||||||||||||||||||
| 70.1.1c6 | C | ||||||||||||||||||||||||||||
| 70.1.1d1 | C | ||||||||||||||||||||||||||||
| 70.1.1d2 | C | ||||||||||||||||||||||||||||
| 70.1.1d3 | C | ||||||||||||||||||||||||||||
| 70.1.1d4 | C | ||||||||||||||||||||||||||||
| 70.1.1d5 | C | ||||||||||||||||||||||||||||
| 70.1.1d6 | C | ||||||||||||||||||||||||||||
| 70.1.2a1 | R | ||||||||||||||||||||||||||||
| 70.1.2a2 | R | ||||||||||||||||||||||||||||
| 70.1.2b1 | C | ||||||||||||||||||||||||||||
| 70.1.2b2 | C | ||||||||||||||||||||||||||||
| 70.1.2b3 | C | ||||||||||||||||||||||||||||
| 70.1.2b4 | C | ||||||||||||||||||||||||||||
| 70.1.2b5 | C | ||||||||||||||||||||||||||||
| 70.1.2b6 | C | ||||||||||||||||||||||||||||
| 70.1.2b7 | C | ||||||||||||||||||||||||||||
| 70.1.2b8 | C | ||||||||||||||||||||||||||||
| 70.1.2b9 | C | ||||||||||||||||||||||||||||
| 70.1.2b10 | C | ||||||||||||||||||||||||||||
| 70.1.2b11 | C | ||||||||||||||||||||||||||||
| 70.1.2b12 | C |
Regular extensions
Data not computed