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Properties

Label 15T68
Order \(12000\)
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 $t$ :  $68$
CHM label :  $[D(5)^{3}:2]S(3)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,11)(2,7)(4,14)(5,10)(8,13), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15), (3,12)(6,9), (1,7,4,13)(2,14,8,11)(3,6,12,9), (3,6,9,12,15)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
6:  $S_3$
8:  $C_4\times C_2$
12:  $D_{6}$
24:  $S_4$, $S_3 \times C_4$
48:  $S_4\times C_2$
96:  12T53

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: None

Low degree siblings

20T454, 30T772, 30T779, 30T780, 30T784, 30T798, 30T799, 30T804, 30T805, 30T806, 40T10422, 40T10423, 40T10424

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 $ $1$ $1$ $()$
$ 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $5$ $( 3, 6, 9,12,15)$
$ 5, 5, 1, 1, 1, 1, 1 $ $24$ $5$ $( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
$ 5, 5, 1, 1, 1, 1, 1 $ $24$ $5$ $( 2, 5, 8,11,14)( 3, 9,15, 6,12)$
$ 5, 5, 5 $ $16$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
$ 5, 5, 5 $ $48$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 9,15, 6,12)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $75$ $2$ $( 5,14)( 6,15)( 8,11)( 9,12)$
$ 5, 2, 2, 2, 2, 1, 1 $ $300$ $10$ $( 1, 4, 7,10,13)( 5,14)( 6,15)( 8,11)( 9,12)$
$ 3, 3, 3, 3, 3 $ $200$ $3$ $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$
$ 15 $ $800$ $15$ $( 1,11, 9, 4,14,12, 7, 2,15,10, 5, 3,13, 8, 6)$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $30$ $2$ $( 1,11)( 2, 7)( 4,14)( 5,10)( 8,13)$
$ 5, 2, 2, 2, 2, 2 $ $120$ $10$ $( 1,11)( 2, 7)( 3, 6, 9,12,15)( 4,14)( 5,10)( 8,13)$
$ 10, 1, 1, 1, 1, 1 $ $120$ $10$ $( 1,14, 4, 2, 7, 5,10, 8,13,11)$
$ 10, 5 $ $240$ $10$ $( 1,14, 4, 2, 7, 5,10, 8,13,11)( 3, 6, 9,12,15)$
$ 10, 5 $ $240$ $10$ $( 1,14, 4, 2, 7, 5,10, 8,13,11)( 3, 9,15, 6,12)$
$ 4, 4, 2, 2, 2, 1 $ $750$ $4$ $( 1, 8,13,11)( 2, 7)( 4, 5,10,14)( 6,15)( 9,12)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $15$ $2$ $( 6,15)( 9,12)$
$ 5, 2, 2, 1, 1, 1, 1, 1, 1 $ $120$ $10$ $( 2, 5, 8,11,14)( 6,15)( 9,12)$
$ 5, 5, 2, 2, 1 $ $120$ $10$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 6,15)( 9,12)$
$ 5, 5, 2, 2, 1 $ $120$ $10$ $( 1, 4, 7,10,13)( 2, 8,14, 5,11)( 6,15)( 9,12)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $125$ $2$ $( 4,13)( 5,14)( 6,15)( 7,10)( 8,11)( 9,12)$
$ 6, 6, 3 $ $1000$ $6$ $( 1,11, 6,10, 5,15)( 2,12, 4,14, 9, 7)( 3,13, 8)$
$ 2, 2, 2, 2, 2, 2, 2, 1 $ $150$ $2$ $( 1,11)( 2, 7)( 4,14)( 5,10)( 6,15)( 8,13)( 9,12)$
$ 10, 2, 2, 1 $ $600$ $10$ $( 1,14, 4, 2, 7, 5,10, 8,13,11)( 6,15)( 9,12)$
$ 4, 4, 2, 1, 1, 1, 1, 1 $ $150$ $4$ $( 1, 8,13,11)( 2, 7)( 4, 5,10,14)$
$ 5, 4, 4, 2 $ $600$ $20$ $( 1, 8,13,11)( 2, 7)( 3, 6, 9,12,15)( 4, 5,10,14)$
$ 4, 4, 4, 1, 1, 1 $ $125$ $4$ $( 4, 7,13,10)( 5, 8,14,11)( 6, 9,15,12)$
$ 4, 4, 4, 1, 1, 1 $ $375$ $4$ $( 4, 7,13,10)( 5,11,14, 8)( 6,12,15, 9)$
$ 12, 3 $ $1000$ $12$ $( 1,11,15, 7, 8, 9,10,14, 6, 4, 2,12)( 3,13, 5)$
$ 4, 4, 4, 2, 1 $ $750$ $4$ $( 1,11,10,14)( 2, 7, 8, 4)( 5,13)( 6, 9,15,12)$
$ 10, 4, 1 $ $600$ $20$ $( 1, 8, 4, 2, 7,11,10, 5,13,14)( 6,12,15, 9)$
$ 4, 2, 2, 2, 2, 2, 1 $ $150$ $4$ $( 1,14)( 2, 7)( 4, 8)( 5,13)( 6,12,15, 9)(10,11)$
$ 4, 4, 4, 1, 1, 1 $ $375$ $4$ $( 4, 7,13,10)( 5, 8,14,11)( 6,12,15, 9)$
$ 4, 4, 4, 1, 1, 1 $ $125$ $4$ $( 4,10,13, 7)( 5,11,14, 8)( 6,12,15, 9)$
$ 12, 3 $ $1000$ $12$ $( 1,11,15, 4, 2,12,10,14, 6, 7, 8, 9)( 3,13, 5)$
$ 4, 4, 4, 2, 1 $ $750$ $4$ $( 1,11,10,14)( 2, 7, 8, 4)( 5,13)( 6,12,15, 9)$
$ 10, 4, 1 $ $600$ $20$ $( 1, 8, 4, 2, 7,11,10, 5,13,14)( 6, 9,15,12)$
$ 4, 2, 2, 2, 2, 2, 1 $ $150$ $4$ $( 1,14)( 2, 7)( 4, 8)( 5,13)( 6, 9,15,12)(10,11)$

Group invariants

Order:  $12000=2^{5} \cdot 3 \cdot 5^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.