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Properties

Label 21T22
Degree $21$
Order $504$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_3\times \GL(3,2)$

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Show commands: Magma

magma: G := TransitiveGroup(21, 22);
 

Group action invariants

Degree $n$:  $21$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $22$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3\times \GL(3,2)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $3$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2,3)(4,9,6,8,5,7)(10,20,12,19,11,21)(13,14,15)(16,17,18), (1,8,17,15,4,21,12)(2,9,18,13,5,19,10)(3,7,16,14,6,20,11)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$
$168$:  $\GL(3,2)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

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

Low degree siblings

21T22, 24T1355 x 2, 24T1356, 42T96 x 2, 42T103 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
$1^{21}$ $1$ $1$ $()$
$3^{7}$ $1$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)$
$3^{7}$ $1$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)$
$6^{2},3^{3}$ $21$ $6$ $( 1, 3, 2)( 4, 6, 5)( 7,13, 8,14, 9,15)(10,12,11)(16,19,17,20,18,21)$
$2^{6},1^{9}$ $21$ $2$ $( 7,14)( 8,15)( 9,13)(16,20)(17,21)(18,19)$
$6^{2},3^{3}$ $21$ $6$ $( 1, 2, 3)( 4, 5, 6)( 7,15, 9,14, 8,13)(10,11,12)(16,21,18,20,17,19)$
$12,6,3$ $42$ $12$ $( 1, 2, 3)( 4, 9,11,21, 5, 7,12,19, 6, 8,10,20)(13,16,15,18,14,17)$
$12,6,3$ $42$ $12$ $( 1, 3, 2)( 4, 7,10,21, 6, 9,12,20, 5, 8,11,19)(13,17,14,18,15,16)$
$4^{3},2^{3},1^{3}$ $42$ $4$ $( 4, 8,12,21)( 5, 9,10,19)( 6, 7,11,20)(13,18)(14,16)(15,17)$
$3^{7}$ $56$ $3$ $( 1, 2, 3)( 4, 9,14)( 5, 7,15)( 6, 8,13)(10,20,17)(11,21,18)(12,19,16)$
$3^{7}$ $56$ $3$ $( 1, 3, 2)( 4, 7,13)( 5, 8,14)( 6, 9,15)(10,21,16)(11,19,17)(12,20,18)$
$3^{6},1^{3}$ $56$ $3$ $( 4, 8,15)( 5, 9,13)( 6, 7,14)(10,19,18)(11,20,16)(12,21,17)$
$7^{3}$ $24$ $7$ $( 1, 4, 8,12,15,17,21)( 2, 5, 9,10,13,18,19)( 3, 6, 7,11,14,16,20)$
$21$ $24$ $21$ $( 1, 5, 7,12,13,16,21, 2, 6, 8,10,14,17,19, 3, 4, 9,11,15,18,20)$
$21$ $24$ $21$ $( 1, 6, 9,12,14,18,21, 3, 5, 8,11,13,17,20, 2, 4, 7,10,15,16,19)$
$7^{3}$ $24$ $7$ $( 1, 4, 8,21,17,12,15)( 2, 5, 9,19,18,10,13)( 3, 6, 7,20,16,11,14)$
$21$ $24$ $21$ $( 1, 5, 7,21,18,11,15, 2, 6, 8,19,16,12,13, 3, 4, 9,20,17,10,14)$
$21$ $24$ $21$ $( 1, 6, 9,21,16,10,15, 3, 5, 8,20,18,12,14, 2, 4, 7,19,17,11,13)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $504=2^{3} \cdot 3^{2} \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  504.157
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 3B 3C1 3C-1 4A 6A1 6A-1 7A1 7A-1 12A1 12A-1 21A1 21A-1 21A2 21A-2
Size 1 21 1 1 56 56 56 42 21 21 24 24 42 42 24 24 24 24
2 P 1A 1A 3A-1 3A1 3B 3C-1 3C1 2A 3A1 3A-1 7A1 7A-1 6A1 6A-1 21A2 21A-2 21A1 21A-1
3 P 1A 2A 1A 1A 1A 1A 1A 4A 2A 2A 7A-1 7A1 4A 4A 7A1 7A-1 7A1 7A-1
7 P 1A 2A 3A1 3A-1 3B 3C1 3C-1 4A 6A1 6A-1 1A 1A 12A1 12A-1 3A1 3A-1 3A-1 3A1
Type
504.157.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
504.157.1b1 C 1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1
504.157.1b2 C 1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3
504.157.3a1 C 3 −1 3 3 0 0 0 1 −1 −1 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 1 1 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
504.157.3a2 C 3 −1 3 3 0 0 0 1 −1 −1 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 1 1 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
504.157.3b1 C 3 −1 3ζ21−7 3ζ217 0 0 0 1 −ζ217 −ζ21−7 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 ζ21−7 ζ217 −ζ21−10+1−ζ212+ζ217−ζ218 ζ21−ζ212+ζ214−ζ219 −ζ21+ζ212−ζ214−ζ217+ζ219 ζ21−10+ζ212+ζ218
504.157.3b2 C 3 −1 3ζ217 3ζ21−7 0 0 0 1 −ζ21−7 −ζ217 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ217 ζ21−7 ζ21−ζ212+ζ214−ζ219 −ζ21−10+1−ζ212+ζ217−ζ218 ζ21−10+ζ212+ζ218 −ζ21+ζ212−ζ214−ζ217+ζ219
504.157.3b3 C 3 −1 3ζ21−7 3ζ217 0 0 0 1 −ζ217 −ζ21−7 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−7 ζ217 ζ21−10+ζ212+ζ218 −ζ21+ζ212−ζ214−ζ217+ζ219 ζ21−ζ212+ζ214−ζ219 −ζ21−10+1−ζ212+ζ217−ζ218
504.157.3b4 C 3 −1 3ζ217 3ζ21−7 0 0 0 1 −ζ21−7 −ζ217 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 ζ217 ζ21−7 −ζ21+ζ212−ζ214−ζ217+ζ219 ζ21−10+ζ212+ζ218 −ζ21−10+1−ζ212+ζ217−ζ218 ζ21−ζ212+ζ214−ζ219
504.157.6a R 6 2 6 6 0 0 0 0 2 2 −1 −1 0 0 −1 −1 −1 −1
504.157.6b1 C 6 2 6ζ3−1 6ζ3 0 0 0 0 2ζ3 2ζ3−1 −1 −1 0 0 −ζ3−1 −ζ3 −ζ3 −ζ3−1
504.157.6b2 C 6 2 6ζ3 6ζ3−1 0 0 0 0 2ζ3−1 2ζ3 −1 −1 0 0 −ζ3 −ζ3−1 −ζ3−1 −ζ3
504.157.7a R 7 −1 7 7 1 1 1 −1 −1 −1 0 0 −1 −1 0 0 0 0
504.157.7b1 C 7 −1 7ζ3−1 7ζ3 1 ζ3 ζ3−1 −1 −ζ3 −ζ3−1 0 0 −ζ3−1 −ζ3 0 0 0 0
504.157.7b2 C 7 −1 7ζ3 7ζ3−1 1 ζ3−1 ζ3 −1 −ζ3−1 −ζ3 0 0 −ζ3 −ζ3−1 0 0 0 0
504.157.8a R 8 0 8 8 −1 −1 −1 0 0 0 1 1 0 0 1 1 1 1
504.157.8b1 C 8 0 8ζ3−1 8ζ3 −1 −ζ3 −ζ3−1 0 0 0 1 1 0 0 ζ3−1 ζ3 ζ3 ζ3−1
504.157.8b2 C 8 0 8ζ3 8ζ3−1 −1 −ζ3−1 −ζ3 0 0 0 1 1 0 0 ζ3 ζ3−1 ζ3−1 ζ3

magma: CharacterTable(G);