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Properties

Label 28T23
Degree $28$
Order $168$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{28}:C_6$

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Show commands: Magma

magma: G := TransitiveGroup(28, 23);
 

Group action invariants

Degree $n$:  $28$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $23$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{28}:C_6$
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:  (1,4,22,16,18,8,2,3,21,15,17,7)(5,12,10,19,26,23,6,11,9,20,25,24)(13,28,14,27), (1,13,21,18,5,26)(2,14,22,17,6,25)(3,20,11,15,27,8)(4,19,12,16,28,7)(23,24)
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: $F_7 \times C_2$

Low degree siblings

28T23

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

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:  168.9
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 28A1 28A5
Size 1 1 14 14 7 7 2 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 14A 14A
3 P 1A 2A 2B 2C 1A 1A 4A 2A 2A 2B 2B 2C 2C 7A 4A 4A 14A 28A1 28A5
7 P 1A 2A 2B 2C 3A1 3A-1 4A 6A1 6A-1 6B-1 6B1 6C-1 6C1 1A 12A1 12A-1 2A 4A 4A
Type
168.9.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
168.9.1b R 1 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1
168.9.1c R 1 1 −1 1 1 1 −1 1 1 −1 −1 1 1 1 −1 −1 1 −1 −1
168.9.1d R 1 1 1 −1 1 1 −1 1 1 1 1 −1 −1 1 −1 −1 1 −1 −1
168.9.1e1 C 1 1 1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 1 ζ3−1 ζ3 1 1 1
168.9.1e2 C 1 1 1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 1 ζ3 ζ3−1 1 1 1
168.9.1f1 C 1 1 −1 −1 ζ3−1 ζ3 1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 1 ζ3−1 ζ3 1 1 1
168.9.1f2 C 1 1 −1 −1 ζ3 ζ3−1 1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 1 ζ3 ζ3−1 1 1 1
168.9.1g1 C 1 1 −1 1 ζ3−1 ζ3 −1 ζ3 ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3 1 −ζ3−1 −ζ3 1 −1 −1
168.9.1g2 C 1 1 −1 1 ζ3 ζ3−1 −1 ζ3−1 ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1 1 −ζ3 −ζ3−1 1 −1 −1
168.9.1h1 C 1 1 1 −1 ζ3−1 ζ3 −1 ζ3 ζ3−1 ζ3 ζ3−1 −ζ3−1 −ζ3 1 −ζ3−1 −ζ3 1 −1 −1
168.9.1h2 C 1 1 1 −1 ζ3 ζ3−1 −1 ζ3−1 ζ3 ζ3−1 ζ3 −ζ3 −ζ3−1 1 −ζ3 −ζ3−1 1 −1 −1
168.9.2a R 2 −2 0 0 2 2 0 −2 −2 0 0 0 0 2 0 0 −2 0 0
168.9.2b1 C 2 −2 0 0 2ζ3−1 2ζ3 0 −2ζ3 −2ζ3−1 0 0 0 0 2 0 0 −2 0 0
168.9.2b2 C 2 −2 0 0 2ζ3 2ζ3−1 0 −2ζ3−1 −2ζ3 0 0 0 0 2 0 0 −2 0 0
168.9.6a R 6 6 0 0 0 0 6 0 0 0 0 0 0 −1 0 0 −1 −1 −1
168.9.6b R 6 6 0 0 0 0 −6 0 0 0 0 0 0 −1 0 0 −1 1 1
168.9.6c1 R 6 −6 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 1 −2ζ28+ζ287−2ζ289+2ζ2811 2ζ28−ζ287+2ζ289−2ζ2811
168.9.6c2 R 6 −6 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 1 2ζ28−ζ287+2ζ289−2ζ2811 −2ζ28+ζ287−2ζ289+2ζ2811

magma: CharacterTable(G);