Properties

Label 28T41
Order \(336\)
n \(28\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_4\times F_7$

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Group action invariants

Degree $n$ :  $28$
Transitive number $t$ :  $41$
Group :  $D_4\times F_7$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,16,11,22,10,17,2,15,12,21,9,18)(3,20,6,24,13,25,4,19,5,23,14,26)(7,27,8,28), (1,16,6,27,13,23,2,15,5,28,14,24)(3,21,9,26,8,20,4,22,10,25,7,19)(11,18,12,17), (1,11,8,4,13,9,5,2,12,7,3,14,10,6)(15,26,22,18,27,24,19)(16,25,21,17,28,23,20)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
3:  $C_3$
4:  $C_2^2$ x 7
6:  $C_6$ x 7
8:  $D_{4}$ x 2, $C_2^3$
12:  $C_6\times C_2$ x 7
16:  $D_4\times C_2$
24:  $D_4 \times C_3$ x 2, 24T3
42:  $F_7$
48:  24T38
84:  $F_7 \times C_2$ x 3
168:  28T24

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 7: $F_7$

Degree 14: $F_7 \times C_2$

Low degree siblings

28T41 x 3

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)$
$ 6, 6, 3, 3, 3, 3, 2, 1, 1 $ $14$ $6$ $( 3, 5,10)( 4, 6, 9)( 7,14,11)( 8,13,12)(15,23,26,16,24,25)(17,27,20,18,28,19) (21,22)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $7$ $3$ $( 3, 5,10)( 4, 6, 9)( 7,14,11)( 8,13,12)(15,24,26)(16,23,25)(17,28,20) (18,27,19)$
$ 6, 6, 6, 6, 2, 1, 1 $ $14$ $6$ $( 3, 8, 5,13,10,12)( 4, 7, 6,14, 9,11)(15,17,24,28,26,20)(16,18,23,27,25,19) (21,22)$
$ 6, 6, 6, 6, 1, 1, 1, 1 $ $7$ $6$ $( 3, 8, 5,13,10,12)( 4, 7, 6,14, 9,11)(15,18,24,27,26,19)(16,17,23,28,25,20)$
$ 6, 6, 3, 3, 3, 3, 2, 1, 1 $ $14$ $6$ $( 3,10, 5)( 4, 9, 6)( 7,11,14)( 8,12,13)(15,25,24,16,26,23)(17,19,28,18,20,27) (21,22)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $7$ $3$ $( 3,10, 5)( 4, 9, 6)( 7,11,14)( 8,12,13)(15,26,24)(16,25,23)(17,20,28) (18,19,27)$
$ 6, 6, 6, 6, 1, 1, 1, 1 $ $7$ $6$ $( 3,12,10,13, 5, 8)( 4,11, 9,14, 6, 7)(15,19,26,27,24,18)(16,20,25,28,23,17)$
$ 6, 6, 6, 6, 2, 1, 1 $ $14$ $6$ $( 3,12,10,13, 5, 8)( 4,11, 9,14, 6, 7)(15,20,26,28,24,17)(16,19,25,27,23,18) (21,22)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $7$ $2$ $( 3,13)( 4,14)( 5,12)( 6,11)( 7, 9)( 8,10)(15,27)(16,28)(17,25)(18,26)(19,24) (20,23)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $14$ $2$ $( 3,13)( 4,14)( 5,12)( 6,11)( 7, 9)( 8,10)(15,28)(16,27)(17,26)(18,25)(19,23) (20,24)(21,22)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)$
$ 6, 6, 6, 6, 2, 2 $ $7$ $6$ $( 1, 2)( 3, 6,10, 4, 5, 9)( 7,13,11, 8,14,12)(15,23,26,16,24,25) (17,27,20,18,28,19)(21,22)$
$ 6, 6, 6, 6, 2, 2 $ $7$ $6$ $( 1, 2)( 3, 7, 5,14,10,11)( 4, 8, 6,13, 9,12)(15,17,24,28,26,20) (16,18,23,27,25,19)(21,22)$
$ 6, 6, 6, 6, 2, 2 $ $7$ $6$ $( 1, 2)( 3, 9, 5, 4,10, 6)( 7,12,14, 8,11,13)(15,25,24,16,26,23) (17,19,28,18,20,27)(21,22)$
$ 6, 6, 6, 6, 2, 2 $ $7$ $6$ $( 1, 2)( 3,11,10,14, 5, 7)( 4,12, 9,13, 6, 8)(15,20,26,28,24,17) (16,19,25,27,23,18)(21,22)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $7$ $2$ $( 1, 2)( 3,14)( 4,13)( 5,11)( 6,12)( 7,10)( 8, 9)(15,28)(16,27)(17,26)(18,25) (19,23)(20,24)(21,22)$
$ 14, 7, 7 $ $12$ $14$ $( 1, 3, 5, 8,10,12,13)( 2, 4, 6, 7, 9,11,14)(15,17,19,21,24,25,27,16,18,20,22, 23,26,28)$
$ 7, 7, 7, 7 $ $6$ $7$ $( 1, 3, 5, 8,10,12,13)( 2, 4, 6, 7, 9,11,14)(15,18,19,22,24,26,27) (16,17,20,21,23,25,28)$
$ 14, 14 $ $6$ $14$ $( 1, 4, 5, 7,10,11,13, 2, 3, 6, 8, 9,12,14)(15,17,19,21,24,25,27,16,18,20,22, 23,26,28)$
$ 14, 14 $ $12$ $14$ $( 1,15, 3,18, 5,19, 8,22,10,24,12,26,13,27)( 2,16, 4,17, 6,20, 7,21, 9,23,11, 25,14,28)$
$ 28 $ $12$ $28$ $( 1,15, 4,17, 5,19, 7,21,10,24,11,25,13,27, 2,16, 3,18, 6,20, 8,22, 9,23,12, 26,14,28)$
$ 12, 12, 4 $ $14$ $12$ $( 1,15,11,21,10,18, 2,16,12,22, 9,17)( 3,19, 6,23,13,26, 4,20, 5,24,14,25) ( 7,28, 8,27)$
$ 6, 6, 6, 6, 2, 2 $ $14$ $6$ $( 1,15,12,22,10,18)( 2,16,11,21, 9,17)( 3,19, 5,24,13,26)( 4,20, 6,23,14,25) ( 7,28)( 8,27)$
$ 6, 6, 6, 6, 2, 2 $ $14$ $6$ $( 1,15, 5,27,13,24)( 2,16, 6,28,14,23)( 3,22,10,26, 8,19)( 4,21, 9,25, 7,20) (11,17)(12,18)$
$ 12, 12, 4 $ $14$ $12$ $( 1,15, 6,28,13,24, 2,16, 5,27,14,23)( 3,22, 9,25, 8,19, 4,21,10,26, 7,20) (11,17,12,18)$
$ 6, 6, 6, 6, 2, 2 $ $14$ $6$ $( 1,15,13,22,10,19)( 2,16,14,21, 9,20)( 3,24)( 4,23)( 5,18, 8,26,12,27) ( 6,17, 7,25,11,28)$
$ 12, 12, 4 $ $14$ $12$ $( 1,15,14,21,10,19, 2,16,13,22, 9,20)( 3,24, 4,23)( 5,18, 7,25,12,27, 6,17, 8, 26,11,28)$
$ 12, 12, 4 $ $14$ $12$ $( 1,15, 7,17, 3,26, 2,16, 8,18, 4,25)( 5,22, 9,28,12,24, 6,21,10,27,11,23) (13,19,14,20)$
$ 6, 6, 6, 6, 2, 2 $ $14$ $6$ $( 1,15, 8,18, 3,26)( 2,16, 7,17, 4,25)( 5,22,10,27,12,24)( 6,21, 9,28,11,23) (13,19)(14,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $14$ $2$ $( 1,15)( 2,16)( 3,27)( 4,28)( 5,26)( 6,25)( 7,23)( 8,24)( 9,21)(10,22)(11,20) (12,19)(13,18)(14,17)$
$ 4, 4, 4, 4, 4, 4, 4 $ $14$ $4$ $( 1,15, 2,16)( 3,27, 4,28)( 5,26, 6,25)( 7,23, 8,24)( 9,21,10,22)(11,20,12,19) (13,18,14,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1,21)( 2,22)( 3,23)( 4,24)( 5,25)( 6,26)( 7,27)( 8,28)( 9,15)(10,16)(11,18) (12,17)(13,20)(14,19)$
$ 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,21, 2,22)( 3,23, 4,24)( 5,25, 6,26)( 7,27, 8,28)( 9,15,10,16)(11,18,12,17) (13,20,14,19)$

Group invariants

Order:  $336=2^{4} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [336, 125]
Character table: Data not available.