Properties

Label 15T33
Order \(810\)
n \(15\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $15$
Transitive number $t$ :  $33$
CHM label :  $[3^{4}:2]5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,6,11)(4,14,9), (1,11)(2,7)(4,14)(5,10)(8,13), (1,4,7,10,13)(2,5,8,11,14)(3,6,9,12,15)
$|\Aut(F/K)|$:  $1$

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

Degree 3: None

Degree 5: $C_5$

Low degree siblings

15T33 x 7, 30T190 x 8, 45T118 x 2, 45T119 x 16, 45T120 x 8

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 $ $1$ $1$ $()$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 1, 6,11)( 4,14, 9)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 1, 6,11)( 2,12, 7)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 2,12, 7)( 4,14, 9)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 4,14, 9)( 5,15,10)$
$ 3, 3, 3, 3, 1, 1, 1 $ $10$ $3$ $( 1, 6,11)( 2,12, 7)( 4, 9,14)( 5,15,10)$
$ 3, 3, 3, 3, 1, 1, 1 $ $10$ $3$ $( 1, 6,11)( 2, 7,12)( 4,14, 9)( 5,15,10)$
$ 3, 3, 3, 3, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 2, 7,12)( 4, 9,14)( 5,15,10)$
$ 3, 3, 3, 3, 3 $ $10$ $3$ $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,10,15)$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $81$ $2$ $( 6,11)( 7,12)( 8,13)( 9,14)(10,15)$
$ 5, 5, 5 $ $81$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
$ 10, 5 $ $81$ $10$ $( 1, 4,12,10, 8, 6,14, 2, 5,13)( 3,11, 9, 7,15)$
$ 5, 5, 5 $ $81$ $5$ $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$
$ 10, 5 $ $81$ $10$ $( 1,12, 3,14, 5, 6, 7, 8, 9,10)( 2,13, 4,15,11)$
$ 5, 5, 5 $ $81$ $5$ $( 1,13,10, 7, 4)( 2,14,11, 8, 5)( 3,15,12, 9, 6)$
$ 10, 5 $ $81$ $10$ $( 1, 8, 5, 2, 9,11,13,15, 7, 4)( 3,10,12,14, 6)$
$ 5, 5, 5 $ $81$ $5$ $( 1,10, 4,13, 7)( 2,11, 5,14, 8)( 3,12, 6,15, 9)$
$ 10, 5 $ $81$ $10$ $( 1,15,14,13,12,11, 5, 9, 3, 7)( 2, 6,10, 4, 8)$

Group invariants

Order:  $810=2 \cdot 3^{4} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [810, 102]
Character table:   
      2  1  .  .  .  .  .  .  .  .  1  1   1  1   1  1   1  1   1
      3  4  4  4  4  4  4  4  4  4  .  .   .  .   .  .   .  .   .
      5  1  .  .  .  .  .  .  .  .  1  1   1  1   1  1   1  1   1

        1a 3a 3b 3c 3d 3e 3f 3g 3h 2a 5a 10a 5b 10b 5c 10c 5d 10d
     2P 1a 3a 3b 3c 3d 3e 3f 3g 3h 1a 5b  5b 5c  5c 5d  5d 5a  5a
     3P 1a 1a 1a 1a 1a 1a 1a 1a 1a 2a 5d 10d 5a 10a 5b 10b 5c 10c
     5P 1a 3a 3b 3c 3d 3e 3f 3g 3h 2a 1a  2a 1a  2a 1a  2a 1a  2a
     7P 1a 3a 3b 3c 3d 3e 3f 3g 3h 2a 5b 10b 5c 10c 5d 10d 5a 10a

X.1      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  -1
X.3      1  1  1  1  1  1  1  1  1 -1  A  -A  B  -B /A -/A /B -/B
X.4      1  1  1  1  1  1  1  1  1 -1  B  -B /A -/A /B -/B  A  -A
X.5      1  1  1  1  1  1  1  1  1 -1 /B -/B  A  -A  B  -B /A -/A
X.6      1  1  1  1  1  1  1  1  1 -1 /A -/A /B -/B  A  -A  B  -B
X.7      1  1  1  1  1  1  1  1  1  1  A   A  B   B /A  /A /B  /B
X.8      1  1  1  1  1  1  1  1  1  1  B   B /A  /A /B  /B  A   A
X.9      1  1  1  1  1  1  1  1  1  1 /B  /B  A   A  B   B /A  /A
X.10     1  1  1  1  1  1  1  1  1  1 /A  /A /B  /B  A   A  B   B
X.11    10 -5  1  4 -2 -2  1  4 -2  .  .   .  .   .  .   .  .   .
X.12    10  1 -5 -2  4  1 -2  4 -2  .  .   .  .   .  .   .  .   .
X.13    10  1 -2  4 -2  1 -5 -2  4  .  .   .  .   .  .   .  .   .
X.14    10  4 -2 -2 -5 -2  4  1  1  .  .   .  .   .  .   .  .   .
X.15    10  4  4  1  1 -2 -2 -2 -5  .  .   .  .   .  .   .  .   .
X.16    10 -2  1 -2  4 -5  1 -2  4  .  .   .  .   .  .   .  .   .
X.17    10 -2  4 -5 -2  4 -2  1  1  .  .   .  .   .  .   .  .   .
X.18    10 -2 -2  1  1  4  4 -5 -2  .  .   .  .   .  .   .  .   .

A = E(5)^4
B = E(5)^3