# Properties

 Label 31T2 Degree $31$ Order $62$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $D_{31}$

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magma: G := TransitiveGroup(31, 2);

## Group action invariants

 Degree $n$: $31$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $2$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $D_{31}$ 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: (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16), (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)
$2$:  $C_2$

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

magma: ConjugacyClasses(G);

## Group invariants

 Order: $62=2 \cdot 31$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 62.1 magma: IdentifyGroup(G);
 Character table:  2 1 1 . . . . . . . . . . . . . . . 31 1 . 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1a 2a 31a 31b 31c 31d 31e 31f 31g 31h 31i 31j 31k 31l 31m 31n 31o 2P 1a 1a 31b 31d 31f 31h 31j 31l 31n 31o 31m 31k 31i 31g 31e 31c 31a 3P 1a 2a 31c 31f 31i 31l 31o 31m 31j 31g 31d 31a 31b 31e 31h 31k 31n 5P 1a 2a 31e 31j 31o 31k 31f 31a 31d 31i 31n 31l 31g 31b 31c 31h 31m 7P 1a 2a 31g 31n 31j 31c 31d 31k 31m 31f 31a 31h 31o 31i 31b 31e 31l 11P 1a 2a 31k 31i 31b 31m 31g 31d 31o 31e 31f 31n 31c 31h 31l 31a 31j 13P 1a 2a 31m 31e 31h 31j 31c 31o 31b 31k 31g 31f 31l 31a 31n 31d 31i 17P 1a 2a 31n 31c 31k 31f 31h 31i 31e 31l 31b 31o 31a 31m 31d 31j 31g 19P 1a 2a 31l 31g 31e 31n 31b 31j 31i 31c 31o 31d 31h 31k 31a 31m 31f 23P 1a 2a 31h 31o 31g 31a 31i 31n 31f 31b 31j 31m 31e 31c 31k 31l 31d 29P 1a 2a 31b 31d 31f 31h 31j 31l 31n 31o 31m 31k 31i 31g 31e 31c 31a 31P 1a 2a 1a 1a 1a 1a 1a 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 1 1 1 1 X.2 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.3 2 . A L H D N E B J I O K F C M G X.4 2 . B M O H D K C E A J G I L N F X.5 2 . C N J O H G L K B E F A M D I X.6 2 . D J F G K B H A N I C M O E L X.7 2 . E F C B A N K M J L D O G I H X.8 2 . F B N M L O I H G D J K A C E X.9 2 . G A M L C H F D K N O E I B J X.10 2 . H E I F G C O B D A L N J K M X.11 2 . I C D N M J A O F H E G B L K X.12 2 . J G B A I M E L O C N H K F D X.13 2 . K I L C B D G N E M H J F A O X.14 2 . L D E J O F M G C K I B N H A X.15 2 . M H K E J I N F L G A C D O B X.16 2 . N O G K E A D I M F B L H J C X.17 2 . O K A I F L J C H B M D E G N A = E(31)^3+E(31)^28 B = E(31)^10+E(31)^21 C = E(31)^8+E(31)^23 D = E(31)^12+E(31)^19 E = E(31)^13+E(31)^18 F = E(31)^5+E(31)^26 G = E(31)^14+E(31)^17 H = E(31)^9+E(31)^22 I = E(31)^4+E(31)^27 J = E(31)^7+E(31)^24 K = E(31)^2+E(31)^29 L = E(31)^6+E(31)^25 M = E(31)^11+E(31)^20 N = E(31)^15+E(31)^16 O = E(31)+E(31)^30 

magma: CharacterTable(G);