Properties

 Label 31T4 Degree $31$ Order $155$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $C_{31}:C_{5}$

Show commands: Magma

magma: G := TransitiveGroup(31, 4);

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$5$:  $C_5$

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

magma: ConjugacyClasses(G);

Group invariants

 Order: $155=5 \cdot 31$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 155.1 magma: IdentifyGroup(G);
 Character table:  5 1 1 1 1 1 . . . . . . 31 1 . . . . 1 1 1 1 1 1 1a 5a 5b 5c 5d 31a 31b 31c 31d 31e 31f 2P 1a 5b 5d 5a 5c 31a 31b 31c 31d 31e 31f 3P 1a 5c 5a 5d 5b 31b 31c 31f 31e 31a 31d 5P 1a 1a 1a 1a 1a 31c 31f 31d 31a 31b 31e 7P 1a 5b 5d 5a 5c 31d 31e 31a 31c 31f 31b 11P 1a 5a 5b 5c 5d 31e 31a 31b 31f 31d 31c 13P 1a 5c 5a 5d 5b 31e 31a 31b 31f 31d 31c 17P 1a 5b 5d 5a 5c 31b 31c 31f 31e 31a 31d 19P 1a 5d 5c 5b 5a 31d 31e 31a 31c 31f 31b 23P 1a 5c 5a 5d 5b 31f 31d 31e 31b 31c 31a 29P 1a 5d 5c 5b 5a 31f 31d 31e 31b 31c 31a 31P 1a 5a 5b 5c 5d 1a 1a 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 A B /B /A 1 1 1 1 1 1 X.3 1 B /A A /B 1 1 1 1 1 1 X.4 1 /B A /A B 1 1 1 1 1 1 X.5 1 /A /B B A 1 1 1 1 1 1 X.6 5 . . . . C E D /E /D /C X.7 5 . . . . D /C /E C E /D X.8 5 . . . . E D /C /D C /E X.9 5 . . . . /C /E /D E D C X.10 5 . . . . /E /D C D /C E X.11 5 . . . . /D C E /C /E D A = E(5)^4 B = E(5)^3 C = E(31)+E(31)^2+E(31)^4+E(31)^8+E(31)^16 D = E(31)^5+E(31)^9+E(31)^10+E(31)^18+E(31)^20 E = E(31)^3+E(31)^6+E(31)^12+E(31)^17+E(31)^24 

magma: CharacterTable(G);