Galois group: 7T4

• Label: 7T4
• Degree: $7$
• Order: $42$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $F_7$

Group invariants

Abstract group:		$F_7$
Order:		$42=2 \cdot 3 \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$7$
Transitive number $t$:		$4$
CHM label:		$F_{42}(7) = 7:6$
Parity:		$-1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		$(1,3,2,6,4,5)$, $(1,2,3,4,5,6,7)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$6$:	$C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

14T4, 21T4, 42T4

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	Index	Representative
1A	$1^{7}$	$1$	$1$	$0$	$()$
2A	$2^{3},1$	$7$	$2$	$3$	$(1,4)(2,3)(5,7)$
3A1	$3^{2},1$	$7$	$3$	$4$	$(1,7,3)(2,4,5)$
3A-1	$3^{2},1$	$7$	$3$	$4$	$(1,3,7)(2,5,4)$
6A1	$6,1$	$7$	$6$	$5$	$(1,2,7,4,3,5)$
6A-1	$6,1$	$7$	$6$	$5$	$(1,5,3,4,7,2)$
7A	$7$	$6$	$7$	$6$	$(1,5,2,6,3,7,4)$

Malle's constant $a(G)$:     $1/3$

Character table

		1A	2A	3A1	3A-1	6A1	6A-1	7A
	Size	1	7	7	7	7	7	6
	2 P	1A	1A	3A-1	3A1	3A1	3A-1	7A
	3 P	1A	2A	1A	1A	2A	2A	7A
	7 P	1A	2A	3A1	3A-1	6A1	6A-1	1A
	Type
42.1.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$
42.1.1b	R	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$
42.1.1c1	C	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$
42.1.1c2	C	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$
42.1.1d1	C	$1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$1$
42.1.1d2	C	$1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$1$
42.1.6a	R	$6$	$0$	$0$	$0$	$0$	$0$	$-1$

Regular extensions

$f_{ 1 } =$

$x^{7} - 21 x^{5} - 70 x^{4} - 105 x^{3} + \left(7 t^{2} + 672\right) x^{2} + \left(42 t^{2} + 4501\right) x + \left(t^{3} + 61 t^{2} + 108 t + 6582\right)$