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Properties

Label	25T10
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, 10);

Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $F_5$ x 2

Low degree siblings

10T10 x 2, 20T27 x 2
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 $	$25$	$4$	$( 2, 3, 5, 4)( 6,16,22,12)( 7,18,21,15)( 8,20,25,13)( 9,17,24,11)(10,19,23,14)$
	$ 4, 4, 4, 4, 4, 4, 1 $	$25$	$4$	$( 2, 4, 5, 3)( 6,12,22,16)( 7,15,21,18)( 8,13,25,20)( 9,11,24,17)(10,14,23,19)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $	$25$	$2$	$( 2, 5)( 3, 4)( 6,22)( 7,21)( 8,25)( 9,24)(10,23)(11,17)(12,16)(13,20)(14,19) (15,18)$
	$ 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 $	$4$	$5$	$( 1, 6,15,18,22)( 2, 7,11,19,23)( 3, 8,12,20,24)( 4, 9,13,16,25) ( 5,10,14,17,21)$
	$ 5, 5, 5, 5, 5 $	$4$	$5$	$( 1, 7,12,16,21)( 2, 8,13,17,22)( 3, 9,14,18,23)( 4,10,15,19,24) ( 5, 6,11,20,25)$
	$ 5, 5, 5, 5, 5 $	$4$	$5$	$( 1, 8,14,19,25)( 2, 9,15,20,21)( 3,10,11,16,22)( 4, 6,12,17,23) ( 5, 7,13,18,24)$
	$ 5, 5, 5, 5, 5 $	$4$	$5$	$( 1, 9,11,17,24)( 2,10,12,18,25)( 3, 6,13,19,21)( 4, 7,14,20,22) ( 5, 8,15,16,23)$
	$ 5, 5, 5, 5, 5 $	$4$	$5$	$( 1,10,13,20,23)( 2, 6,14,16,24)( 3, 7,15,17,25)( 4, 8,11,18,21) ( 5, 9,12,19,22)$

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:		100.12	magma: IdentifyGroup(G);
Character table:

		1A	2A	4A1	4A-1	5A	5B	5C1	5C2	5D1	5D2
	Size	1	25	25	25	4	4	4	4	4	4
	2 P	1A	1A	2A	2A	5C2	5B	5A	5D1	5D2	5C1
	5 P	1A	2A	4A1	4A-1	1A	1A	1A	1A	1A	1A
	Type
100.12.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
100.12.1b	R	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$
100.12.1c1	C	$1$	$- 1$	$- i$	$i$	$1$	$1$	$1$	$1$	$1$	$1$
100.12.1c2	C	$1$	$- 1$	$i$	$- i$	$1$	$1$	$1$	$1$	$1$	$1$
100.12.4a	R	$4$	$0$	$0$	$0$	$- 1$	$4$	$- 1$	$- 1$	$- 1$	$- 1$
100.12.4b	R	$4$	$0$	$0$	$0$	$4$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$
100.12.4c1	R	$4$	$0$	$0$	$0$	$- 1$	$- 1$	$2 ζ_{5}^{- 2} + 2 ζ_{5}^{2}$	$2 ζ_{5}^{- 1} + 2 ζ_{5}$	$ζ_{5}^{- 2} + 2 + ζ_{5}^{2}$	$- ζ_{5}^{- 2} + 1 - ζ_{5}^{2}$
100.12.4c2	R	$4$	$0$	$0$	$0$	$- 1$	$- 1$	$2 ζ_{5}^{- 1} + 2 ζ_{5}$	$2 ζ_{5}^{- 2} + 2 ζ_{5}^{2}$	$- ζ_{5}^{- 2} + 1 - ζ_{5}^{2}$	$ζ_{5}^{- 2} + 2 + ζ_{5}^{2}$
100.12.4d1	R	$4$	$0$	$0$	$0$	$- 1$	$- 1$	$ζ_{5}^{- 2} + 2 + ζ_{5}^{2}$	$- ζ_{5}^{- 2} + 1 - ζ_{5}^{2}$	$2 ζ_{5}^{- 1} + 2 ζ_{5}$	$2 ζ_{5}^{- 2} + 2 ζ_{5}^{2}$
100.12.4d2	R	$4$	$0$	$0$	$0$	$- 1$	$- 1$	$- ζ_{5}^{- 2} + 1 - ζ_{5}^{2}$	$ζ_{5}^{- 2} + 2 + ζ_{5}^{2}$	$2 ζ_{5}^{- 2} + 2 ζ_{5}^{2}$	$2 ζ_{5}^{- 1} + 2 ζ_{5}$

magma: CharacterTable(G);