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Properties

Label 15T55
Order \(4860\)
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 $t$ :  $55$
CHM label :  $[3^{5}:2]D(5)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4)(2,8)(3,12)(6,9)(7,13)(11,14), (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), (5,10,15)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
10:  $D_{5}$
12:  $D_{6}$
20:  $D_{10}$
60:  $D_5\times S_3$
1620:  15T43

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: $D_{5}$

Low degree siblings

15T55 x 7, 30T553 x 8, 30T554 x 8, 30T561 x 8, 30T563 x 4, 45T368 x 4, 45T371 x 4, 45T378 x 16, 45T379 x 16, 45T380 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, 3, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 3,13, 8)( 5,15,10)$
$ 3, 3, 3, 3, 3 $ $10$ $3$ $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,10,15)$
$ 3, 3, 3, 3, 1, 1, 1 $ $10$ $3$ $( 1, 6,11)( 2,12, 7)( 3, 8,13)( 4,14, 9)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 3,13, 8)( 4, 9,14)$
$ 3, 3, 3, 3, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 3, 8,13)( 4, 9,14)( 5,15,10)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 1, 6,11)( 4, 9,14)( 5,10,15)$
$ 3, 3, 3, 3, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 2,12, 7)( 3, 8,13)( 5,10,15)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 2,12, 7)( 4, 9,14)$
$ 5, 5, 5 $ $162$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
$ 5, 5, 5 $ $162$ $5$ $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $45$ $2$ $( 2, 5)( 3, 9)( 4,13)( 7,10)( 8,14)(12,15)$
$ 6, 6, 3 $ $90$ $6$ $( 1,11, 6)( 2,15,12,10, 7, 5)( 3, 9,13, 4, 8,14)$
$ 6, 6, 1, 1, 1 $ $90$ $6$ $( 2,15, 7, 5,12,10)( 3, 4,13,14, 8, 9)$
$ 6, 3, 2, 2, 2 $ $90$ $6$ $( 1,11, 6)( 2,10)( 3, 4, 8, 9,13,14)( 5,12)( 7,15)$
$ 6, 3, 2, 2, 2 $ $90$ $6$ $( 1, 6,11)( 2, 5,12,15, 7,10)( 3, 4)( 8, 9)(13,14)$
$ 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 5,15,10)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $20$ $3$ $( 1,11, 6)( 3,13, 8)( 5,10,15)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 1, 6,11)( 3, 8,13)$
$ 3, 3, 3, 3, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)$
$ 3, 3, 3, 3, 3 $ $10$ $3$ $( 1, 6,11)( 2,12, 7)( 3, 8,13)( 4,14, 9)( 5,15,10)$
$ 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 3,13, 8)( 4, 9,14)( 5,15,10)$
$ 3, 3, 3, 3, 1, 1, 1 $ $20$ $3$ $( 1,11, 6)( 3, 8,13)( 4, 9,14)( 5,10,15)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 2,12, 7)( 3,13, 8)( 5,10,15)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $20$ $3$ $( 1,11, 6)( 2,12, 7)( 3, 8,13)$
$ 3, 3, 3, 3, 3 $ $10$ $3$ $( 1,11, 6)( 2, 7,12)( 3, 8,13)( 4,14, 9)( 5,15,10)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 2, 7,12)( 3, 8,13)$
$ 3, 3, 3, 3, 1, 1, 1 $ $20$ $3$ $( 1, 6,11)( 3, 8,13)( 4,14, 9)( 5,10,15)$
$ 15 $ $324$ $15$ $( 1, 4, 7,10, 8,11,14, 2, 5, 3, 6, 9,12,15,13)$
$ 15 $ $324$ $15$ $( 1, 7,13, 4,10,11, 2, 8,14, 5, 6,12, 3, 9,15)$
$ 6, 2, 2, 2, 1, 1, 1 $ $90$ $6$ $( 2, 5,12,15, 7,10)( 3, 9)( 4,13)( 8,14)$
$ 6, 6, 3 $ $90$ $6$ $( 1,11, 6)( 2,15, 7, 5,12,10)( 3, 9,13, 4, 8,14)$
$ 6, 3, 2, 2, 2 $ $90$ $6$ $( 1, 6,11)( 2,10)( 3, 9, 8,14,13, 4)( 5,12)( 7,15)$
$ 6, 2, 2, 2, 1, 1, 1 $ $90$ $6$ $( 2,15)( 3, 4,13,14, 8, 9)( 5, 7)(10,12)$
$ 6, 6, 3 $ $90$ $6$ $( 1,11, 6)( 2,10,12, 5, 7,15)( 3, 4, 8, 9,13,14)$
$ 6, 3, 2, 2, 2 $ $90$ $6$ $( 1, 6,11)( 2, 5, 7,10,12,15)( 3, 4)( 8, 9)(13,14)$
$ 6, 6, 1, 1, 1 $ $90$ $6$ $( 2,10, 7,15,12, 5)( 3,14, 8, 4,13, 9)$
$ 3, 2, 2, 2, 2, 2, 2 $ $90$ $6$ $( 1,11, 6)( 2, 5)( 3,14)( 4, 8)( 7,10)( 9,13)(12,15)$
$ 6, 6, 3 $ $90$ $6$ $( 1, 6,11)( 2,15,12,10, 7, 5)( 3,14,13, 9, 8, 4)$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $243$ $2$ $( 6,11)( 7,12)( 8,13)( 9,14)(10,15)$
$ 10, 5 $ $486$ $10$ $( 1, 4, 7,15,13,11, 9, 2, 5, 8)( 3, 6,14,12,10)$
$ 10, 5 $ $486$ $10$ $( 1, 7, 3, 9, 5,11,12,13,14,15)( 2, 8, 4,10, 6)$
$ 6, 2, 2, 2, 2, 1 $ $270$ $6$ $( 2, 5)( 3, 9, 8, 4,13,14)( 6,11)( 7,15)(10,12)$
$ 6, 2, 2, 2, 2, 1 $ $270$ $6$ $( 1,11)( 2,15, 7,10,12, 5)( 3, 9)( 4, 8)(13,14)$
$ 6, 6, 2, 1 $ $270$ $6$ $( 1, 6)( 2,10,12,15, 7, 5)( 3, 9,13,14, 8, 4)$
$ 2, 2, 2, 2, 2, 2, 2, 1 $ $135$ $2$ $( 2,10)( 3,14)( 4,13)( 5, 7)( 6,11)( 8, 9)(12,15)$
$ 6, 6, 2, 1 $ $270$ $6$ $( 1,11)( 2, 5, 7,15,12,10)( 3,14,13, 4, 8, 9)$

Group invariants

Order:  $4860=2^{2} \cdot 3^{5} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.