Properties

Label 28T38
Order \(224\)
n \(28\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^2\times F_8$

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Group action invariants

Degree $n$ :  $28$
Transitive number $t$ :  $38$
Group :  $C_2^2\times F_8$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,24,18,11,5,27,21,16,9,3,26,19,13,7)(2,23,17,12,6,28,22,15,10,4,25,20,14,8), (1,15)(2,16)(5,6)(7,21)(8,22)(11,26)(12,25)(13,28)(14,27)(19,20)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
7:  $C_7$
14:  $C_{14}$ x 3
28:  28T2
56:  $C_2^3:C_7$
112:  14T9 x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 7: $C_7$

Degree 14: 14T9 x 3

Low degree siblings

28T38 x 5, 28T39 x 3, 32T2228

Siblings are shown with degree $\leq 47$

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

Group invariants

Order:  $224=2^{5} \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [224, 195]
Character table: Data not available.