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Properties

Label 16T725
Order \(384\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^4.(C_4\times S_3)$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $725$
Group :  $C_2^4.(C_4\times S_3)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,13,5,8,3,11,2,14,6,7,4,12)(9,15,10,16), (1,11,16,14)(2,12,15,13)(3,7,5,9)(4,8,6,10), (1,2)(3,5,15,4,6,16)(7,12,13,8,11,14)(9,10)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
6:  $S_3$
8:  $C_4\times C_2$
12:  $D_{6}$
24:  $S_4$, $S_3 \times C_4$
48:  $S_4\times C_2$
96:  12T53
192:  $V_4^2:(S_3\times C_2)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $V_4^2:(S_3\times C_2)$

Low degree siblings

12T153 x 2, 12T155 x 2, 16T725, 24T720, 24T836, 24T860, 24T1136, 24T1142, 24T1145, 24T1146, 24T1265 x 2, 24T1266, 24T1267, 24T1274, 24T1275, 24T1276 x 2, 24T1277 x 2, 24T1278 x 2, 24T1279 x 2, 32T9333, 32T9454

Siblings are shown with degree $\leq 47$

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

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 7,10)( 8, 9)(11,13)(12,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $2$ $( 5,16)( 6,15)( 9,12)(10,11)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $24$ $4$ $( 5,16)( 6,15)( 7,10,13,11)( 8, 9,14,12)$
$ 3, 3, 3, 3, 1, 1, 1, 1 $ $32$ $3$ $( 3, 6,15)( 4, 5,16)( 9,14,12)(10,13,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 9)( 8,10)(11,14)(12,13)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 2)( 3, 4)( 5,15)( 6,16)( 7, 8)( 9,11)(10,12)(13,14)$
$ 4, 4, 2, 2, 2, 2 $ $24$ $4$ $( 1, 2)( 3, 4)( 5,15)( 6,16)( 7, 9,13,12)( 8,10,14,11)$
$ 6, 6, 2, 2 $ $32$ $6$ $( 1, 2)( 3, 5,15, 4, 6,16)( 7, 8)( 9,13,12,10,14,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 3)( 2, 4)( 5,16)( 6,15)( 7,10)( 8, 9)(11,13)(12,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 3)( 2, 4)( 5,16)( 6,15)( 7,13)( 8,14)( 9,12)(10,11)$
$ 4, 4, 4, 4 $ $12$ $4$ $( 1, 3, 6,15)( 2, 4, 5,16)( 7,10,11,13)( 8, 9,12,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 4)( 2, 3)( 5,15)( 6,16)( 7, 9)( 8,10)(11,14)(12,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 4)( 2, 3)( 5,15)( 6,16)( 7,14)( 8,13)( 9,11)(10,12)$
$ 4, 4, 4, 4 $ $12$ $4$ $( 1, 4, 6,16)( 2, 3, 5,15)( 7, 9,11,14)( 8,10,12,13)$
$ 4, 4, 4, 4 $ $12$ $4$ $( 1, 7, 2, 8)( 3,10, 4, 9)( 5,12, 6,11)(13,16,14,15)$
$ 4, 4, 4, 4 $ $24$ $4$ $( 1, 7, 4, 9)( 2, 8, 3,10)( 5,12,15,13)( 6,11,16,14)$
$ 4, 4, 4, 4 $ $12$ $4$ $( 1, 7, 5,12)( 2, 8, 6,11)( 3,10,16,14)( 4, 9,15,13)$
$ 12, 4 $ $32$ $12$ $( 1, 7, 2, 8)( 3,10, 5,14,15,11, 4, 9, 6,13,16,12)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 7, 2, 8)( 3,13, 4,14)( 5,12, 6,11)( 9,15,10,16)$
$ 4, 4, 4, 4 $ $12$ $4$ $( 1, 7, 4,14)( 2, 8, 3,13)( 5,12,15,10)( 6,11,16, 9)$
$ 4, 4, 4, 4 $ $12$ $4$ $( 1, 8, 2, 7)( 3, 9, 4,10)( 5,11, 6,12)(13,15,14,16)$
$ 4, 4, 4, 4 $ $24$ $4$ $( 1, 8, 4,10)( 2, 7, 3, 9)( 5,11,15,14)( 6,12,16,13)$
$ 4, 4, 4, 4 $ $12$ $4$ $( 1, 8, 5,11)( 2, 7, 6,12)( 3, 9,16,13)( 4,10,15,14)$
$ 12, 4 $ $32$ $12$ $( 1, 8, 2, 7)( 3, 9, 5,13,15,12, 4,10, 6,14,16,11)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 8, 2, 7)( 3,14, 4,13)( 5,11, 6,12)( 9,16,10,15)$
$ 4, 4, 4, 4 $ $12$ $4$ $( 1, 8, 4,13)( 2, 7, 3,14)( 5,11,15, 9)( 6,12,16,10)$

Group invariants

Order:  $384=2^{7} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [384, 5566]
Character table: Data not available.