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Properties

Label 16T1577
Order \(4096\)
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 $t$ :  $1577$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $5$
Generators:  (1,7,9,15)(2,8,10,16)(3,6,12,14,4,5,11,13), (1,9)(2,10)(3,4)(5,14,6,13)(15,16), (15,16), (1,5,10,14,2,6,9,13)(3,7,11,15)(4,8,12,16)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 15
4:  $C_2^2$ x 35
8:  $D_{4}$ x 12, $C_2^3$ x 15
16:  $D_4\times C_2$ x 18, $Q_8:C_2$ x 4, $C_2^4$
32:  $C_2^2 \wr C_2$ x 4, $C_2^3 : D_4 $ x 2, $C_2 \times (C_4\times C_2):C_2$ x 2, $C_2^2 \times D_4$ x 3
64:  $(((C_4 \times C_2): C_2):C_2):C_2$ x 4, 16T73, 16T105, 16T115 x 4, 16T117
128:  16T245 x 2, 32T1074
256:  16T473, 16T494 x 2
512:  32T15122
1024:  16T1123 x 2, 16T1125
2048:  64T?

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 8: $Q_8:C_2$

Low degree siblings

16T1566 x 16, 16T1577 x 15, 32T207150 x 8, 32T207151 x 8, 32T207152 x 16, 32T207153 x 16, 32T207154 x 8, 32T207155 x 8, 32T207156 x 16, 32T207157 x 8, 32T207158 x 16, 32T207159 x 16, 32T207160 x 8, 32T207161 x 8, 32T207162 x 8, 32T207163 x 8, 32T207164 x 16, 32T207165 x 16, 32T207166 x 8, 32T207167 x 8, 32T207168 x 16, 32T207169 x 8, 32T207170 x 8, 32T207171 x 16, 32T207172 x 8, 32T207288 x 8, 32T207289 x 16, 32T207290 x 16, 32T207291 x 32, 32T207292 x 32, 32T207293 x 8, 32T207294 x 8, 32T207295 x 16, 32T207296 x 16, 32T207297 x 16, 32T207298 x 16, 32T207299 x 32, 32T207300 x 32, 32T207301 x 8, 32T207302 x 16, 32T207303 x 16, 32T207304 x 8, 32T207305 x 16, 32T207306 x 16, 32T207307 x 8, 32T207308 x 8, 32T207309 x 16, 32T207310 x 16, 32T207311 x 16, 32T207312 x 16, 32T207313 x 8, 32T207314 x 8, 32T207315 x 8, 32T207316 x 16, 32T207317 x 16, 32T207318 x 8, 32T207319 x 8, 32T207320 x 8, 32T207321 x 8, 32T220344 x 8, 32T220901 x 16, 32T220908 x 16, 32T224441 x 16, 32T245158 x 4, 32T245209 x 4, 32T262075 x 4, 32T322055 x 4, 32T375661 x 4, 32T396477 x 4

Siblings are shown with degree $\leq 47$

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

Conjugacy Classes

There are 88 conjugacy classes of elements. Data not shown.

Group invariants

Order:  $4096=2^{12}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.