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Properties

Label	25T17
Degree	$25$
Order	$200$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	yes
$p$-group	no
Group:	$C_5^2:Q_8$

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Show commands: Magma

magma: G := TransitiveGroup(25, 17);

Group action invariants

Degree $n$:		$25$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:		$17$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:		$C_5^2:Q_8$
Parity:		$1$	magma: IsEven(G);
Primitive:		yes	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:		$1$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:		(1,7,24,18)(2,9,23,16)(3,6,22,19)(4,8,21,17)(5,10,25,20)(11,12,14,13), (1,7,11,10)(2,12,15,5)(3,17,14,25)(4,22,13,20)(8,16,9,21)(18,19,24,23)	magma: Generators(G);

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Low degree siblings

10T20 x 3, 20T47 x 3, 40T166 x 3
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	Representative
	$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 4, 4, 4, 4, 4, 4, 1 $	$50$	$4$	$( 2, 3, 5, 4)( 6,16,21,11)( 7,18,25,14)( 8,20,24,12)( 9,17,23,15)(10,19,22,13)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $	$25$	$2$	$( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$
	$ 4, 4, 4, 4, 4, 4, 1 $	$50$	$4$	$( 2, 6, 5,21)( 3,11, 4,16)( 7,10,25,22)( 8,15,24,17)( 9,20,23,12)(13,14,19,18)$
	$ 4, 4, 4, 4, 4, 4, 1 $	$50$	$4$	$( 2,11, 5,16)( 3,21, 4, 6)( 7,13,25,19)( 8,23,24, 9)(10,18,22,14)(12,15,20,17)$
	$ 5, 5, 5, 5, 5 $	$8$	$5$	$( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
	$ 5, 5, 5, 5, 5 $	$8$	$5$	$( 1, 7,13,19,25)( 2, 8,14,20,21)( 3, 9,15,16,22)( 4,10,11,17,23) ( 5, 6,12,18,24)$
	$ 5, 5, 5, 5, 5 $	$8$	$5$	$( 1, 8,15,17,24)( 2, 9,11,18,25)( 3,10,12,19,21)( 4, 6,13,20,22) ( 5, 7,14,16,23)$

magma: ConjugacyClasses(G);

Group invariants

Order:		$200=2^{3} \cdot 5^{2}$	magma: Order(G);
Cyclic:		no	magma: IsCyclic(G);
Abelian:		no	magma: IsAbelian(G);
Solvable:		yes	magma: IsSolvable(G);
Nilpotency class:		not nilpotent
Label:		200.44	magma: IdentifyGroup(G);
Character table:

		1A	2A	4A	4B	4C	5A	5B	5C
	Size	1	25	50	50	50	8	8	8
	2 P	1A	1A	2A	2A	2A	5C	5A	5B
	5 P	1A	2A	4A	4B	4C	1A	1A	1A
	Type
200.44.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
200.44.1b	R	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$
200.44.1c	R	$1$	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$1$
200.44.1d	R	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$
200.44.2a	S	$2$	$- 2$	$0$	$0$	$0$	$2$	$2$	$2$
200.44.8a	R	$8$	$0$	$0$	$0$	$0$	$- 2$	$- 2$	$3$
200.44.8b	R	$8$	$0$	$0$	$0$	$0$	$- 2$	$3$	$- 2$
200.44.8c	R	$8$	$0$	$0$	$0$	$0$	$3$	$- 2$	$- 2$

magma: CharacterTable(G);