# Properties

 Label 31T6 Degree $31$ Order $310$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $C_{31}:C_{10}$

Show commands: Magma

magma: G := TransitiveGroup(31, 6);

## Group action invariants

 Degree $n$: $31$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $6$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_{31}:C_{10}$ Parity: $-1$ magma: IsEven(G); Primitive: yes magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $1$ magma: Order(Centralizer(SymmetricGroup(n), G)); 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) magma: Generators(G);

## 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $5, 5, 5, 5, 5, 5, 1$ $31$ $5$ $( 2, 3, 5, 9,17)( 4, 7,13,25,18)( 6,11,21,10,19)( 8,15,29,26,20) (12,23,14,27,22)(16,31,30,28,24)$ $5, 5, 5, 5, 5, 5, 1$ $31$ $5$ $( 2, 5,17, 3, 9)( 4,13,18, 7,25)( 6,21,19,11,10)( 8,29,20,15,26) (12,14,22,23,27)(16,30,24,31,28)$ $5, 5, 5, 5, 5, 5, 1$ $31$ $5$ $( 2, 9, 3,17, 5)( 4,25, 7,18,13)( 6,10,11,19,21)( 8,26,15,20,29) (12,27,23,22,14)(16,28,31,24,30)$ $10, 10, 10, 1$ $31$ $10$ $( 2,16, 9,28, 3,31,17,24, 5,30)( 4,15,25,20, 7,29,18, 8,13,26)( 6,14,10,12,11, 27,19,23,21,22)$ $5, 5, 5, 5, 5, 5, 1$ $31$ $5$ $( 2,17, 9, 5, 3)( 4,18,25,13, 7)( 6,19,10,21,11)( 8,20,26,29,15) (12,22,27,14,23)(16,24,28,30,31)$ $10, 10, 10, 1$ $31$ $10$ $( 2,24, 3,16, 5,31, 9,30,17,28)( 4, 8, 7,15,13,29,25,26,18,20)( 6,23,11,14,21, 27,10,22,19,12)$ $10, 10, 10, 1$ $31$ $10$ $( 2,28,17,30, 9,31, 5,16, 3,24)( 4,20,18,26,25,29,13,15, 7, 8)( 6,12,19,22,10, 27,21,14,11,23)$ $10, 10, 10, 1$ $31$ $10$ $( 2,30, 5,24,17,31, 3,28, 9,16)( 4,26,13, 8,18,29, 7,20,25,15)( 6,22,21,23,19, 27,11,12,10,14)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1$ $31$ $2$ $( 2,31)( 3,30)( 4,29)( 5,28)( 6,27)( 7,26)( 8,25)( 9,24)(10,23)(11,22)(12,21) (13,20)(14,19)(15,18)(16,17)$ $31$ $10$ $31$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31)$ $31$ $10$ $31$ $( 1, 4, 7,10,13,16,19,22,25,28,31, 3, 6, 9,12,15,18,21,24,27,30, 2, 5, 8,11, 14,17,20,23,26,29)$ $31$ $10$ $31$ $( 1, 6,11,16,21,26,31, 5,10,15,20,25,30, 4, 9,14,19,24,29, 3, 8,13,18,23,28, 2, 7,12,17,22,27)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $310=2 \cdot 5 \cdot 31$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 310.1 magma: IdentifyGroup(G);
 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 

magma: CharacterTable(G);