Properties

Label 15T28
Order \(720\)
n \(15\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $15$
Transitive number $t$ :  $28$
CHM label :  $S_{6}(15)$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,10,3,14)(2,15,7,12,6)(4,5,11,13,8), (1,5)(2,7)(3,6)(4,15)(8,9)(12,13), (1,4)(2,6)(3,7)(5,15)(8,9)(12,13)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: None

Low degree siblings

6T16 x 2, 10T32, 12T183 x 2, 15T28, 20T145, 20T149 x 2, 30T164 x 2, 30T166 x 2, 30T176 x 2, 36T1252, 40T589, 40T592 x 2, 45T96

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, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $15$ $2$ $( 4, 6)( 5, 7)( 8,10)( 9,11)$
$ 3, 3, 3, 3, 1, 1, 1 $ $40$ $3$ $( 3, 8,10)( 4,13, 6)( 5, 7,14)( 9,15,11)$
$ 4, 4, 4, 2, 1 $ $90$ $4$ $( 2, 5, 7,14)( 3, 8,13, 6)( 4,10)( 9,15,12,11)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $45$ $2$ $( 2, 5)( 3,13)( 4,10)( 7,14)( 9,15)(11,12)$
$ 6, 3, 3, 2, 1 $ $120$ $6$ $( 2, 9, 5,12, 7,11)( 3, 6, 4)( 8,10,13)(14,15)$
$ 4, 4, 4, 2, 1 $ $90$ $4$ $( 2, 9, 5,15)( 3,10,13, 4)( 6, 8)( 7,11,14,12)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $15$ $2$ $( 2, 9)( 3,13)( 5,15)( 6, 8)( 7,12)(11,14)$
$ 5, 5, 5 $ $144$ $5$ $( 1, 2, 3, 4,11)( 5,12,14, 8,15)( 6,13, 9, 7,10)$
$ 6, 6, 3 $ $120$ $6$ $( 1, 2, 3, 4,13, 9)( 5,10,15)( 6,11, 7,12,14, 8)$
$ 3, 3, 3, 3, 3 $ $40$ $3$ $( 1, 2,12)( 3, 9, 5)( 4, 6,13)( 7, 8,15)(10,11,14)$

Group invariants

Order:  $720=2^{4} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [720, 763]
Character table:   
      2  4  4  1  3  4  1  3  4  .  1  1
      3  2  1  2  .  .  1  .  1  .  1  2
      5  1  .  .  .  .  .  .  .  1  .  .

        1a 2a 3a 4a 2b 6a 4b 2c 5a 6b 3b
     2P 1a 1a 3a 2b 1a 3a 2b 1a 5a 3b 3b
     3P 1a 2a 1a 4a 2b 2a 4b 2c 5a 2c 1a
     5P 1a 2a 3a 4a 2b 6a 4b 2c 1a 6b 3b

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      5 -3  2 -1  1  . -1  1  .  1 -1
X.4      5  3  2  1  1  . -1 -1  . -1 -1
X.5      5 -1 -1  1  1 -1 -1  3  .  .  2
X.6      5  1 -1 -1  1  1 -1 -3  .  .  2
X.7      9 -3  .  1  1  .  1 -3 -1  .  .
X.8      9  3  . -1  1  .  1  3 -1  .  .
X.9     10 -2  1  . -2  1  .  2  . -1  1
X.10    10  2  1  . -2 -1  . -2  .  1  1
X.11    16  . -2  .  .  .  .  .  1  . -2