Galois group: 28T393

• Label: 28T393
• Degree: $28$
• Order: $12096$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: yes
• $p$-group: no
• Group: $G(2,2)$

Group invariants

Abstract group:		$G(2,2)$
Order:		$12096=2^{6} \cdot 3^{3} \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$28$
Transitive number $t$:		$393$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		$(1,7,12)(2,10,22,25,5,14)(3,23,13,15,9,28)(4,18,19,26,6,27)(8,16,20,17,11,24)$, $(1,27,15,10,24,20)(3,22,17,26,11,7)(4,6,19,16,13,14)(5,9,18,23,12,25)(8,28,21)$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 7: None

Degree 14: None

Low degree siblings

36T9590

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
	$1^{28}$	$1$	$1$	$0$	$()$
	$7^{4}$	$1728$	$7$	$24$	$( 1,12, 6,13, 3,24,21)( 2,19,23,11, 5,22,16)( 4, 8,10,15,27,28,14)( 7,18, 9,26,17,25,20)$
	$2^{12},1^{4}$	$63$	$2$	$12$	$( 1,27)( 2,17)( 3,11)( 4,18)( 5,22)( 6,25)( 8,20)( 9,24)(10,12)(13,16)(14,21)(23,26)$
	$3^{9},1$	$56$	$3$	$18$	$( 1,10,14)( 2,13, 4)( 3,23, 8)( 5,25,24)( 6, 9,22)( 7,15,28)(11,26,20)(12,21,27)(16,18,17)$
	$4^{6},1^{4}$	$126$	$4$	$18$	$( 1, 5,27,22)( 2,11,17, 3)( 4,20,18, 8)( 6,10,25,12)( 9,14,24,21)(13,26,16,23)$
	$6^{4},3,1$	$504$	$6$	$22$	$( 1,21,10,27,14,12)( 2,18,13,17, 4,16)( 3,20,23,11, 8,26)( 5, 9,25,22,24, 6)( 7,28,15)$
	$12^{2},3,1$	$1008$	$12$	$24$	$( 1, 6,21, 5,10, 9,27,25,14,22,12,24)( 2,23,18,11,13, 8,17,26, 4, 3,16,20)( 7,15,28)$
	$3^{9},1$	$672$	$3$	$18$	$( 1, 8, 4)( 2,10, 3)( 5,15, 9)( 6,24, 7)(11,27,16)(12,18,26)(13,14,23)(17,20,21)(22,25,28)$
	$2^{12},1^{4}$	$252$	$2$	$12$	$( 1, 7)( 2,25)( 3,22)( 4,24)( 5,13)( 6, 8)( 9,23)(10,28)(12,21)(14,15)(17,18)(20,26)$
	$6^{4},3,1$	$2016$	$6$	$22$	$( 1,24, 8, 7, 4, 6)( 2,22,10,25, 3,28)( 5,23,15,13, 9,14)(11,16,27)(12,20,18,21,26,17)$
	$4^{6},1^{4}$	$252$	$4$	$18$	$( 1,13,23, 8)( 2,15,20,21)( 3,17, 9,16)( 5,10,11,27)( 6,22,28,24)(12,18,26,19)$
	$12^{2},3,1$	$1008$	$12$	$24$	$( 1,16, 2,13, 3,15,23,17,20, 8, 9,21)( 4, 7,14)( 5,28,18,10,24,26,11, 6,19,27,22,12)$
	$12^{2},3,1$	$1008$	$12$	$24$	$( 1,15, 9,13,20,16,23,21, 3, 8, 2,17)( 4,14, 7)( 5,26,22,10,19,28,11,12,24,27,18, 6)$
	$8^{3},2,1^{2}$	$1512$	$8$	$22$	$( 1,23,19,13,10,16,14,26)( 2, 5,21, 6, 8, 4,28, 3)( 7,27,20, 9,24,17,12,15)(11,18)$
	$4^{6},2^{2}$	$378$	$4$	$20$	$( 1,25, 3,20)( 2,19, 4,13)( 5, 6,15,24)( 7,27,10,14)( 8,18, 9,23)(11,26,21,17)(12,22)(16,28)$
	$8^{3},4$	$1512$	$8$	$24$	$( 1,19,25, 4, 3,13,20, 2)( 5, 7, 6,27,15,10,24,14)( 8,17,18,11, 9,26,23,21)(12,16,22,28)$

Malle's constant $a(G)$:     $1/12$

Character table

Character table not computed

Regular extensions

$f_{ 1 } =$

$\left(x^{6}-6 x^{5}-435 x^{4}-308 x^{3}+15 x^{2}+66 x+19\right)^{4} \left(x^{4}+20 x^{3}+114 x^{2}+68 x+13\right)-2^{2} 3^{9} \left(x^{2}+4 x+1\right)^{12} \left(2 x+1\right) t$