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Properties

Label 21T27
Degree $21$
Order $1008$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $S_3\times \GL(3,2)$

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Show commands: Magma

magma: G := TransitiveGroup(21, 27);
 

Group action invariants

Degree $n$:  $21$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $27$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3\times \GL(3,2)$
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:  (1,16,19)(2,18,20,3,17,21)(4,7,13)(5,9,14,6,8,15)(11,12), (1,18,8)(2,16,9)(3,17,7)(4,6,5)(10,21,14)(11,19,15)(12,20,13)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$
$168$:  $\GL(3,2)$
$336$:  14T17

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 7: $\GL(3,2)$

Low degree siblings

21T27, 24T2671, 42T169 x 2, 42T170 x 2, 42T171 x 2, 42T175 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)$
$ 6, 6, 3, 3, 3 $ $42$ $6$ $( 1, 2, 3)( 4,11, 6,10, 5,12)( 7,20, 9,19, 8,21)(13,14,15)(16,17,18)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $21$ $2$ $( 4,10)( 5,11)( 6,12)( 7,19)( 8,20)( 9,21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $63$ $2$ $( 1, 3)( 4,12)( 5,11)( 6,10)( 7,21)( 8,20)( 9,19)(13,15)(16,18)$
$ 12, 6, 3 $ $84$ $12$ $( 1, 3, 2)( 4,12, 5,10, 6,11)( 7,18,20,13, 9,17,19,15, 8,16,21,14)$
$ 4, 4, 4, 2, 2, 2, 1, 1, 1 $ $42$ $4$ $( 4,10)( 5,11)( 6,12)( 7,16,19,13)( 8,17,20,14)( 9,18,21,15)$
$ 4, 4, 4, 2, 2, 2, 2, 1 $ $126$ $4$ $( 1, 2)( 4,11)( 5,10)( 6,12)( 7,17,19,14)( 8,16,20,13)( 9,18,21,15)$
$ 3, 3, 3, 3, 3, 3, 3 $ $112$ $3$ $( 1, 2, 3)( 4,17,21)( 5,18,19)( 6,16,20)( 7,11,15)( 8,12,13)( 9,10,14)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $56$ $3$ $( 4,16,19)( 5,17,20)( 6,18,21)( 7,10,13)( 8,11,14)( 9,12,15)$
$ 6, 6, 3, 3, 2, 1 $ $168$ $6$ $( 1, 3)( 4,18,19, 6,16,21)( 5,17,20)( 7,12,13, 9,10,15)( 8,11,14)$
$ 7, 7, 7 $ $24$ $7$ $( 1,16,19, 7,10,13, 4)( 2,17,20, 8,11,14, 5)( 3,18,21, 9,12,15, 6)$
$ 21 $ $48$ $21$ $( 1,18,20, 7,12,14, 4, 3,17,19, 9,11,13, 6, 2,16,21, 8,10,15, 5)$
$ 14, 7 $ $72$ $14$ $( 1,16,19, 7,10,13, 4)( 2,18,20, 9,11,15, 5, 3,17,21, 8,12,14, 6)$
$ 21 $ $48$ $21$ $( 1,17,21,13, 5,12, 7, 2,18,19,14, 6,10, 8, 3,16,20,15, 4,11, 9)$
$ 7, 7, 7 $ $24$ $7$ $( 1,16,19,13, 4,10, 7)( 2,17,20,14, 5,11, 8)( 3,18,21,15, 6,12, 9)$
$ 14, 7 $ $72$ $14$ $( 1,18,19,15, 4,12, 7, 3,16,21,13, 6,10, 9)( 2,17,20,14, 5,11, 8)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $1008=2^{4} \cdot 3^{2} \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  1008.883
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 3C 4A 4B 6A 6B 7A1 7A-1 12A 14A1 14A-1 21A1 21A-1
Size 1 3 21 63 2 56 112 42 126 42 168 24 24 84 72 72 48 48
2 P 1A 1A 1A 1A 3A 3B 3C 2B 2B 3A 3B 7A1 7A-1 6A 7A1 7A-1 21A1 21A-1
3 P 1A 2A 2B 2C 1A 1A 1A 4A 4B 2B 2A 7A-1 7A1 4A 14A-1 14A1 7A1 7A-1
7 P 1A 2A 2B 2C 3A 3B 3C 4A 4B 6A 6B 1A 1A 12A 2A 2A 3A 3A
Type
1008.883.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1008.883.1b R 1 −1 1 −1 1 1 1 1 −1 1 −1 1 1 1 −1 −1 1 1
1008.883.2a R 2 0 2 0 −1 2 −1 2 0 −1 0 2 2 −1 0 0 −1 −1
1008.883.3a1 C 3 3 −1 −1 3 0 0 1 1 −1 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 1 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
1008.883.3a2 C 3 3 −1 −1 3 0 0 1 1 −1 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 1 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
1008.883.3b1 C 3 −3 −1 1 3 0 0 1 −1 −1 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 1 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
1008.883.3b2 C 3 −3 −1 1 3 0 0 1 −1 −1 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 1 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
1008.883.6a R 6 6 2 2 6 0 0 0 0 2 0 −1 −1 0 −1 −1 −1 −1
1008.883.6b R 6 −6 2 −2 6 0 0 0 0 2 0 −1 −1 0 1 1 −1 −1
1008.883.6c1 C 6 0 −2 0 −3 0 0 2 0 1 0 −2ζ7−3−2−2ζ7−2ζ72 2ζ7−3+2ζ7+2ζ72 −1 0 0 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72
1008.883.6c2 C 6 0 −2 0 −3 0 0 2 0 1 0 2ζ7−3+2ζ7+2ζ72 −2ζ7−3−2−2ζ7−2ζ72 −1 0 0 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72
1008.883.7a R 7 7 −1 −1 7 1 1 −1 −1 −1 1 0 0 −1 0 0 0 0
1008.883.7b R 7 −7 −1 1 7 1 1 −1 1 −1 −1 0 0 −1 0 0 0 0
1008.883.8a R 8 8 0 0 8 −1 −1 0 0 0 −1 1 1 0 1 1 1 1
1008.883.8b R 8 −8 0 0 8 −1 −1 0 0 0 1 1 1 0 −1 −1 1 1
1008.883.12a R 12 0 4 0 −6 0 0 0 0 −2 0 −2 −2 0 0 0 1 1
1008.883.14a R 14 0 −2 0 −7 2 −1 −2 0 1 0 0 0 1 0 0 0 0
1008.883.16a R 16 0 0 0 −8 −2 1 0 0 0 0 2 2 0 0 0 −1 −1

magma: CharacterTable(G);