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Properties

Label	25T9
Degree	$25$
Order	$100$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_5:F_5$

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Show commands: Magma

magma: G := TransitiveGroup(25, 9);

Group action invariants

Degree $n$:		$25$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:		$9$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:		$C_5:F_5$
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:		(1,14,24,6,17)(2,15,25,7,18)(3,11,21,8,19)(4,12,22,9,20)(5,13,23,10,16), (1,25,19,12,10)(2,21,20,13,6)(3,22,16,14,7)(4,23,17,15,8)(5,24,18,11,9), (1,19,25,10)(2,16,24,8)(3,18,23,6)(4,20,22,9)(5,17,21,7)(11,15,13,14)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$
$4$: $C_4$
$20$: $F_5$ x 6

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $F_5$ x 6

Low degree siblings

There are no siblings 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 $	$25$	$4$	$( 2, 3, 5, 4)( 6,14,24,17)( 7,11,23,20)( 8,13,22,18)( 9,15,21,16)(10,12,25,19)$
	$ 4, 4, 4, 4, 4, 4, 1 $	$25$	$4$	$( 2, 4, 5, 3)( 6,17,24,14)( 7,20,23,11)( 8,18,22,13)( 9,16,21,15)(10,19,25,12)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $	$25$	$2$	$( 2, 5)( 3, 4)( 6,24)( 7,23)( 8,22)( 9,21)(10,25)(11,20)(12,19)(13,18)(14,17) (15,16)$
	$ 5, 5, 5, 5, 5 $	$4$	$5$	$( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
	$ 5, 5, 5, 5, 5 $	$4$	$5$	$( 1, 6,14,17,24)( 2, 7,15,18,25)( 3, 8,11,19,21)( 4, 9,12,20,22) ( 5,10,13,16,23)$
	$ 5, 5, 5, 5, 5 $	$4$	$5$	$( 1, 7,11,20,23)( 2, 8,12,16,24)( 3, 9,13,17,25)( 4,10,14,18,21) ( 5, 6,15,19,22)$
	$ 5, 5, 5, 5, 5 $	$4$	$5$	$( 1, 8,13,18,22)( 2, 9,14,19,23)( 3,10,15,20,24)( 4, 6,11,16,25) ( 5, 7,12,17,21)$
	$ 5, 5, 5, 5, 5 $	$4$	$5$	$( 1, 9,15,16,21)( 2,10,11,17,22)( 3, 6,12,18,23)( 4, 7,13,19,24) ( 5, 8,14,20,25)$
	$ 5, 5, 5, 5, 5 $	$4$	$5$	$( 1,10,12,19,25)( 2, 6,13,20,21)( 3, 7,14,16,22)( 4, 8,15,17,23) ( 5, 9,11,18,24)$

magma: ConjugacyClasses(G);

Group invariants

Order:		$100=2^{2} \cdot 5^{2}$	magma: Order(G);
Cyclic:		no	magma: IsCyclic(G);
Abelian:		no	magma: IsAbelian(G);
Solvable:		yes	magma: IsSolvable(G);
Nilpotency class:		not nilpotent
Label:		100.11	magma: IdentifyGroup(G);
Character table:

		1A	2A	4A1	4A-1	5A	5B	5C	5D	5E	5F
	Size	1	25	25	25	4	4	4	4	4	4
	2 P	1A	1A	2A	2A	5D	5E	5C	5A	5B	5F
	5 P	1A	2A	4A1	4A-1	1A	1A	1A	1A	1A	1A
	Type
100.11.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
100.11.1b	R	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$
100.11.1c1	C	$1$	$- 1$	$- i$	$i$	$1$	$1$	$1$	$1$	$1$	$1$
100.11.1c2	C	$1$	$- 1$	$i$	$- i$	$1$	$1$	$1$	$1$	$1$	$1$
100.11.4a	R	$4$	$0$	$0$	$0$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$4$
100.11.4b	R	$4$	$0$	$0$	$0$	$- 1$	$- 1$	$- 1$	$- 1$	$4$	$- 1$
100.11.4c	R	$4$	$0$	$0$	$0$	$- 1$	$- 1$	$- 1$	$4$	$- 1$	$- 1$
100.11.4d	R	$4$	$0$	$0$	$0$	$- 1$	$- 1$	$4$	$- 1$	$- 1$	$- 1$
100.11.4e	R	$4$	$0$	$0$	$0$	$- 1$	$4$	$- 1$	$- 1$	$- 1$	$- 1$
100.11.4f	R	$4$	$0$	$0$	$0$	$4$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$

magma: CharacterTable(G);