Properties

 Label 10T43 Degree $10$ Order $28800$ Cyclic no Abelian no Solvable no Primitive no $p$-group no Group: $S_5^2 \wr C_2$

Related objects

Show commands: Magma

magma: G := TransitiveGroup(10, 43);

Group action invariants

 Degree $n$: $10$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $43$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $S_5^2 \wr C_2$ CHM label: $[S(5)^{2}]2$ Parity: $-1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $1$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (2,4,6,8,10), (1,6)(2,7)(3,8)(4,9)(5,10), (2,10) magma: Generators(G);

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

 Label Cycle Type Size Order Index Representative 1A $1^{10}$ $1$ $1$ $0$ $()$ 2A $2,1^{8}$ $20$ $2$ $1$ $(5,7)$ 2B $2^{2},1^{6}$ $30$ $2$ $2$ $(2,4)(6,8)$ 2C $2^{2},1^{6}$ $100$ $2$ $2$ $( 2,10)( 5, 7)$ 2D $2^{5}$ $120$ $2$ $5$ $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$ 2E $2^{4},1^{2}$ $225$ $2$ $4$ $(1,3)(2,4)(6,8)(7,9)$ 2F $2^{3},1^{4}$ $300$ $2$ $3$ $(2,4)(5,7)(6,8)$ 3A $3,1^{7}$ $40$ $3$ $2$ $(2,4,6)$ 3B $3^{2},1^{4}$ $400$ $3$ $4$ $(1,7,9)(2,4,6)$ 4A $4,1^{6}$ $60$ $4$ $3$ $(1,7,5,9)$ 4B $4,2,1^{4}$ $600$ $4$ $4$ $( 2,10, 6, 8)( 5, 7)$ 4C $4^{2},1^{2}$ $900$ $4$ $6$ $( 2,10, 6, 8)( 3, 7, 5, 9)$ 4D $4,2^{2},1^{2}$ $900$ $4$ $5$ $(1,7,5,9)(2,4)(6,8)$ 4E $4,2^{3}$ $1200$ $4$ $6$ $( 1, 6)( 2, 7,10, 5)( 3, 8)( 4, 9)$ 4F $4^{2},2$ $1800$ $4$ $7$ $( 1, 8, 3, 6)( 2, 7, 4, 9)( 5,10)$ 5A $5,1^{5}$ $48$ $5$ $4$ $( 2, 4, 6, 8,10)$ 5B $5^{2}$ $576$ $5$ $8$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$ 6A $3,2,1^{5}$ $40$ $6$ $3$ $(1,3)(5,9,7)$ 6B $3^{2},2^{2}$ $400$ $6$ $6$ $( 1, 3)( 2,10, 6)( 4, 8)( 5, 9, 7)$ 6C $3,2^{2},1^{3}$ $400$ $6$ $4$ $( 2,10, 6)( 4, 8)( 5, 7)$ 6D $3,2,1^{5}$ $400$ $6$ $3$ $(2,4,6)(5,7)$ 6E $3,2^{2},1^{3}$ $600$ $6$ $4$ $(1,3)(2,4,6)(7,9)$ 6F $3,2^{3},1$ $600$ $6$ $5$ $(1,3)(2,4)(5,9,7)(6,8)$ 6G $3^{2},2,1^{2}$ $800$ $6$ $5$ $(1,3)(2,4,6)(5,9,7)$ 6H $6,2^{2}$ $2400$ $6$ $7$ $( 1, 2, 7, 4, 9, 6)( 3, 8)( 5,10)$ 8A $8,2$ $3600$ $8$ $8$ $( 1, 2, 7,10, 5, 4, 9, 6)( 3, 8)$ 10A $5,2,1^{3}$ $480$ $10$ $5$ $( 2, 4, 6, 8,10)( 5, 7)$ 10B $5,2^{2},1$ $720$ $10$ $6$ $( 1, 3)( 2, 4, 6, 8,10)( 7, 9)$ 10C $10$ $2880$ $10$ $9$ $( 1, 8, 3,10, 5, 2, 7, 4, 9, 6)$ 12A $4,3,1^{3}$ $1200$ $12$ $5$ $(1,7,5,9)(2,4,6)$ 12B $4,3,2,1$ $1200$ $12$ $6$ $( 2,10, 6)( 3, 7, 5, 9)( 4, 8)$ 12C $6,4$ $2400$ $12$ $8$ $( 1, 8, 3, 6)( 2, 7,10, 5, 4, 9)$ 15A $5,3,1^{2}$ $960$ $15$ $6$ $( 1, 7, 9)( 2, 4, 6, 8,10)$ 20A $5,4,1$ $1440$ $20$ $7$ $( 1, 7, 5, 9)( 2, 4, 6, 8,10)$ 30A $5,3,2$ $960$ $30$ $7$ $( 1, 3)( 2, 4, 6, 8,10)( 5, 9, 7)$

magma: ConjugacyClasses(G);

Malle's constant $a(G)$:     $1$

Group invariants

 Order: $28800=2^{7} \cdot 3^{2} \cdot 5^{2}$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: no magma: IsSolvable(G); Nilpotency class: not nilpotent Label: 28800.r magma: IdentifyGroup(G); Character table: 35 x 35 character table

magma: CharacterTable(G);