# Properties

 Label 15T33 Degree $15$ Order $810$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no

# Related objects

## 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) $|\Aut(F/K)|$: $1$ 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)

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

## 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