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Properties

Label 10T30
Order \(720\)
n \(10\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $\PGL(2,9)$

Related objects

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Group action invariants

Degree $n$ :  $10$
Transitive number $t$ :  $30$
Group :  $\PGL(2,9)$
CHM label :  $L(10):2=PGL(2,9)$
Parity:  $-1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2)(4,7)(5,8)(9,10), (1,2,10)(3,4,5)(6,7,8), (1,7,3,4,2,5,6,8)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 5: None

Low degree siblings

12T182, 20T146, 30T171, 36T1254, 40T590, 45T110

Siblings are shown 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 $ $1$ $1$ $()$
$ 8, 1, 1 $ $90$ $8$ $( 3, 4, 9, 8, 6, 5,10, 7)$
$ 8, 1, 1 $ $90$ $8$ $( 3, 5, 9, 7, 6, 4,10, 8)$
$ 2, 2, 2, 2, 1, 1 $ $45$ $2$ $( 3, 6)( 4, 5)( 7, 8)( 9,10)$
$ 4, 4, 1, 1 $ $90$ $4$ $( 3, 9, 6,10)( 4, 8, 5, 7)$
$ 3, 3, 3, 1 $ $80$ $3$ $( 2, 3, 6)( 4, 9, 7)( 5, 8,10)$
$ 2, 2, 2, 2, 2 $ $36$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 9)( 8,10)$
$ 5, 5 $ $72$ $5$ $( 1, 2, 3, 4, 9)( 5, 7,10, 6, 8)$
$ 10 $ $72$ $10$ $( 1, 2, 3, 5, 4, 8, 6,10, 9, 7)$
$ 10 $ $72$ $10$ $( 1, 2, 3, 7, 8, 4, 6, 9,10, 5)$
$ 5, 5 $ $72$ $5$ $( 1, 2, 3, 8,10)( 4, 7, 5, 9, 6)$

Group invariants

Order:  $720=2^{4} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [720, 764]
Character table:    
      2  4  3  3  4  3  .  2  1   1   1  1
      3  2  .  .  .  .  2  .  .   .   .  .
      5  1  .  .  .  .  .  1  1   1   1  1

        1a 8a 8b 2a 4a 3a 2b 5a 10a 10b 5b
     2P 1a 4a 4a 1a 2a 3a 1a 5b  5a  5b 5a
     3P 1a 8b 8a 2a 4a 1a 2b 5b 10b 10a 5a
     5P 1a 8b 8a 2a 4a 3a 2b 1a  2b  2b 1a
     7P 1a 8a 8b 2a 4a 3a 2b 5b 10b 10a 5a

X.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
X.3      8  .  .  .  . -1 -2  B  *B   B *B
X.4      8  .  .  .  . -1 -2 *B   B  *B  B
X.5      8  .  .  .  . -1  2  B -*B  -B *B
X.6      8  .  .  .  . -1  2 *B  -B -*B  B
X.7      9  1  1  1  1  . -1 -1  -1  -1 -1
X.8      9 -1 -1  1  1  .  1 -1   1   1 -1
X.9     10  .  .  2 -2  1  .  .   .   .  .
X.10    10  A -A -2  .  1  .  .   .   .  .
X.11    10 -A  A -2  .  1  .  .   .   .  .

A = -E(8)+E(8)^3
  = -Sqrt(2) = -r2
B = -E(5)-E(5)^4
  = (1-Sqrt(5))/2 = -b5