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Properties

Label 28T21
Degree $28$
Order $168$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{14}:C_6$

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Show commands: Magma

magma: G := TransitiveGroup(28, 21);
 

Group action invariants

Degree $n$:  $28$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $21$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_{14}:C_6$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,12,10)(2,11,9)(3,5,13)(4,6,14)(15,21,18,16,22,17)(19,23,26,20,24,25)(27,28), (1,19,8,24,5,18)(2,20,7,23,6,17)(3,26,12,22,13,27)(4,25,11,21,14,28)(9,16)(10,15)
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
$8$:  $D_{4}$
$12$:  $C_6\times C_2$
$24$:  $D_4 \times C_3$
$42$:  $F_7$
$84$:  $F_7 \times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 7: $F_7$

Degree 14: $F_7$

Low degree siblings

28T25

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

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:  168.11
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A1 3A-1 4A 6A1 6A-1 6B1 6B-1 6C1 6C-1 7A 12A1 12A-1 14A 14B1 14B-1
Size 1 1 2 14 7 7 14 7 7 14 14 14 14 6 14 14 6 6 6
2 P 1A 1A 1A 1A 3A-1 3A1 2A 3A1 3A-1 3A-1 3A1 3A1 3A-1 7A 6A1 6A-1 7A 7A 7A
3 P 1A 2A 2B 2C 1A 1A 4A 2A 2A 2B 2B 2C 2C 7A 4A 4A 14A 14B-1 14B1
7 P 1A 2A 2B 2C 3A1 3A-1 4A 6A1 6A-1 6B1 6B-1 6C1 6C-1 1A 12A1 12A-1 2A 2B 2B
Type
168.11.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
168.11.1b R 1 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 −1 −1
168.11.1c R 1 1 −1 1 1 1 −1 1 1 −1 −1 1 1 1 −1 −1 1 −1 −1
168.11.1d R 1 1 1 −1 1 1 −1 1 1 1 1 −1 −1 1 −1 −1 1 1 1
168.11.1e1 C 1 1 1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1 ζ3−1 ζ3 1 1 1
168.11.1e2 C 1 1 1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1 ζ3 ζ3−1 1 1 1
168.11.1f1 C 1 1 −1 −1 ζ3−1 ζ3 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 1 ζ3−1 ζ3 1 −1 −1
168.11.1f2 C 1 1 −1 −1 ζ3 ζ3−1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 1 ζ3 ζ3−1 1 −1 −1
168.11.1g1 C 1 1 −1 1 ζ3−1 ζ3 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1 1 −ζ3−1 −ζ3 1 −1 −1
168.11.1g2 C 1 1 −1 1 ζ3 ζ3−1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3 1 −ζ3 −ζ3−1 1 −1 −1
168.11.1h1 C 1 1 1 −1 ζ3−1 ζ3 −1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 1 −ζ3−1 −ζ3 1 1 1
168.11.1h2 C 1 1 1 −1 ζ3 ζ3−1 −1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 1 −ζ3 −ζ3−1 1 1 1
168.11.2a R 2 −2 0 0 2 2 0 −2 −2 0 0 0 0 2 0 0 −2 0 0
168.11.2b1 C 2 −2 0 0 2ζ3−1 2ζ3 0 −2ζ3 −2ζ3−1 0 0 0 0 2 0 0 −2 0 0
168.11.2b2 C 2 −2 0 0 2ζ3 2ζ3−1 0 −2ζ3−1 −2ζ3 0 0 0 0 2 0 0 −2 0 0
168.11.6a R 6 6 6 0 0 0 0 0 0 0 0 0 0 −1 0 0 −1 −1 −1
168.11.6b R 6 6 −6 0 0 0 0 0 0 0 0 0 0 −1 0 0 −1 1 1
168.11.6c1 C 6 −6 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 1 −2ζ7−3−1−2ζ7−2ζ72 2ζ7−3+1+2ζ7+2ζ72
168.11.6c2 C 6 −6 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 1 2ζ7−3+1+2ζ7+2ζ72 −2ζ7−3−1−2ζ7−2ζ72

magma: CharacterTable(G);