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Properties

Label 28T28
Degree $28$
Order $168$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_7:A_4$

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Show commands: Magma

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

Group action invariants

Degree $n$:  $28$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $28$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_7:A_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,24,26,12,19,14)(2,21,28,11,18,16)(3,23,27,10,20,15)(4,22,25,9,17,13)(5,7,8), (1,10,15)(2,11,13)(3,9,14)(4,12,16)(5,25,23)(6,28,21)(7,26,22)(8,27,24)(17,18,19)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$12$:  $A_4$
$24$:  $A_4\times C_2$
$42$:  $F_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $A_4$

Degree 7: $F_7$

Degree 14: None

Low degree siblings

42T30, 42T31

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $168=2^{3} \cdot 3 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  168.49
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A1 3A-1 6A1 6A-1 7A 14A1 14A3 14A5
Size 1 3 7 21 28 28 28 28 6 6 6 6
2 P 1A 1A 1A 1A 3A-1 3A1 3A1 3A-1 7A 7A 7A 7A
3 P 1A 2A 2B 2C 1A 1A 2B 2B 7A 14A3 14A5 14A1
7 P 1A 2A 2B 2C 3A1 3A-1 6A1 6A-1 1A 2A 2A 2A
Type
168.49.1a R 1 1 1 1 1 1 1 1 1 1 1 1
168.49.1b R 1 1 −1 −1 1 1 −1 −1 1 1 1 1
168.49.1c1 C 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1
168.49.1c2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1
168.49.1d1 C 1 1 −1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1 1 1
168.49.1d2 C 1 1 −1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1 1 1
168.49.3a R 3 −1 3 −1 0 0 0 0 3 −1 −1 −1
168.49.3b R 3 −1 −3 1 0 0 0 0 3 −1 −1 −1
168.49.6a R 6 6 0 0 0 0 0 0 −1 −1 −1 −1
168.49.6b1 R 6 −2 0 0 0 0 0 0 −1 2ζ7−3+1+2ζ73 2ζ7−2+1+2ζ72 2ζ7−1+1+2ζ7
168.49.6b2 R 6 −2 0 0 0 0 0 0 −1 2ζ7−2+1+2ζ72 2ζ7−1+1+2ζ7 2ζ7−3+1+2ζ73
168.49.6b3 R 6 −2 0 0 0 0 0 0 −1 2ζ7−1+1+2ζ7 2ζ7−3+1+2ζ73 2ζ7−2+1+2ζ72

magma: CharacterTable(G);