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Properties

Label 28T19
Degree $28$
Order $112$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times F_8$

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Show commands: Magma

magma: G := TransitiveGroup(28, 19);
 

Group action invariants

Degree $n$:  $28$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $19$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\times F_8$
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:  (1,15)(2,16)(3,18)(4,17)(7,22)(8,21)(9,10)(11,12)(23,24)(25,26), (1,5,9,14,18,22,25)(2,6,10,13,17,21,26)(3,7,12,16,20,24,27)(4,8,11,15,19,23,28)
magma: Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 7: $C_7$

Degree 14: 14T6, 14T9

Low degree siblings

14T9, 16T196, 28T19 x 2, 28T20

Siblings are shown with degree $\leq 47$

A number field with this Galois group has 2 arithmetically equivalent fields.

Conjugacy classes

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $112=2^{4} \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  112.41
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 7A1 7A-1 7A2 7A-2 7A3 7A-3 14A1 14A-1 14A3 14A-3 14A5 14A-5
Size 1 1 7 7 8 8 8 8 8 8 8 8 8 8 8 8
2 P 1A 1A 1A 1A 7A2 7A-2 7A-3 7A3 7A-1 7A1 7A1 7A3 7A2 7A-3 7A-2 7A-1
7 P 1A 2A 2B 2C 7A3 7A-3 7A-1 7A1 7A2 7A-2 14A3 14A-5 14A-1 14A5 14A1 14A-3
Type
112.41.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
112.41.1b R 1 −1 1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
112.41.1c1 C 1 1 1 1 ζ7−3 ζ73 ζ7 ζ7−1 ζ7−2 ζ72 ζ72 ζ7−2 ζ7−1 ζ7 ζ73 ζ7−3
112.41.1c2 C 1 1 1 1 ζ73 ζ7−3 ζ7−1 ζ7 ζ72 ζ7−2 ζ7−2 ζ72 ζ7 ζ7−1 ζ7−3 ζ73
112.41.1c3 C 1 1 1 1 ζ7−2 ζ72 ζ73 ζ7−3 ζ7 ζ7−1 ζ7−1 ζ7 ζ7−3 ζ73 ζ72 ζ7−2
112.41.1c4 C 1 1 1 1 ζ72 ζ7−2 ζ7−3 ζ73 ζ7−1 ζ7 ζ7 ζ7−1 ζ73 ζ7−3 ζ7−2 ζ72
112.41.1c5 C 1 1 1 1 ζ7−1 ζ7 ζ7−2 ζ72 ζ7−3 ζ73 ζ73 ζ7−3 ζ72 ζ7−2 ζ7 ζ7−1
112.41.1c6 C 1 1 1 1 ζ7 ζ7−1 ζ72 ζ7−2 ζ73 ζ7−3 ζ7−3 ζ73 ζ7−2 ζ72 ζ7−1 ζ7
112.41.1d1 C 1 −1 1 −1 ζ7−3 ζ73 ζ7 ζ7−1 ζ7−2 ζ72 −ζ72 −ζ7−2 −ζ7−1 −ζ7 −ζ73 −ζ7−3
112.41.1d2 C 1 −1 1 −1 ζ73 ζ7−3 ζ7−1 ζ7 ζ72 ζ7−2 −ζ7−2 −ζ72 −ζ7 −ζ7−1 −ζ7−3 −ζ73
112.41.1d3 C 1 −1 1 −1 ζ7−2 ζ72 ζ73 ζ7−3 ζ7 ζ7−1 −ζ7−1 −ζ7 −ζ7−3 −ζ73 −ζ72 −ζ7−2
112.41.1d4 C 1 −1 1 −1 ζ72 ζ7−2 ζ7−3 ζ73 ζ7−1 ζ7 −ζ7 −ζ7−1 −ζ73 −ζ7−3 −ζ7−2 −ζ72
112.41.1d5 C 1 −1 1 −1 ζ7−1 ζ7 ζ7−2 ζ72 ζ7−3 ζ73 −ζ73 −ζ7−3 −ζ72 −ζ7−2 −ζ7 −ζ7−1
112.41.1d6 C 1 −1 1 −1 ζ7 ζ7−1 ζ72 ζ7−2 ζ73 ζ7−3 −ζ7−3 −ζ73 −ζ7−2 −ζ72 −ζ7−1 −ζ7
112.41.7a R 7 7 −1 −1 0 0 0 0 0 0 0 0 0 0 0 0
112.41.7b R 7 −7 −1 1 0 0 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);