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Properties

Label 16T724
Order \(384\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_2^2:S_4:C_2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $724$
Group :  $C_2\times C_2^2:S_4:C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9)(2,10)(3,8,6,11,15,13)(4,7,5,12,16,14), (1,14,6,7)(2,13,5,8)(3,10,15,12)(4,9,16,11), (1,2)(3,4)(5,6)(7,13)(8,14)(9,12)(10,11)(15,16)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
6:  $S_3$
8:  $C_2^3$
12:  $D_{6}$ x 3
24:  $S_4$, $S_3 \times C_2^2$
48:  $S_4\times C_2$ x 3
96:  12T48
192:  $V_4^2:(S_3\times C_2)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

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

Low degree siblings

12T136 x 4, 12T137 x 4, 16T724 x 3, 24T1143 x 2, 24T1144 x 2, 24T1184 x 2, 24T1185 x 4, 24T1186 x 2, 24T1187 x 4, 24T1188 x 4, 24T1189 x 4, 24T1190 x 4, 24T1191 x 2, 24T1192 x 4, 24T1193 x 2, 24T1194 x 4, 24T1195 x 8, 24T1196 x 8, 24T1197 x 8, 24T1198 x 8, 32T9332 x 2, 32T9457 x 4

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

Group invariants

Order:  $384=2^{7} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [384, 17948]
Character table: Data not available.