Properties

Label 16T1720
Order \(8192\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $1720$
Parity:  $1$
Primitive:  No
Nilpotency class:  $7$
Generators:  (1,10,2,9)(3,4)(5,13,6,14)(7,16)(8,15)(11,12), (1,8,14,3,9,15,6,12)(2,7,13,4,10,16,5,11), (1,7,13,4,2,8,14,3)(5,12,9,16,6,11,10,15)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_4$ x 4, $C_2^2$ x 7
8:  $D_{4}$ x 12, $C_4\times C_2$ x 6, $C_2^3$
16:  $D_4\times C_2$ x 6, $C_2^2:C_4$ x 12, $C_4\times C_2^2$
32:  $C_2^2 \wr C_2$ x 4, $C_2^3 : C_4 $ x 12, $C_2 \times (C_2^2:C_4)$ x 3
64:  $((C_8 : C_2):C_2):C_2$ x 8, 16T76 x 6, 16T79
128:  16T227 x 4, 16T240 x 3
256:  16T502 x 2, 16T581
512:  16T842 x 2, 16T911
1024:  32T40133
2048:  16T1363
4096:  32T231839

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $((C_8 : C_2):C_2):C_2$

Low degree siblings

16T1720 x 31, 16T1726 x 32, 32T399994 x 32, 32T399995 x 32, 32T399996 x 32, 32T399997 x 16, 32T399998 x 32, 32T399999 x 32, 32T400000 x 32, 32T400001 x 16, 32T400002 x 32, 32T400003 x 32, 32T400004 x 32, 32T400005 x 32, 32T400006 x 32, 32T400007 x 32, 32T400008 x 16, 32T400009 x 32, 32T400010 x 32, 32T400011 x 32, 32T400012 x 16, 32T400013 x 32, 32T400014 x 32, 32T400015 x 16, 32T400016 x 32, 32T400017 x 16, 32T400018 x 32, 32T400019 x 32, 32T400020 x 16, 32T400021 x 16, 32T400022 x 32, 32T400023 x 16, 32T400024 x 16, 32T400025 x 16, 32T400026 x 32, 32T400027 x 32, 32T400028 x 32, 32T400029 x 32, 32T400030 x 16, 32T400031 x 16, 32T400032 x 16, 32T400162 x 32, 32T400163 x 16, 32T400164 x 32, 32T400165 x 32, 32T400166 x 32, 32T400167 x 32, 32T400168 x 32, 32T400169 x 32, 32T400170 x 32, 32T400171 x 16, 32T400172 x 16, 32T400173 x 32, 32T400174 x 32, 32T400175 x 32, 32T400176 x 32, 32T400177 x 16, 32T400178 x 32, 32T400179 x 16, 32T400180 x 32, 32T400181 x 32, 32T400182 x 32, 32T400183 x 32, 32T400184 x 16, 32T400185 x 32, 32T400186 x 32, 32T400187 x 32, 32T400188 x 16, 32T400189 x 16, 32T400190 x 32, 32T400191 x 32, 32T400192 x 16, 32T400193 x 16, 32T400194 x 32, 32T400195 x 32, 32T400196 x 16, 32T400197 x 16, 32T400198 x 16, 32T400199 x 16, 32T404892 x 16, 32T407661 x 16

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

There are 116 conjugacy classes of elements. Data not shown.

Group invariants

Order:  $8192=2^{13}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.