Properties

 Label 10T27 Degree $10$ Order $400$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $(D_5 \wr C_2):C_2$

Related objects

Show commands: Magma

magma: G := TransitiveGroup(10, 27);

Group action invariants

 Degree $n$: $10$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $27$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $(D_5 \wr C_2):C_2$ CHM label: $[1/2.F(5)^{2}]2$ Parity: $-1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $1$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,7,9,3)(2,4,8,6), (2,4,6,8,10), (2,8)(4,6), (1,6)(2,7)(3,8)(4,9)(5,10) magma: Generators(G);

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $C_2^3$
$16$:  $Q_8:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: None

Low degree siblings

10T27 x 2, 20T90 x 3, 20T96 x 3, 20T97 x 3, 25T30, 40T393 x 3, 40T394 x 3, 40T395 x 3

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 1, 1, 1, 1, 1, 1$ $10$ $2$ $( 4,10)( 6, 8)$ $4, 4, 1, 1$ $25$ $4$ $( 3, 5, 9, 7)( 4, 6,10, 8)$ $4, 4, 1, 1$ $50$ $4$ $( 3, 5, 9, 7)( 4, 8,10, 6)$ $4, 4, 1, 1$ $25$ $4$ $( 3, 7, 9, 5)( 4, 8,10, 6)$ $2, 2, 2, 2, 1, 1$ $25$ $2$ $( 3, 9)( 4,10)( 5, 7)( 6, 8)$ $5, 1, 1, 1, 1, 1$ $8$ $5$ $( 2, 4, 6, 8,10)$ $5, 2, 2, 1$ $40$ $10$ $( 2, 4, 6, 8,10)( 3, 9)( 5, 7)$ $2, 2, 2, 2, 2$ $10$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)$ $4, 4, 2$ $50$ $4$ $( 1, 2)( 3, 4, 9,10)( 5, 6, 7, 8)$ $4, 4, 2$ $50$ $4$ $( 1, 2)( 3, 6, 9, 8)( 4, 5,10, 7)$ $2, 2, 2, 2, 2$ $10$ $2$ $( 1, 2)( 3, 6)( 4, 7)( 5,10)( 8, 9)$ $10$ $40$ $10$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10)$ $10$ $40$ $10$ $( 1, 2, 3, 6, 5,10, 7, 4, 9, 8)$ $5, 5$ $8$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$ $5, 5$ $8$ $5$ $( 1, 3, 5, 7, 9)( 2, 6,10, 4, 8)$

magma: ConjugacyClasses(G);

Group invariants

 Order: $400=2^{4} \cdot 5^{2}$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 400.207 magma: IdentifyGroup(G);
 Character table:  2 4 3 4 3 4 4 1 1 3 3 3 3 1 1 1 1 5 2 1 . . . . 2 1 1 . . 1 1 1 2 2 1a 2a 4a 4b 4c 2b 5a 10a 2c 4d 4e 2d 10b 10c 5b 5c 2P 1a 1a 2b 2b 2b 1a 5a 5a 1a 2b 2b 1a 5b 5c 5b 5c 3P 1a 2a 4c 4b 4a 2b 5a 10a 2c 4d 4e 2d 10b 10c 5b 5c 5P 1a 2a 4a 4b 4c 2b 1a 2a 2c 4d 4e 2d 2c 2d 1a 1a 7P 1a 2a 4c 4b 4a 2b 5a 10a 2c 4d 4e 2d 10b 10c 5b 5c X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 -1 1 1 X.3 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 1 1 1 1 X.4 1 -1 1 -1 1 1 1 -1 -1 1 -1 1 -1 1 1 1 X.5 1 -1 1 -1 1 1 1 -1 1 -1 1 -1 1 -1 1 1 X.6 1 1 -1 -1 -1 1 1 1 -1 -1 1 1 -1 1 1 1 X.7 1 1 -1 -1 -1 1 1 1 1 1 -1 -1 1 -1 1 1 X.8 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 X.9 2 . A . -A -2 2 . . . . . . . 2 2 X.10 2 . -A . A -2 2 . . . . . . . 2 2 X.11 8 -4 . . . . 3 1 . . . . . . -2 -2 X.12 8 4 . . . . 3 -1 . . . . . . -2 -2 X.13 8 . . . . . -2 . -4 . . . 1 . 3 -2 X.14 8 . . . . . -2 . . . . -4 . 1 -2 3 X.15 8 . . . . . -2 . . . . 4 . -1 -2 3 X.16 8 . . . . . -2 . 4 . . . -1 . 3 -2 A = -2*E(4) = -2*Sqrt(-1) = -2i 

magma: CharacterTable(G);