# Properties

 Label 31T7 Degree $31$ Order $465$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $C_{31}:C_{15}$

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magma: G := TransitiveGroup(31, 7);

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$
$5$:  $C_5$
$15$:  $C_{15}$

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

magma: ConjugacyClasses(G);

## Group invariants

 Order: $465=3 \cdot 5 \cdot 31$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 465.1 magma: IdentifyGroup(G);
 Character table:  3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 . . 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 . . 31 1 . . . . . . . . . . . . . . 1 1 1a 5a 5b 3a 15a 5c 15b 15c 15d 5d 15e 15f 15g 3b 15h 31a 31b 2P 1a 5b 5d 3b 15e 5a 15f 15a 15c 5c 15d 15g 15h 3a 15b 31a 31b 3P 1a 5c 5a 1a 5a 5d 5d 5c 5d 5b 5b 5c 5a 1a 5b 31b 31a 5P 1a 1a 1a 3b 3a 1a 3b 3b 3a 1a 3b 3a 3b 3a 3a 31a 31b 7P 1a 5b 5d 3a 15h 5a 15c 15g 15f 5c 15b 15a 15e 3b 15d 31a 31b 11P 1a 5a 5b 3b 15g 5c 15d 15f 15b 5d 15h 15c 15a 3a 15e 31b 31a 13P 1a 5c 5a 3a 15f 5d 15e 15b 15h 5b 15g 15d 15c 3b 15a 31b 31a 17P 1a 5b 5d 3b 15e 5a 15f 15a 15c 5c 15d 15g 15h 3a 15b 31b 31a 19P 1a 5d 5c 3a 15d 5b 15g 15e 15a 5a 15c 15h 15b 3b 15f 31a 31b 23P 1a 5c 5a 3b 15c 5d 15h 15d 15e 5b 15a 15b 15f 3a 15g 31b 31a 29P 1a 5d 5c 3b 15b 5b 15a 15h 15g 5a 15f 15e 15d 3a 15c 31b 31a 31P 1a 5a 5b 3a 15a 5c 15b 15c 15d 5d 15e 15f 15g 3b 15h 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 C /C 1 C C /C 1 C /C C /C /C 1 1 X.3 1 1 1 /C C 1 /C /C C 1 /C C /C C C 1 1 X.4 1 A B 1 B /B /B A /B /A /A A B 1 /A 1 1 X.5 1 B /A 1 /A A A B A /B /B B /A 1 /B 1 1 X.6 1 /B A 1 A /A /A /B /A B B /B A 1 B 1 1 X.7 1 /A /B 1 /B B B /A B A A /A /B 1 A 1 1 X.8 1 A B C D /B /D /F /E /A G /G E /C F 1 1 X.9 1 A B /C E /B /E /G /D /A F /F D C G 1 1 X.10 1 B /A C F A /F E /G /B /D D G /C /E 1 1 X.11 1 B /A /C G A /G D /F /B /E E F C /D 1 1 X.12 1 /B A C /G /A G /D F B E /E /F /C D 1 1 X.13 1 /B A /C /F /A F /E G B D /D /G C E 1 1 X.14 1 /A /B C /E B E G D A /F F /D /C /G 1 1 X.15 1 /A /B /C /D B D F E A /G G /E C /F 1 1 X.16 15 . . . . . . . . . . . . . . H /H X.17 15 . . . . . . . . . . . . . . /H H A = E(5)^4 B = E(5)^3 C = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 D = E(15)^14 E = E(15)^4 F = E(15)^8 G = E(15)^13 H = E(31)+E(31)^2+E(31)^4+E(31)^5+E(31)^7+E(31)^8+E(31)^9+E(31)^10+E(31)^14+E(31)^16+E(31)^18+E(31)^19+E(31)^20+E(31)^25+E(31)^28 = (-1+Sqrt(-31))/2 = b31 

magma: CharacterTable(G);