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Properties

Label 28T37
Degree $28$
Order $224$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_4\times F_8$

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Show commands: Magma

magma: G := TransitiveGroup(28, 37);
 

Group action invariants

Degree $n$:  $28$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $37$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_4\times F_8$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (5,6)(7,8)(9,10)(11,12)(21,22)(23,24)(25,26)(27,28), (1,18,8,26,21,15,12,3,19,5,28,23,14,9,2,17,7,25,22,16,11,4,20,6,27,24,13,10)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$7$:  $C_7$
$14$:  $C_{14}$
$28$:  $C_{28}$
$56$:  $C_2^3:C_7$
$112$:  14T9

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 7: $C_7$

Degree 14: $C_{14}$

Low degree siblings

32T2229

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $224=2^{5} \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  224.173
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);