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Properties

Label	21T20
Degree	$21$
Order	$336$
Cyclic	no
Abelian	no
Solvable	no
Primitive	yes
$p$-group	no
Group:	$\PGL(2,7)$

Related objects

Number fields with this Galois group

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Show commands: Magma

magma: G := TransitiveGroup(21, 20);

Group action invariants

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

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None
Degree 7: None

Low degree siblings

8T43, 14T16, 16T713, 24T707, 28T42, 28T46, 42T81, 42T82, 42T83
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 $	$1$	$1$	$()$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $	$28$	$2$	$( 3,14)( 4, 7)( 6,10)( 8,20)( 9,11)(12,13)(15,16)(17,21)(18,19)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $	$21$	$2$	$( 2, 4)( 5, 6)( 7, 9)(10,11)(12,16)(14,20)(15,19)(17,18)$
	$ 8, 8, 4, 1 $	$42$	$8$	$( 2, 4, 9,10, 5, 6,11, 7)( 3,20, 8,14)(12,13,16,19,17,21,18,15)$
	$ 8, 8, 4, 1 $	$42$	$8$	$( 2, 6, 9, 7, 5, 4,11,10)( 3,20, 8,14)(12,21,16,15,17,13,18,19)$
	$ 4, 4, 4, 4, 2, 2, 1 $	$42$	$4$	$( 2, 9, 5,11)( 3, 8)( 4,10, 6, 7)(12,16,17,18)(13,19,21,15)(14,20)$
	$ 6, 6, 6, 3 $	$56$	$6$	$( 1, 2, 3,16, 8,10)( 4,12,17, 9, 7, 5)( 6,13,18,19,11,14)(15,21,20)$
	$ 3, 3, 3, 3, 3, 3, 3 $	$56$	$3$	$( 1, 2, 4)( 3, 6, 5)( 7,13, 9)( 8,15,19)(10,12,20)(11,14,16)(17,18,21)$
	$ 7, 7, 7 $	$48$	$7$	$( 1, 2,12,18,21,19, 9)( 3,15,17,20,10, 4, 7)( 5,13, 8, 6,14,16,11)$

magma: ConjugacyClasses(G);

Group invariants

Order:		$336=2^{4} \cdot 3 \cdot 7$	magma: Order(G);
Cyclic:		no	magma: IsCyclic(G);
Abelian:		no	magma: IsAbelian(G);
Solvable:		no	magma: IsSolvable(G);
Nilpotency class:		not nilpotent
Label:		336.208	magma: IdentifyGroup(G);
Character table:

		1A	2A	2B	3A	4A	6A	7A	8A1	8A3
	Size	1	21	28	56	42	56	48	42	42
	2 P	1A	1A	1A	3A	2A	3A	7A	4A	4A
	3 P	1A	2A	2B	1A	4A	2B	7A	8A3	8A1
	7 P	1A	2A	2B	3A	4A	6A	1A	8A1	8A3
	Type
336.208.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
336.208.1b	R	$1$	$1$	$- 1$	$1$	$1$	$- 1$	$1$	$- 1$	$- 1$
336.208.6a	R	$6$	$- 2$	$0$	$0$	$2$	$0$	$- 1$	$0$	$0$
336.208.6b1	R	$6$	$2$	$0$	$0$	$0$	$0$	$- 1$	$- ζ_{8}^{- 1} - ζ_{8}$	$ζ_{8}^{- 1} + ζ_{8}$
336.208.6b2	R	$6$	$2$	$0$	$0$	$0$	$0$	$- 1$	$ζ_{8}^{- 1} + ζ_{8}$	$- ζ_{8}^{- 1} - ζ_{8}$
336.208.7a	R	$7$	$- 1$	$1$	$1$	$- 1$	$1$	$0$	$- 1$	$- 1$
336.208.7b	R	$7$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$0$	$1$	$1$
336.208.8a	R	$8$	$0$	$2$	$- 1$	$0$	$- 1$	$1$	$0$	$0$
336.208.8b	R	$8$	$0$	$- 2$	$- 1$	$0$	$1$	$1$	$0$	$0$

magma: CharacterTable(G);