Group action invariants
| Degree $n$ : | $31$ | |
| Transitive number $t$ : | $6$ | |
| Group : | $C_{31}:C_{10}$ | |
| Parity: | $-1$ | |
| Primitive: | Yes | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (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), (1,27,16,29,8,30,4,15,2,23)(3,19,17,25,24,28,12,14,6,7)(5,11,18,21,9,26,20,13,10,22) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 5: $C_5$ 10: $C_{10}$ Resolvents shown for degrees $\leq 47$
Subfields
Prime degree - none
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 $ | $1$ | $1$ | $()$ |
| $ 5, 5, 5, 5, 5, 5, 1 $ | $31$ | $5$ | $( 2, 3, 5, 9,17)( 4, 7,13,25,18)( 6,11,21,10,19)( 8,15,29,26,20) (12,23,14,27,22)(16,31,30,28,24)$ |
| $ 5, 5, 5, 5, 5, 5, 1 $ | $31$ | $5$ | $( 2, 5,17, 3, 9)( 4,13,18, 7,25)( 6,21,19,11,10)( 8,29,20,15,26) (12,14,22,23,27)(16,30,24,31,28)$ |
| $ 5, 5, 5, 5, 5, 5, 1 $ | $31$ | $5$ | $( 2, 9, 3,17, 5)( 4,25, 7,18,13)( 6,10,11,19,21)( 8,26,15,20,29) (12,27,23,22,14)(16,28,31,24,30)$ |
| $ 10, 10, 10, 1 $ | $31$ | $10$ | $( 2,16, 9,28, 3,31,17,24, 5,30)( 4,15,25,20, 7,29,18, 8,13,26)( 6,14,10,12,11, 27,19,23,21,22)$ |
| $ 5, 5, 5, 5, 5, 5, 1 $ | $31$ | $5$ | $( 2,17, 9, 5, 3)( 4,18,25,13, 7)( 6,19,10,21,11)( 8,20,26,29,15) (12,22,27,14,23)(16,24,28,30,31)$ |
| $ 10, 10, 10, 1 $ | $31$ | $10$ | $( 2,24, 3,16, 5,31, 9,30,17,28)( 4, 8, 7,15,13,29,25,26,18,20)( 6,23,11,14,21, 27,10,22,19,12)$ |
| $ 10, 10, 10, 1 $ | $31$ | $10$ | $( 2,28,17,30, 9,31, 5,16, 3,24)( 4,20,18,26,25,29,13,15, 7, 8)( 6,12,19,22,10, 27,21,14,11,23)$ |
| $ 10, 10, 10, 1 $ | $31$ | $10$ | $( 2,30, 5,24,17,31, 3,28, 9,16)( 4,26,13, 8,18,29, 7,20,25,15)( 6,22,21,23,19, 27,11,12,10,14)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $31$ | $2$ | $( 2,31)( 3,30)( 4,29)( 5,28)( 6,27)( 7,26)( 8,25)( 9,24)(10,23)(11,22)(12,21) (13,20)(14,19)(15,18)(16,17)$ |
| $ 31 $ | $10$ | $31$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31)$ |
| $ 31 $ | $10$ | $31$ | $( 1, 4, 7,10,13,16,19,22,25,28,31, 3, 6, 9,12,15,18,21,24,27,30, 2, 5, 8,11, 14,17,20,23,26,29)$ |
| $ 31 $ | $10$ | $31$ | $( 1, 6,11,16,21,26,31, 5,10,15,20,25,30, 4, 9,14,19,24,29, 3, 8,13,18,23,28, 2, 7,12,17,22,27)$ |
Group invariants
| Order: | $310=2 \cdot 5 \cdot 31$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [310, 1] |
| Character table: |
2 1 1 1 1 1 1 1 1 1 1 . . .
5 1 1 1 1 1 1 1 1 1 1 . . .
31 1 . . . . . . . . . 1 1 1
1a 5a 5b 5c 10a 5d 10b 10c 10d 2a 31a 31b 31c
2P 1a 5b 5d 5a 5c 5c 5a 5d 5b 1a 31a 31b 31c
3P 1a 5c 5a 5d 10c 5b 10a 10d 10b 2a 31b 31c 31a
5P 1a 1a 1a 1a 2a 1a 2a 2a 2a 2a 31c 31a 31b
7P 1a 5b 5d 5a 10b 5c 10d 10a 10c 2a 31b 31c 31a
11P 1a 5a 5b 5c 10a 5d 10b 10c 10d 2a 31c 31a 31b
13P 1a 5c 5a 5d 10c 5b 10a 10d 10b 2a 31c 31a 31b
17P 1a 5b 5d 5a 10b 5c 10d 10a 10c 2a 31b 31c 31a
19P 1a 5d 5c 5b 10d 5a 10c 10b 10a 2a 31b 31c 31a
23P 1a 5c 5a 5d 10c 5b 10a 10d 10b 2a 31a 31b 31c
29P 1a 5d 5c 5b 10d 5a 10c 10b 10a 2a 31a 31b 31c
31P 1a 5a 5b 5c 10a 5d 10b 10c 10d 2a 1a 1a 1a
X.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
X.3 1 A B /B -/A /A -/B -B -A -1 1 1 1
X.4 1 B /A A -/B /B -A -/A -B -1 1 1 1
X.5 1 /B A /A -B B -/A -A -/B -1 1 1 1
X.6 1 /A /B B -A A -B -/B -/A -1 1 1 1
X.7 1 A B /B /A /A /B B A 1 1 1 1
X.8 1 B /A A /B /B A /A B 1 1 1 1
X.9 1 /B A /A B B /A A /B 1 1 1 1
X.10 1 /A /B B A A B /B /A 1 1 1 1
X.11 10 . . . . . . . . . C E D
X.12 10 . . . . . . . . . D C E
X.13 10 . . . . . . . . . E D C
A = E(5)^4
B = E(5)^3
C = E(31)+E(31)^2+E(31)^4+E(31)^8+E(31)^15+E(31)^16+E(31)^23+E(31)^27+E(31)^29+E(31)^30
D = E(31)^5+E(31)^9+E(31)^10+E(31)^11+E(31)^13+E(31)^18+E(31)^20+E(31)^21+E(31)^22+E(31)^26
E = E(31)^3+E(31)^6+E(31)^7+E(31)^12+E(31)^14+E(31)^17+E(31)^19+E(31)^24+E(31)^25+E(31)^28
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