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Properties

Label	8T43
Degree	$8$
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(8, 43);

Group action invariants

Degree $n$:		$8$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:		$43$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:		$\PGL(2,7)$
CHM label:		$L(8):2=PGL(2,7)$
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:		(1,3,2,6,4,5), (1,2,3,4,5,6,8), (1,6)(2,3)(4,5)(7,8)	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 2: None
Degree 4: None

Low degree siblings

14T16, 16T713, 21T20, 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 $	$1$	$1$	$()$
	$ 2, 2, 2, 1, 1 $	$28$	$2$	$(3,4)(5,7)(6,8)$
	$ 6, 1, 1 $	$56$	$6$	$(3,5,8,4,7,6)$
	$ 3, 3, 1, 1 $	$56$	$3$	$(3,7,8)(4,5,6)$
	$ 7, 1 $	$48$	$7$	$(2,3,7,6,8,5,4)$
	$ 2, 2, 2, 2 $	$21$	$2$	$(1,2)(3,4)(5,8)(6,7)$
	$ 8 $	$42$	$8$	$(1,2,3,6,4,7,8,5)$
	$ 4, 4 $	$42$	$4$	$(1,2,3,7)(4,6,5,8)$
	$ 8 $	$42$	$8$	$(1,2,3,8,6,7,5,4)$

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);