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Properties

Label 28T22
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, 22);
 

Group action invariants

Degree $n$:  $28$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $22$
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:  (1,27,4,21,8,24,2,28,3,22,7,23)(5,15,11,25,10,18,6,16,12,26,9,17)(13,19,14,20), (1,27,5,22,8,26)(2,28,6,21,7,25)(3,18,13,24,12,19)(4,17,14,23,11,20)(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
$8$:  $D_{4}$
$12$:  $C_6\times C_2$
$21$:  $C_7:C_3$
$24$:  $D_4 \times C_3$
$42$:  $(C_7:C_3) \times C_2$ x 3
$84$:  28T14

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 7: $C_7:C_3$

Degree 14: $(C_7:C_3) \times C_2$

Low degree siblings

28T22

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

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:  168.20
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A1 3A-1 4A 6A1 6A-1 6B1 6B-1 6C1 6C-1 7A1 7A-1 12A1 12A-1 14A1 14A-1 14B1 14B-1 14C1 14C-1 28A1 28A-1
Size 1 1 2 2 7 7 2 7 7 14 14 14 14 3 3 14 14 3 3 6 6 6 6 6 6
2 P 1A 1A 1A 1A 3A-1 3A1 2A 3A1 3A-1 3A1 3A-1 3A-1 3A1 7A1 7A-1 6A1 6A-1 7A-1 7A1 7A1 7A-1 7A1 7A-1 14A1 14A-1
3 P 1A 2A 2B 2C 1A 1A 4A 2A 2A 2C 2C 2B 2B 7A-1 7A1 4A 4A 14A-1 14A1 14B-1 14B1 14C-1 14C1 28A-1 28A1
7 P 1A 2A 2B 2C 3A1 3A-1 4A 6A1 6A-1 6C-1 6C1 6B-1 6B1 1A 1A 12A1 12A-1 2A 2A 2C 2C 2B 2B 4A 4A
Type
168.20.1a R 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
168.20.1b R 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
168.20.1c R 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
168.20.1d R 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
168.20.1e1 C 1 1 1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 1 1 1 1 1 1 1 1
168.20.1e2 C 1 1 1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3 ζ3−1 1 1 1 1 1 1 1 1
168.20.1f1 C 1 1 −1 −1 ζ3−1 ζ3 1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 1 1 ζ3−1 ζ3 1 1 −1 −1 −1 −1 1 1
168.20.1f2 C 1 1 −1 −1 ζ3 ζ3−1 1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 1 1 ζ3 ζ3−1 1 1 −1 −1 −1 −1 1 1
168.20.1g1 C 1 1 −1 1 ζ3−1 ζ3 −1 ζ3 ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3 1 1 −ζ3−1 −ζ3 1 1 1 1 −1 −1 −1 −1
168.20.1g2 C 1 1 −1 1 ζ3 ζ3−1 −1 ζ3−1 ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1 1 1 −ζ3 −ζ3−1 1 1 1 1 −1 −1 −1 −1
168.20.1h1 C 1 1 1 −1 ζ3−1 ζ3 −1 ζ3 ζ3−1 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1 −ζ3−1 −ζ3 1 1 −1 −1 1 1 −1 −1
168.20.1h2 C 1 1 1 −1 ζ3 ζ3−1 −1 ζ3−1 ζ3 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1 −ζ3 −ζ3−1 1 1 −1 −1 1 1 −1 −1
168.20.2a R 2 −2 0 0 2 2 0 −2 −2 0 0 0 0 2 2 0 0 −2 −2 0 0 0 0 0 0
168.20.2b1 C 2 −2 0 0 2ζ3−1 2ζ3 0 −2ζ3 −2ζ3−1 0 0 0 0 2 2 0 0 −2 −2 0 0 0 0 0 0
168.20.2b2 C 2 −2 0 0 2ζ3 2ζ3−1 0 −2ζ3−1 −2ζ3 0 0 0 0 2 2 0 0 −2 −2 0 0 0 0 0 0
168.20.3a1 C 3 3 3 3 0 0 3 0 0 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
168.20.3a2 C 3 3 3 3 0 0 3 0 0 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
168.20.3b1 C 3 3 −3 −3 0 0 3 0 0 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
168.20.3b2 C 3 3 −3 −3 0 0 3 0 0 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
168.20.3c1 C 3 3 −3 3 0 0 −3 0 0 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72
168.20.3c2 C 3 3 −3 3 0 0 −3 0 0 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72
168.20.3d1 C 3 3 3 −3 0 0 −3 0 0 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72
168.20.3d2 C 3 3 3 −3 0 0 −3 0 0 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72
168.20.6a1 C 6 −6 0 0 0 0 0 0 0 0 0 0 0 −2ζ7−3−2−2ζ7−2ζ72 2ζ7−3+2ζ7+2ζ72 0 0 −2ζ7−3−2ζ7−2ζ72 2ζ7−3+2+2ζ7+2ζ72 0 0 0 0 0 0
168.20.6a2 C 6 −6 0 0 0 0 0 0 0 0 0 0 0 2ζ7−3+2ζ7+2ζ72 −2ζ7−3−2−2ζ7−2ζ72 0 0 2ζ7−3+2+2ζ7+2ζ72 −2ζ7−3−2ζ7−2ζ72 0 0 0 0 0 0

magma: CharacterTable(G);