Galois group: 16T1116

• Label: 16T1116
• Degree: $16$
• Order: $1024$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $C_2^7:D_4$

Group invariants

Abstract group:		$C_2^7:D_4$
Order:		$1024=2^{10}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$4$

Group action invariants

Degree $n$:		$16$
Transitive number $t$:		$1116$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$4$
Generators:		$(5,9)(6,10)(7,12)(8,11)$, $(9,10)(11,12)(13,14)(15,16)$, $(1,11,3,10)(2,12,4,9)(5,13,7,16)(6,14,8,15)$, $(1,3)(2,4)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 15
$4$:	$C_2^2$ x 35
$8$:	$D_{4}$ x 20, $C_2^3$ x 15
$16$:	$D_4\times C_2$ x 30, $C_2^4$
$32$:	$C_2^2 \wr C_2$ x 8, $Q_8:C_2^2$ x 2, $C_2^2 \times D_4$ x 5
$64$:	$(((C_4 \times C_2): C_2):C_2):C_2$ x 8, 16T87, 16T105 x 2, 16T109 x 4
$128$:	$C_2 \wr C_2\wr C_2$ x 4, 16T245 x 4, 32T1237
$256$:	16T477 x 2, 16T509 x 2, 16T531 x 2, 16T536
$512$:	32T12264 x 2, 32T13404

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $(((C_4 \times C_2): C_2):C_2):C_2$, $C_2 \wr C_2\wr C_2$ x 2

Low degree siblings

16T1116 x 127, 32T35541 x 32, 32T35542 x 32, 32T35543 x 64, 32T35544 x 64, 32T35545 x 32, 32T35546 x 32, 32T35547 x 64, 32T35548 x 32, 32T35549 x 32, 32T35550 x 64, 32T35551 x 32, 32T44704 x 16, 32T44706 x 16, 32T44742 x 32, 32T45272 x 16, 32T46107 x 16, 32T46112 x 16, 32T46125 x 32, 32T49347 x 32, 32T49348 x 32, 32T49509 x 32, 32T51999 x 32, 32T55778 x 16, 32T55814 x 32, 32T56666 x 16

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

Conjugacy classes not computed

Character table

76 x 76 character table

Regular extensions

Data not computed