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Properties

Label	28T70
Degree	$28$
Order	$504$
Cyclic	no
Abelian	no
Solvable	no
Primitive	yes
$p$-group	no
Group:	$\PSL(2,8)$

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Show commands: Magma

magma: G := TransitiveGroup(28, 70);

Group action invariants

Degree $n$:		$28$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:		$70$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:		$\PSL(2,8)$
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:		(1,21,27,15,25,10,24,7,20)(2,19,8,26,17,16,6,5,3)(9,28,18,13,11,22,14,23,12), (1,28,21,4,25,18,10)(2,26,19,17,12,7,11)(3,24,20,16,13,14,9)(5,6,23,15,8,27,22)	magma: Generators(G);

Low degree resolvents

none
Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None
Degree 4: None
Degree 7: None
Degree 14: None

Low degree siblings

9T27, 36T712
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
	$1^{28}$	$1$	$1$	$()$
	$2^{12},1^{4}$	$63$	$2$	$( 3, 7)( 4,28)( 5,14)( 6,27)( 8,11)( 9,10)(12,13)(15,21)(16,22)(17,24)(18,20) (19,23)$
	$9^{3},1$	$56$	$9$	$( 2, 3,27, 8,13,12,11, 6, 7)( 4,14,19, 9,25,10,23, 5,28)(15,20,22,26,16,18,21, 24,17)$
	$9^{3},1$	$56$	$9$	$( 2, 6,12, 8, 3, 7,11,13,27)( 4, 5,10, 9,14,28,23,25,19)(15,24,18,26,20,17,21, 16,22)$
	$3^{9},1$	$56$	$3$	$( 2, 8,11)( 3,13, 6)( 4, 9,23)( 5,14,25)( 7,27,12)(10,28,19)(15,26,21) (16,24,20)(17,22,18)$
	$9^{3},1$	$56$	$9$	$( 2,12, 3,11,27, 6, 8, 7,13)( 4,10,14,23,19, 5, 9,28,25)(15,18,20,21,22,24,26, 17,16)$
	$7^{4}$	$72$	$7$	$( 1, 2, 3, 4, 5, 6, 7)( 8,21,24,17,12,23,14)( 9,26,22,25,18,13,15) (10,11,28,27,19,20,16)$
	$7^{4}$	$72$	$7$	$( 1, 2,15,10, 5,20,13)( 3,23, 8, 7,21,16,12)( 4,18, 9, 6,19,14,11) (17,22,24,25,27,28,26)$
	$7^{4}$	$72$	$7$	$( 1, 4, 7, 3, 6, 2, 5)( 8,17,14,24,23,21,12)( 9,25,15,22,13,26,18) (10,27,16,28,20,11,19)$

magma: ConjugacyClasses(G);

Group invariants

Order:		$504=2^{3} \cdot 3^{2} \cdot 7$	magma: Order(G);
Cyclic:		no	magma: IsCyclic(G);
Abelian:		no	magma: IsAbelian(G);
Solvable:		no	magma: IsSolvable(G);
Nilpotency class:		not nilpotent
Label:		504.156	magma: IdentifyGroup(G);
Character table:

		1A	2A	3A	7A1	7A2	7A3	9A1	9A2	9A4
	Size	1	63	56	72	72	72	56	56	56
	2 P	1A	1A	3A	7A2	7A3	7A1	9A1	9A2	9A4
	3 P	1A	2A	1A	7A3	7A1	7A2	3A	3A	3A
	7 P	1A	2A	3A	1A	1A	1A	9A1	9A2	9A4
	Type
504.156.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
504.156.7a	R	$7$	$- 1$	$- 2$	$0$	$0$	$0$	$1$	$1$	$1$
504.156.7b1	R	$7$	$- 1$	$1$	$0$	$0$	$0$	$- ζ_{9}^{- 1} - ζ_{9}$	$- ζ_{9}^{- 2} - ζ_{9}^{2}$	$- ζ_{9}^{- 4} - ζ_{9}^{4}$
504.156.7b2	R	$7$	$- 1$	$1$	$0$	$0$	$0$	$- ζ_{9}^{- 2} - ζ_{9}^{2}$	$- ζ_{9}^{- 4} - ζ_{9}^{4}$	$- ζ_{9}^{- 1} - ζ_{9}$
504.156.7b3	R	$7$	$- 1$	$1$	$0$	$0$	$0$	$- ζ_{9}^{- 4} - ζ_{9}^{4}$	$- ζ_{9}^{- 1} - ζ_{9}$	$- ζ_{9}^{- 2} - ζ_{9}^{2}$
504.156.8a	R	$8$	$0$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$
504.156.9a1	R	$9$	$1$	$0$	$ζ_{7}^{- 3} + ζ_{7}^{3}$	$ζ_{7}^{- 1} + ζ_{7}$	$ζ_{7}^{- 2} + ζ_{7}^{2}$	$0$	$0$	$0$
504.156.9a2	R	$9$	$1$	$0$	$ζ_{7}^{- 2} + ζ_{7}^{2}$	$ζ_{7}^{- 3} + ζ_{7}^{3}$	$ζ_{7}^{- 1} + ζ_{7}$	$0$	$0$	$0$
504.156.9a3	R	$9$	$1$	$0$	$ζ_{7}^{- 1} + ζ_{7}$	$ζ_{7}^{- 2} + ζ_{7}^{2}$	$ζ_{7}^{- 3} + ζ_{7}^{3}$	$0$	$0$	$0$

magma: CharacterTable(G);