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Properties

Label	25T28
Degree	$25$
Order	$300$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	yes
$p$-group	no
Group:	$C_5^2:C_3:C_4$

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Show commands: Magma

magma: G := TransitiveGroup(25, 28);

Group action invariants

Degree $n$:		$25$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:		$28$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:		$C_5^2:C_3:C_4$
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:		(1,10,13,9)(2,17,12,22)(3,4,11,15)(5,23,14,16)(6,25,8,19)(18,24,21,20), (1,7,25,19)(2,22,24,4)(3,12,23,14)(5,17,21,9)(6,10,20,16)(8,15,18,11)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$
$4$: $C_4$
$6$: $S_3$
$12$: $C_3 : C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Low degree siblings

15T17 x 2, 30T71 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 $	$75$	$4$	$( 2, 3, 5, 4)( 6,19,21,13)( 7,16,25,11)( 8,18,24,14)( 9,20,23,12)(10,17,22,15)$
	$ 4, 4, 4, 4, 4, 4, 1 $	$75$	$4$	$( 2, 4, 5, 3)( 6,13,21,19)( 7,11,25,16)( 8,14,24,18)( 9,12,23,20)(10,15,22,17)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $	$25$	$2$	$( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 1 $	$50$	$3$	$( 2, 6,25)( 3,11,19)( 4,16,13)( 5,21, 7)( 8,10,20)( 9,15,14)(12,24,22) (17,18,23)$
	$ 6, 6, 6, 6, 1 $	$50$	$6$	$( 2, 7, 6, 5,25,21)( 3,13,11, 4,19,16)( 8,12,10,24,20,22)( 9,18,15,23,14,17)$
	$ 5, 5, 5, 5, 5 $	$12$	$5$	$( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
	$ 5, 5, 5, 5, 5 $	$12$	$5$	$( 1, 8,15,17,24)( 2, 9,11,18,25)( 3,10,12,19,21)( 4, 6,13,20,22) ( 5, 7,14,16,23)$

magma: ConjugacyClasses(G);

Group invariants

Order:		$300=2^{2} \cdot 3 \cdot 5^{2}$	magma: Order(G);
Cyclic:		no	magma: IsCyclic(G);
Abelian:		no	magma: IsAbelian(G);
Solvable:		yes	magma: IsSolvable(G);
Nilpotency class:		not nilpotent
Label:		300.23	magma: IdentifyGroup(G);
Character table:

		1A	2A	3A	4A1	4A-1	5A	5B	6A
	Size	1	25	50	75	75	12	12	50
	2 P	1A	1A	3A	2A	2A	5A	5B	3A
	3 P	1A	2A	1A	4A-1	4A1	5A	5B	2A
	5 P	1A	2A	3A	4A1	4A-1	1A	1A	6A
	Type
300.23.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
300.23.1b	R	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$
300.23.1c1	C	$1$	$- 1$	$1$	$- i$	$i$	$1$	$1$	$- 1$
300.23.1c2	C	$1$	$- 1$	$1$	$i$	$- i$	$1$	$1$	$- 1$
300.23.2a	R	$2$	$2$	$- 1$	$0$	$0$	$2$	$2$	$- 1$
300.23.2b	S	$2$	$- 2$	$- 1$	$0$	$0$	$2$	$2$	$1$
300.23.12a	R	$12$	$0$	$0$	$0$	$0$	$- 3$	$2$	$0$
300.23.12b	R	$12$	$0$	$0$	$0$	$0$	$2$	$- 3$	$0$

magma: CharacterTable(G);