# Properties

 Label 31T3 Degree $31$ Order $93$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $C_{31}:C_{3}$

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magma: G := TransitiveGroup(31, 3);

## Group action invariants

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

## Low degree resolvents

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

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

magma: ConjugacyClasses(G);

## Group invariants

 Order: $93=3 \cdot 31$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 93.1 magma: IdentifyGroup(G);
 Character table:  3 1 1 1 . . . . . . . . . . 31 1 . . 1 1 1 1 1 1 1 1 1 1 1a 3a 3b 31a 31b 31c 31d 31e 31f 31g 31h 31i 31j 2P 1a 3b 3a 31b 31d 31e 31f 31h 31i 31j 31g 31a 31c 3P 1a 1a 1a 31c 31e 31f 31h 31i 31g 31b 31a 31j 31d 5P 1a 3b 3a 31a 31b 31c 31d 31e 31f 31g 31h 31i 31j 7P 1a 3a 3b 31d 31f 31h 31i 31g 31a 31c 31j 31b 31e 11P 1a 3b 3a 31g 31j 31b 31c 31d 31e 31i 31f 31h 31a 13P 1a 3a 3b 31c 31e 31f 31h 31i 31g 31b 31a 31j 31d 17P 1a 3b 3a 31j 31c 31d 31e 31f 31h 31a 31i 31g 31b 19P 1a 3a 3b 31b 31d 31e 31f 31h 31i 31j 31g 31a 31c 23P 1a 3b 3a 31j 31c 31d 31e 31f 31h 31a 31i 31g 31b 29P 1a 3b 3a 31h 31g 31a 31j 31b 31c 31f 31d 31e 31i 31P 1a 3a 3b 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 A /A 1 1 1 1 1 1 1 1 1 1 X.3 1 /A A 1 1 1 1 1 1 1 1 1 1 X.4 3 . . B F E /C /B /D C /F /E D X.5 3 . . C D F E /C /B /E /D /F B X.6 3 . . D E /C /B /D /F B /E C F X.7 3 . . E /B /D /F /E C F B D /C X.8 3 . . /C /D /F /E C B E D F /B X.9 3 . . F /C /B /D /F /E D C B E X.10 3 . . /E B D F E /C /F /B /D C X.11 3 . . /D /E C B D F /B E /C /F X.12 3 . . /B /F /E C B D /C F E /D X.13 3 . . /F C B D F E /D /C /B /E A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = E(31)^2+E(31)^10+E(31)^19 C = E(31)^17+E(31)^22+E(31)^23 D = E(31)^3+E(31)^13+E(31)^15 E = E(31)^6+E(31)^26+E(31)^30 F = E(31)^4+E(31)^7+E(31)^20 

magma: CharacterTable(G);