↑

Properties

Label 28T44
Degree $28$
Order $336$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $F_8:C_6$

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(28, 44);
 

Group action invariants

Degree $n$:  $28$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $44$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $F_8:C_6$
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:  (1,10,18,26,20,13,22,15,24,4,12,6,27,8)(2,9,17,25,19,14,21,16,23,3,11,5,28,7), (1,13,9,15,27,23)(2,14,10,16,28,24)(3,17)(4,18)(5,8,12,19,22,26)(6,7,11,20,21,25)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$21$:  $C_7:C_3$
$42$:  $(C_7:C_3) \times C_2$
$168$:  $C_2^3:(C_7: C_3)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 7: $C_7:C_3$

Degree 14: $(C_7:C_3) \times C_2$, 14T11, 14T18

Low degree siblings

14T18, 16T712, 42T67

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $336=2^{4} \cdot 3 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  336.210
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A1 3A-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 7A1 7A-1 14A1 14A-1
Size 1 1 7 7 28 28 28 28 28 28 28 28 24 24 24 24
2 P 1A 1A 1A 1A 3A-1 3A1 3A1 3A-1 3A-1 3A1 3A-1 3A1 7A1 7A-1 7A1 7A-1
3 P 1A 2A 2B 2C 1A 1A 2B 2C 2B 2A 2A 2C 7A-1 7A1 14A-1 14A1
7 P 1A 2A 2B 2C 3A1 3A-1 6B1 6C-1 6B-1 6A1 6A-1 6C1 1A 1A 2A 2A
Type
336.210.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
336.210.1b R 1 −1 −1 1 1 1 −1 −1 −1 −1 1 1 1 1 −1 −1
336.210.1c1 C 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 1 1
336.210.1c2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 1 1 1 1
336.210.1d1 C 1 −1 −1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3 ζ3−1 1 1 −1 −1
336.210.1d2 C 1 −1 −1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3−1 ζ3 1 1 −1 −1
336.210.3a1 C 3 3 3 3 0 0 0 0 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
336.210.3a2 C 3 3 3 3 0 0 0 0 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
336.210.3b1 C 3 −3 −3 3 0 0 0 0 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72
336.210.3b2 C 3 −3 −3 3 0 0 0 0 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72
336.210.7a R 7 7 −1 −1 1 1 1 1 −1 −1 −1 −1 0 0 0 0
336.210.7b R 7 −7 1 −1 1 1 −1 −1 1 1 −1 −1 0 0 0 0
336.210.7c1 C 7 7 −1 −1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 0 0 0 0
336.210.7c2 C 7 7 −1 −1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 0 0 0 0
336.210.7d1 C 7 −7 1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 ζ3 ζ3−1 −ζ3 −ζ3−1 0 0 0 0
336.210.7d2 C 7 −7 1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 ζ3−1 ζ3 −ζ3−1 −ζ3 0 0 0 0

magma: CharacterTable(G);