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Properties

Label 15T71
Order \(19440\)
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 $t$ :  $71$
CHM label :  $[1/2.S(3)^{5}]5$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4,7,10,13)(2,5,8,11,14)(3,6,9,12,15), (5,10,15), (1,11)(4,14)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
5:  $C_5$
80:  $C_2^4 : C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: $C_5$

Low degree siblings

30T908 x 4, 30T909, 30T911 x 4, 30T912 x 4, 30T920 x 2, 30T924 x 2, 45T627 x 2

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 5,15,10)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $3$ $( 3,13, 8)( 5,15,10)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $3$ $( 1,11, 6)( 5,15,10)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $40$ $3$ $( 1,11, 6)( 3,13, 8)( 5,15,10)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $40$ $3$ $( 3,13, 8)( 4,14, 9)( 5,15,10)$
$ 3, 3, 3, 3, 1, 1, 1 $ $80$ $3$ $( 1,11, 6)( 3,13, 8)( 4,14, 9)( 5,15,10)$
$ 3, 3, 3, 3, 3 $ $16$ $3$ $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$
$ 3, 3, 3, 3, 3 $ $16$ $3$ $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,10,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $405$ $2$ $( 6,11)( 7,12)( 9,14)(10,15)$
$ 3, 2, 2, 2, 2, 1, 1, 1, 1 $ $810$ $6$ $( 3,13, 8)( 6,11)( 7,12)( 9,14)(10,15)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $45$ $2$ $( 6,11)( 8,13)$
$ 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $90$ $6$ $( 5,15,10)( 6,11)( 8,13)$
$ 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $90$ $6$ $( 4,14, 9)( 6,11)( 8,13)$
$ 3, 3, 2, 2, 1, 1, 1, 1, 1 $ $180$ $6$ $( 4,14, 9)( 5,15,10)( 6,11)( 8,13)$
$ 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $90$ $6$ $( 2,12, 7)( 6,11)( 8,13)$
$ 3, 3, 2, 2, 1, 1, 1, 1, 1 $ $180$ $6$ $( 2,12, 7)( 5,15,10)( 6,11)( 8,13)$
$ 3, 3, 2, 2, 1, 1, 1, 1, 1 $ $180$ $6$ $( 2,12, 7)( 4,14, 9)( 6,11)( 8,13)$
$ 3, 3, 3, 2, 2, 1, 1 $ $360$ $6$ $( 2,12, 7)( 4,14, 9)( 5,15,10)( 6,11)( 8,13)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $45$ $2$ $( 8,13)( 9,14)$
$ 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $90$ $6$ $( 5,15,10)( 8,13)( 9,14)$
$ 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $90$ $6$ $( 1,11, 6)( 8,13)( 9,14)$
$ 3, 3, 2, 2, 1, 1, 1, 1, 1 $ $180$ $6$ $( 1,11, 6)( 5,15,10)( 8,13)( 9,14)$
$ 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $90$ $6$ $( 2,12, 7)( 8,13)( 9,14)$
$ 3, 3, 2, 2, 1, 1, 1, 1, 1 $ $180$ $6$ $( 2,12, 7)( 5,15,10)( 8,13)( 9,14)$
$ 3, 3, 2, 2, 1, 1, 1, 1, 1 $ $180$ $6$ $( 1,11, 6)( 2,12, 7)( 8,13)( 9,14)$
$ 3, 3, 3, 2, 2, 1, 1 $ $360$ $6$ $( 1,11, 6)( 2,12, 7)( 5,15,10)( 8,13)( 9,14)$
$ 5, 5, 5 $ $1296$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
$ 15 $ $1296$ $15$ $( 1, 4, 7, 5, 8,11,14, 2,15, 3, 6, 9,12,10,13)$
$ 15 $ $1296$ $15$ $( 1, 4, 7,15, 3, 6, 9,12, 5, 8,11,14, 2,10,13)$
$ 5, 5, 5 $ $1296$ $5$ $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$
$ 15 $ $1296$ $15$ $( 1, 7,13, 4, 5,11, 2, 8,14,15, 6,12, 3, 9,10)$
$ 15 $ $1296$ $15$ $( 1, 7,13, 4,15, 6,12, 3, 9, 5,11, 2, 8,14,10)$
$ 5, 5, 5 $ $1296$ $5$ $( 1,13,10, 7, 4)( 2,14,11, 8, 5)( 3,15,12, 9, 6)$
$ 15 $ $1296$ $15$ $( 1,13, 5, 2,14,11, 8,15,12, 9, 6, 3,10, 7, 4)$
$ 15 $ $1296$ $15$ $( 1,13,15,12, 9, 6, 3, 5, 2,14,11, 8,10, 7, 4)$
$ 5, 5, 5 $ $1296$ $5$ $( 1,10, 4,13, 7)( 2,11, 5,14, 8)( 3,12, 6,15, 9)$
$ 15 $ $1296$ $15$ $( 1, 5,14, 8, 2,11,15, 9, 3,12, 6,10, 4,13, 7)$
$ 15 $ $1296$ $15$ $( 1,15, 9, 3,12, 6, 5,14, 8, 2,11,10, 4,13, 7)$

Group invariants

Order:  $19440=2^{4} \cdot 3^{5} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.