↑

Properties

Label 28T15
Degree $28$
Order $84$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times F_7$

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(28, 15);
 

Group action invariants

Degree $n$:  $28$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $15$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\times F_7$
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:  (1,2)(3,8,19,27,23,12)(4,7,20,28,24,11)(5,14,9,26,17,22)(6,13,10,25,18,21)(15,16), (1,12,18,16,26,4)(2,11,17,15,25,3)(5,19)(6,20)(7,9,28,21,23,13)(8,10,27,22,24,14)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $C_6$ x 3
$12$:  $C_6\times C_2$
$42$:  $F_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 7: $F_7$

Degree 14: $F_7$, $F_7 \times C_2$ x 2

Low degree siblings

14T7 x 2, 42T10 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no 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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $7$ $3$ $( 3,19,23)( 4,20,24)( 5, 9,17)( 6,10,18)( 7,28,11)( 8,27,12)(13,25,21) (14,26,22)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $7$ $3$ $( 3,23,19)( 4,24,20)( 5,17, 9)( 6,18,10)( 7,11,28)( 8,12,27)(13,21,25) (14,22,26)$
$ 6, 6, 6, 6, 2, 2 $ $7$ $6$ $( 1, 2)( 3, 8,19,27,23,12)( 4, 7,20,28,24,11)( 5,14, 9,26,17,22) ( 6,13,10,25,18,21)(15,16)$
$ 6, 6, 6, 6, 2, 2 $ $7$ $6$ $( 1, 2)( 3,12,23,27,19, 8)( 4,11,24,28,20, 7)( 5,22,17,26, 9,14) ( 6,21,18,25,10,13)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $7$ $2$ $( 1, 2)( 3,27)( 4,28)( 5,26)( 6,25)( 7,24)( 8,23)( 9,22)(10,21)(11,20)(12,19) (13,18)(14,17)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $7$ $2$ $( 1, 3)( 2, 4)( 5,27)( 6,28)( 7,26)( 8,25)( 9,24)(10,23)(11,22)(12,21)(13,20) (14,19)(15,18)(16,17)$
$ 6, 6, 6, 6, 2, 2 $ $7$ $6$ $( 1, 3,10,28,26,19)( 2, 4, 9,27,25,20)( 5,16,17,24,13,12)( 6,15,18,23,14,11) ( 7,22)( 8,21)$
$ 6, 6, 6, 6, 2, 2 $ $7$ $6$ $( 1, 3,14, 7, 6,23)( 2, 4,13, 8, 5,24)( 9,16,17,27,21,20)(10,15,18,28,22,19) (11,26)(12,25)$
$ 14, 14 $ $6$ $14$ $( 1, 4, 6, 8,10,12,14,16,18,20,22,24,26,27)( 2, 3, 5, 7, 9,11,13,15,17,19,21, 23,25,28)$
$ 6, 6, 6, 6, 2, 2 $ $7$ $6$ $( 1, 4,22,16,18, 8)( 2, 3,21,15,17, 7)( 5,11, 9,19,25,23)( 6,12,10,20,26,24) (13,28)(14,27)$
$ 6, 6, 6, 6, 2, 2 $ $7$ $6$ $( 1, 4,26,16,18,12)( 2, 3,25,15,17,11)( 5,19)( 6,20)( 7,13,23,21,28, 9) ( 8,14,24,22,27,10)$
$ 7, 7, 7, 7 $ $6$ $7$ $( 1, 6,10,14,18,22,26)( 2, 5, 9,13,17,21,25)( 3, 7,11,15,19,23,28) ( 4, 8,12,16,20,24,27)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,16)( 2,15)( 3,17)( 4,18)( 5,19)( 6,20)( 7,21)( 8,22)( 9,23)(10,24)(11,25) (12,26)(13,28)(14,27)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $84=2^{2} \cdot 3 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  84.7
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A1 3A-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 7A 14A
Size 1 1 7 7 7 7 7 7 7 7 7 7 6 6
2 P 1A 1A 1A 1A 3A-1 3A1 3A-1 3A1 3A1 3A-1 3A1 3A-1 7A 7A
3 P 1A 2A 2B 2C 1A 1A 2A 2A 2B 2B 2C 2C 7A 14A
7 P 1A 2A 2B 2C 3A1 3A-1 6C-1 6C1 6A1 6A-1 6B-1 6B1 1A 2A
Type
84.7.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
84.7.1b R 1 −1 −1 1 1 1 −1 −1 1 1 −1 −1 1 −1
84.7.1c R 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1 1 −1
84.7.1d R 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1 1 1
84.7.1e1 C 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1 1
84.7.1e2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1 1
84.7.1f1 C 1 −1 −1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 1 −1
84.7.1f2 C 1 −1 −1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 1 −1
84.7.1g1 C 1 −1 1 −1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 1 −1
84.7.1g2 C 1 −1 1 −1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 1 −1
84.7.1h1 C 1 1 −1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1 1 1
84.7.1h2 C 1 1 −1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3 1 1
84.7.6a R 6 6 0 0 0 0 0 0 0 0 0 0 −1 −1
84.7.6b R 6 −6 0 0 0 0 0 0 0 0 0 0 −1 1

magma: CharacterTable(G);