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Properties

Label 15T84
Order \(77760\)
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 $t$ :  $84$
CHM label :  $1/2[S(3)^{5}]F(5)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,4,8)(3,6,12,9)(5,10)(7,14,13,11), (1,4)(2,8)(3,12)(6,9)(7,13)(11,14), (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)
2:  $C_2$
4:  $C_4$
20:  $F_5$
320:  $(C_2^4 : C_5):C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: $F_5$

Low degree siblings

30T1334, 30T1340, 30T1342

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 $ $40$ $3$ $( 3,13, 8)( 5,15,10)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $80$ $3$ $( 2,12, 7)( 3,13, 8)( 5,15,10)$
$ 3, 3, 3, 3, 1, 1, 1 $ $80$ $3$ $( 2,12, 7)( 3,13, 8)( 4, 9,14)( 5,15,10)$
$ 3, 3, 3, 3, 3 $ $32$ $3$ $( 1, 6,11)( 2,12, 7)( 3,13, 8)( 4, 9,14)( 5,15,10)$
$ 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 $ $90$ $2$ $( 6,11)( 8,13)$
$ 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $360$ $6$ $( 5,15,10)( 6,11)( 8,13)$
$ 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $180$ $6$ $( 2,12, 7)( 6,11)( 8,13)$
$ 3, 3, 2, 2, 1, 1, 1, 1, 1 $ $720$ $6$ $( 2,12, 7)( 5,15,10)( 6,11)( 8,13)$
$ 3, 3, 2, 2, 1, 1, 1, 1, 1 $ $360$ $6$ $( 4, 9,14)( 5,15,10)( 6,11)( 8,13)$
$ 3, 3, 3, 2, 2, 1, 1 $ $720$ $6$ $( 2,12, 7)( 4, 9,14)( 5,15,10)( 6,11)( 8,13)$
$ 5, 5, 5 $ $5184$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
$ 15 $ $5184$ $15$ $( 1, 4, 7, 5, 8,11,14, 2,15, 3, 6, 9,12,10,13)$
$ 15 $ $5184$ $15$ $( 1, 4, 7,15, 3, 6, 9,12, 5, 8,11,14, 2,10,13)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $180$ $2$ $( 1, 4)( 2, 8)( 3,12)( 6, 9)( 7,13)(11,14)$
$ 3, 2, 2, 2, 2, 2, 2 $ $360$ $6$ $( 1, 4)( 2, 8)( 3,12)( 5,15,10)( 6, 9)( 7,13)(11,14)$
$ 6, 2, 2, 2, 1, 1, 1 $ $720$ $6$ $( 1, 4)( 2, 3,12,13, 7, 8)( 6, 9)(11,14)$
$ 6, 3, 2, 2, 2 $ $720$ $6$ $( 1, 4)( 2, 3,12,13, 7, 8)( 5,15,10)( 6, 9)(11,14)$
$ 6, 3, 2, 2, 2 $ $720$ $6$ $( 1, 4)( 2, 3,12,13, 7, 8)( 5,10,15)( 6, 9)(11,14)$
$ 6, 6, 1, 1, 1 $ $720$ $6$ $( 1, 9, 6,14,11, 4)( 2, 3,12,13, 7, 8)$
$ 6, 6, 3 $ $1440$ $6$ $( 1, 9, 6,14,11, 4)( 2, 3,12,13, 7, 8)( 5,15,10)$
$ 4, 2, 2, 2, 2, 2, 1 $ $3240$ $4$ $( 1, 4)( 2, 8)( 3, 7,13,12)( 6,14)( 9,11)(10,15)$
$ 6, 4, 2, 2, 1 $ $6480$ $12$ $( 1, 9,11,14, 6, 4)( 2, 8)( 3, 7,13,12)(10,15)$
$ 4, 4, 2, 2, 1, 1, 1 $ $1620$ $4$ $( 1, 4)( 2,13, 7, 8)( 3,12)( 6, 9,11,14)$
$ 4, 4, 3, 2, 2 $ $1620$ $12$ $( 1, 4)( 2,13, 7, 8)( 3,12)( 5,15,10)( 6, 9,11,14)$
$ 4, 4, 3, 2, 2 $ $1620$ $12$ $( 1, 4)( 2,13, 7, 8)( 3,12)( 5,10,15)( 6, 9,11,14)$
$ 4, 4, 4, 2, 1 $ $3240$ $4$ $( 1, 2, 4, 8)( 3, 6,12, 9)( 7,14,13,11)(10,15)$
$ 12, 2, 1 $ $6480$ $12$ $( 1, 2, 4, 3, 6,12, 9,13,11, 7,14, 8)(10,15)$
$ 8, 4, 1, 1, 1 $ $3240$ $8$ $( 1, 2, 4, 8)( 3,11,12,14,13, 6, 7, 9)$
$ 8, 4, 3 $ $3240$ $24$ $( 1, 2, 4, 8)( 3,11,12,14,13, 6, 7, 9)( 5,15,10)$
$ 8, 4, 3 $ $3240$ $24$ $( 1, 2, 4, 8)( 3,11,12,14,13, 6, 7, 9)( 5,10,15)$
$ 4, 4, 4, 2, 1 $ $3240$ $4$ $( 1, 8, 4, 2)( 3, 9,12, 6)( 7,11,13,14)(10,15)$
$ 12, 2, 1 $ $6480$ $12$ $( 1, 3, 9,12, 6,13,14, 7,11, 8, 4, 2)(10,15)$
$ 8, 4, 1, 1, 1 $ $3240$ $8$ $( 1, 8, 4, 2)( 3,14,12,11,13, 9, 7, 6)$
$ 8, 4, 3 $ $3240$ $24$ $( 1, 8, 4, 2)( 3,14,12,11,13, 9, 7, 6)( 5,15,10)$
$ 8, 4, 3 $ $3240$ $24$ $( 1, 8, 4, 2)( 3,14,12,11,13, 9, 7, 6)( 5,10,15)$

Group invariants

Order:  $77760=2^{6} \cdot 3^{5} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.