Properties

 Label 31T5 Degree $31$ Order $186$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $C_{31}:C_{6}$

Learn more

Show commands: Magma

magma: G := TransitiveGroup(31, 5);

Group action invariants

 Degree $n$: $31$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $5$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_{31}:C_{6}$ 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: (1,26,25,30,5,6)(2,21,19,29,10,12)(3,16,13,28,15,18)(4,11,7,27,20,24)(8,22,14,23,9,17), (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) magma: Generators(G);

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$

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

magma: ConjugacyClasses(G);

Group invariants

 Order: $186=2 \cdot 3 \cdot 31$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 186.1 magma: IdentifyGroup(G);
 Character table:  2 1 1 1 1 1 1 . . . . . 3 1 1 1 1 1 1 . . . . . 31 1 . . . . . 1 1 1 1 1 1a 3a 6a 3b 6b 2a 31a 31b 31c 31d 31e 2P 1a 3b 3a 3a 3b 1a 31b 31d 31a 31e 31c 3P 1a 1a 2a 1a 2a 2a 31c 31a 31e 31b 31d 5P 1a 3b 6b 3a 6a 2a 31a 31b 31c 31d 31e 7P 1a 3a 6a 3b 6b 2a 31d 31e 31b 31c 31a 11P 1a 3b 6b 3a 6a 2a 31d 31e 31b 31c 31a 13P 1a 3a 6a 3b 6b 2a 31c 31a 31e 31b 31d 17P 1a 3b 6b 3a 6a 2a 31e 31c 31d 31a 31b 19P 1a 3a 6a 3b 6b 2a 31b 31d 31a 31e 31c 23P 1a 3b 6b 3a 6a 2a 31e 31c 31d 31a 31b 29P 1a 3b 6b 3a 6a 2a 31b 31d 31a 31e 31c 31P 1a 3a 6a 3b 6b 2a 1a 1a 1a 1a 1a X.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 X.3 1 A -/A /A -A -1 1 1 1 1 1 X.4 1 /A -A A -/A -1 1 1 1 1 1 X.5 1 A /A /A A 1 1 1 1 1 1 X.6 1 /A A A /A 1 1 1 1 1 1 X.7 6 . . . . . B F E C D X.8 6 . . . . . C D F E B X.9 6 . . . . . D E C B F X.10 6 . . . . . E B D F C X.11 6 . . . . . F C B D E A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = E(31)^4+E(31)^7+E(31)^11+E(31)^20+E(31)^24+E(31)^27 C = E(31)^3+E(31)^13+E(31)^15+E(31)^16+E(31)^18+E(31)^28 D = E(31)+E(31)^5+E(31)^6+E(31)^25+E(31)^26+E(31)^30 E = E(31)^2+E(31)^10+E(31)^12+E(31)^19+E(31)^21+E(31)^29 F = E(31)^8+E(31)^9+E(31)^14+E(31)^17+E(31)^22+E(31)^23 

magma: CharacterTable(G);