Properties

Label 31T6
Order \(310\)
n \(31\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_{31}:C_{10}$

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Group action invariants

Degree $n$ :  $31$
Transitive number $t$ :  $6$
Group :  $C_{31}:C_{10}$
Parity:  $-1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
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), (1,27,16,29,8,30,4,15,2,23)(3,19,17,25,24,28,12,14,6,7)(5,11,18,21,9,26,20,13,10,22)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
5:  $C_5$
10:  $C_{10}$

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

Group invariants

Order:  $310=2 \cdot 5 \cdot 31$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [310, 1]
Character table:   
      2  1  1  1  1   1  1   1   1   1  1   .   .   .
      5  1  1  1  1   1  1   1   1   1  1   .   .   .
     31  1  .  .  .   .  .   .   .   .  .   1   1   1

        1a 5a 5b 5c 10a 5d 10b 10c 10d 2a 31a 31b 31c
     2P 1a 5b 5d 5a  5c 5c  5a  5d  5b 1a 31a 31b 31c
     3P 1a 5c 5a 5d 10c 5b 10a 10d 10b 2a 31b 31c 31a
     5P 1a 1a 1a 1a  2a 1a  2a  2a  2a 2a 31c 31a 31b
     7P 1a 5b 5d 5a 10b 5c 10d 10a 10c 2a 31b 31c 31a
    11P 1a 5a 5b 5c 10a 5d 10b 10c 10d 2a 31c 31a 31b
    13P 1a 5c 5a 5d 10c 5b 10a 10d 10b 2a 31c 31a 31b
    17P 1a 5b 5d 5a 10b 5c 10d 10a 10c 2a 31b 31c 31a
    19P 1a 5d 5c 5b 10d 5a 10c 10b 10a 2a 31b 31c 31a
    23P 1a 5c 5a 5d 10c 5b 10a 10d 10b 2a 31a 31b 31c
    29P 1a 5d 5c 5b 10d 5a 10c 10b 10a 2a 31a 31b 31c
    31P 1a 5a 5b 5c 10a 5d 10b 10c 10d 2a  1a  1a  1a

X.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
X.3      1  A  B /B -/A /A -/B  -B  -A -1   1   1   1
X.4      1  B /A  A -/B /B  -A -/A  -B -1   1   1   1
X.5      1 /B  A /A  -B  B -/A  -A -/B -1   1   1   1
X.6      1 /A /B  B  -A  A  -B -/B -/A -1   1   1   1
X.7      1  A  B /B  /A /A  /B   B   A  1   1   1   1
X.8      1  B /A  A  /B /B   A  /A   B  1   1   1   1
X.9      1 /B  A /A   B  B  /A   A  /B  1   1   1   1
X.10     1 /A /B  B   A  A   B  /B  /A  1   1   1   1
X.11    10  .  .  .   .  .   .   .   .  .   C   E   D
X.12    10  .  .  .   .  .   .   .   .  .   D   C   E
X.13    10  .  .  .   .  .   .   .   .  .   E   D   C

A = E(5)^4
B = E(5)^3
C = E(31)+E(31)^2+E(31)^4+E(31)^8+E(31)^15+E(31)^16+E(31)^23+E(31)^27+E(31)^29+E(31)^30
D = E(31)^5+E(31)^9+E(31)^10+E(31)^11+E(31)^13+E(31)^18+E(31)^20+E(31)^21+E(31)^22+E(31)^26
E = E(31)^3+E(31)^6+E(31)^7+E(31)^12+E(31)^14+E(31)^17+E(31)^19+E(31)^24+E(31)^25+E(31)^28