Properties

Label 10T43
Order \(28800\)
n \(10\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $S_5^2 \wr C_2$

Related objects

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Group action invariants

Degree $n$ :  $10$
Transitive number $t$ :  $43$
Group :  $S_5^2 \wr C_2$
CHM label :  $[S(5)^{2}]2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,4,6,8,10), (1,6)(2,7)(3,8)(4,9)(5,10), (2,10)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8:  $D_{4}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: None

Low degree siblings

12T288, 20T539, 20T540, 20T542, 20T544, 24T13996, 24T13997, 24T13998, 25T106, 30T1011, 36T13308, 40T14374, 40T14375, 40T14376, 40T14377, 40T14378, 40T14379, 40T14380, 40T14381

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 $ $1$ $1$ $()$
$ 2, 2, 1, 1, 1, 1, 1, 1 $ $100$ $2$ $( 1, 3)( 6, 8)$
$ 2, 2, 2, 2, 1, 1 $ $225$ $2$ $( 1, 3)( 2,10)( 5, 7)( 6, 8)$
$ 3, 3, 1, 1, 1, 1 $ $400$ $3$ $( 1, 3, 5)( 6, 8,10)$
$ 3, 3, 2, 2 $ $400$ $6$ $( 1, 3, 5)( 2, 4)( 6, 8,10)( 7, 9)$
$ 4, 4, 1, 1 $ $900$ $4$ $( 1, 3, 5, 7)( 2, 6, 8,10)$
$ 5, 5 $ $576$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$
$ 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $2$ $( 6, 8)$
$ 2, 2, 1, 1, 1, 1, 1, 1 $ $30$ $2$ $( 2,10)( 6, 8)$
$ 3, 1, 1, 1, 1, 1, 1, 1 $ $40$ $3$ $( 6, 8,10)$
$ 3, 2, 1, 1, 1, 1, 1 $ $40$ $6$ $( 2, 4)( 6, 8,10)$
$ 4, 1, 1, 1, 1, 1, 1 $ $60$ $4$ $( 2, 6, 8,10)$
$ 5, 1, 1, 1, 1, 1 $ $48$ $5$ $( 2, 4, 6, 8,10)$
$ 2, 2, 2, 1, 1, 1, 1 $ $300$ $2$ $( 1, 3)( 2,10)( 6, 8)$
$ 3, 2, 1, 1, 1, 1, 1 $ $400$ $6$ $( 1, 3)( 6, 8,10)$
$ 3, 2, 2, 1, 1, 1 $ $400$ $6$ $( 1, 3)( 2, 4)( 6, 8,10)$
$ 4, 2, 1, 1, 1, 1 $ $600$ $4$ $( 1, 3)( 2, 6, 8,10)$
$ 5, 2, 1, 1, 1 $ $480$ $10$ $( 1, 3)( 2, 4, 6, 8,10)$
$ 3, 2, 2, 1, 1, 1 $ $600$ $6$ $( 1, 3)( 5, 7)( 6, 8,10)$
$ 3, 2, 2, 2, 1 $ $600$ $6$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8,10)$
$ 4, 2, 2, 1, 1 $ $900$ $4$ $( 1, 3)( 2, 6, 8,10)( 5, 7)$
$ 5, 2, 2, 1 $ $720$ $10$ $( 1, 3)( 2, 4, 6, 8,10)( 5, 7)$
$ 3, 3, 2, 1, 1 $ $800$ $6$ $( 1, 3, 5)( 2, 4)( 6, 8,10)$
$ 4, 3, 1, 1, 1 $ $1200$ $12$ $( 1, 3, 5)( 2, 6, 8,10)$
$ 5, 3, 1, 1 $ $960$ $15$ $( 1, 3, 5)( 2, 4, 6, 8,10)$
$ 4, 3, 2, 1 $ $1200$ $12$ $( 1, 3, 5)( 2, 6, 8,10)( 7, 9)$
$ 5, 3, 2 $ $960$ $30$ $( 1, 3, 5)( 2, 4, 6, 8,10)( 7, 9)$
$ 5, 4, 1 $ $1440$ $20$ $( 1, 3, 5, 7)( 2, 4, 6, 8,10)$
$ 2, 2, 2, 2, 2 $ $120$ $2$ $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$
$ 4, 2, 2, 2 $ $1200$ $4$ $( 1, 8, 3, 6)( 2, 7)( 4, 9)( 5,10)$
$ 4, 4, 2 $ $1800$ $4$ $( 1, 8, 3, 6)( 2, 7,10, 5)( 4, 9)$
$ 6, 2, 2 $ $2400$ $6$ $( 1, 8, 3,10, 5, 6)( 2, 7)( 4, 9)$
$ 6, 4 $ $2400$ $12$ $( 1, 8, 3,10, 5, 6)( 2, 7, 4, 9)$
$ 10 $ $2880$ $10$ $( 1, 8, 3,10, 5, 2, 7, 4, 9, 6)$
$ 8, 2 $ $3600$ $8$ $( 1, 8, 3,10, 5, 2, 7, 6)( 4, 9)$

Group invariants

Order:  $28800=2^{7} \cdot 3^{2} \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table: Data not available.