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Properties

Label 21T21
Degree $21$
Order $441$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7^2:C_3^2$

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Show commands: Magma

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

Group action invariants

Degree $n$:  $21$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $21$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_7^2:C_3^2$
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:  (1,2,4)(3,6,5)(8,12,13)(9,14,10)(15,18,17)(16,20,21), (1,8,21,7,10,17,6,12,20,5,14,16,4,9,19,3,11,15,2,13,18)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$ x 4
$9$:  $C_3^2$
$21$:  $C_7:C_3$ x 2
$63$:  21T7 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: None

Low degree siblings

21T21

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 $ $1$ $1$ $()$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $9$ $7$ $( 8, 9,10,11,12,13,14)(15,16,17,18,19,20,21)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $9$ $7$ $( 8,11,14,10,13, 9,12)(15,18,21,17,20,16,19)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $49$ $3$ $( 2, 3, 5)( 4, 7, 6)( 9,10,12)(11,14,13)(16,17,19)(18,21,20)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $49$ $3$ $( 2, 5, 3)( 4, 6, 7)( 9,12,10)(11,13,14)(16,19,17)(18,20,21)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,10,12,14, 9,11,13)(15,16,17,18,19,20,21)$
$ 7, 7, 7 $ $3$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,11,14,10,13, 9,12)(15,17,19,21,16,18,20)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,12, 9,13,10,14,11)(15,18,21,17,20,16,19)$
$ 7, 7, 7 $ $3$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,13,11, 9,14,12,10)(15,19,16,20,17,21,18)$
$ 7, 7, 7 $ $3$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8, 9,10,11,12,13,14)(15,20,18,16,21,19,17)$
$ 7, 7, 7 $ $3$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8,10,12,14, 9,11,13)(15,21,20,19,18,17,16)$
$ 3, 3, 3, 3, 3, 3, 3 $ $7$ $3$ $( 1, 8,15)( 2,11,17)( 3,14,19)( 4,10,21)( 5,13,16)( 6, 9,18)( 7,12,20)$
$ 21 $ $21$ $21$ $( 1, 8,16, 4,10,15, 7,12,21, 3,14,20, 6, 9,19, 2,11,18, 5,13,17)$
$ 21 $ $21$ $21$ $( 1, 8,18, 3,14,15, 5,13,19, 7,12,16, 2,11,20, 4,10,17, 6, 9,21)$
$ 3, 3, 3, 3, 3, 3, 3 $ $7$ $3$ $( 1, 8,15)( 2,13,19)( 3,11,16)( 4, 9,20)( 5,14,17)( 6,12,21)( 7,10,18)$
$ 21 $ $21$ $21$ $( 1, 8,16, 2,13,20, 3,11,17, 4, 9,21, 5,14,18, 6,12,15, 7,10,19)$
$ 21 $ $21$ $21$ $( 1, 8,18, 4, 9,16, 7,10,21, 3,11,19, 6,12,17, 2,13,15, 5,14,20)$
$ 3, 3, 3, 3, 3, 3, 3 $ $49$ $3$ $( 1, 8,15)( 2,14,16)( 3,13,17)( 4,12,18)( 5,11,19)( 6,10,20)( 7, 9,21)$
$ 3, 3, 3, 3, 3, 3, 3 $ $49$ $3$ $( 1,15, 8)( 2,16,14)( 3,17,13)( 4,18,12)( 5,19,11)( 6,20,10)( 7,21, 9)$
$ 3, 3, 3, 3, 3, 3, 3 $ $7$ $3$ $( 1,15, 8)( 2,17,11)( 3,19,14)( 4,21,10)( 5,16,13)( 6,18, 9)( 7,20,12)$
$ 21 $ $21$ $21$ $( 1,15, 9, 7,20,13, 6,18,10, 5,16,14, 4,21,11, 3,19, 8, 2,17,12)$
$ 21 $ $21$ $21$ $( 1,15,11, 5,16, 9, 2,17,14, 6,18,12, 3,19,10, 7,20, 8, 4,21,13)$
$ 3, 3, 3, 3, 3, 3, 3 $ $7$ $3$ $( 1,15, 8)( 2,19,13)( 3,16,11)( 4,20, 9)( 5,17,14)( 6,21,12)( 7,18,10)$
$ 21 $ $21$ $21$ $( 1,15, 9, 5,17, 8, 2,19,14, 6,21,13, 3,16,12, 7,18,11, 4,20,10)$
$ 21 $ $21$ $21$ $( 1,15,11, 6,21, 8, 4,20,12, 2,19, 9, 7,18,13, 5,17,10, 3,16,14)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $441=3^{2} \cdot 7^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  441.9
magma: IdentifyGroup(G);
 
Character table:

1A 3A1 3A-1 3B1 3B-1 3C1 3C-1 3D1 3D-1 7A1 7A-1 7B1 7B-1 7C1 7C-1 7D1 7D-1 21A1 21A-1 21A2 21A-2 21B1 21B-1 21B2 21B-2
Size 1 7 7 7 7 49 49 49 49 3 3 3 3 9 9 9 9 21 21 21 21 21 21 21 21
3 P 1A 3A-1 3A1 3B-1 3B1 3C1 3C-1 3D1 3D-1 7A1 7A-1 7B1 7B-1 7D-1 7C-1 7C1 7D1 21A-1 21A2 21B-1 21A1 21B2 21B-2 21B1 21A-2
7 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 7A-1 7A1 7B-1 7B1 7D1 7C1 7C-1 7D-1 7A-1 7A1 7B-1 7A1 7B1 7B-1 7B1 7A-1
Type
441.9.1a 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
441.9.1b1 C 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 1 1 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3
441.9.1b2 C 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1
441.9.1c1 C 1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3 ζ3−1 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
441.9.1c2 C 1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
441.9.1d1 C 1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1
441.9.1d2 C 1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1
441.9.1e1 C 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1
441.9.1e2 C 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3
441.9.3a1 C 3 0 0 3 3 0 0 0 0 3 3 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
441.9.3a2 C 3 0 0 3 3 0 0 0 0 3 3 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
441.9.3b1 C 3 3 3 0 0 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 3 3 −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 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 0 0 0 0
441.9.3b2 C 3 3 3 0 0 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 3 3 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 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 0 0 0 0
441.9.3c1 C 3 3ζ21−7 3ζ217 0 0 0 0 0 0 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 3 3 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10+1−ζ212+ζ217−ζ218 ζ21−ζ212+ζ214−ζ219 −ζ21+ζ212−ζ214−ζ217+ζ219 ζ21−10+ζ212+ζ218 0 0 0 0
441.9.3c2 C 3 3ζ217 3ζ21−7 0 0 0 0 0 0 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 3 3 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−ζ212+ζ214−ζ219 −ζ21−10+1−ζ212+ζ217−ζ218 ζ21−10+ζ212+ζ218 −ζ21+ζ212−ζ214−ζ217+ζ219 0 0 0 0
441.9.3c3 C 3 3ζ21−7 3ζ217 0 0 0 0 0 0 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 3 3 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10+ζ212+ζ218 −ζ21+ζ212−ζ214−ζ217+ζ219 ζ21−ζ212+ζ214−ζ219 −ζ21−10+1−ζ212+ζ217−ζ218 0 0 0 0
441.9.3c4 C 3 3ζ217 3ζ21−7 0 0 0 0 0 0 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 3 3 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21+ζ212−ζ214−ζ217+ζ219 ζ21−10+ζ212+ζ218 −ζ21−10+1−ζ212+ζ217−ζ218 ζ21−ζ212+ζ214−ζ219 0 0 0 0
441.9.3d1 C 3 0 0 3ζ21−7 3ζ217 0 0 0 0 3 3 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 0 0 0 0 −ζ21−10+1−ζ212+ζ217−ζ218 ζ21−ζ212+ζ214−ζ219 −ζ21+ζ212−ζ214−ζ217+ζ219 ζ21−10+ζ212+ζ218
441.9.3d2 C 3 0 0 3ζ217 3ζ21−7 0 0 0 0 3 3 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 0 0 0 0 ζ21−ζ212+ζ214−ζ219 −ζ21−10+1−ζ212+ζ217−ζ218 ζ21−10+ζ212+ζ218 −ζ21+ζ212−ζ214−ζ217+ζ219
441.9.3d3 C 3 0 0 3ζ21−7 3ζ217 0 0 0 0 3 3 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 0 0 0 0 ζ21−10+ζ212+ζ218 −ζ21+ζ212−ζ214−ζ217+ζ219 ζ21−ζ212+ζ214−ζ219 −ζ21−10+1−ζ212+ζ217−ζ218
441.9.3d4 C 3 0 0 3ζ217 3ζ21−7 0 0 0 0 3 3 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 0 0 0 0 −ζ21+ζ212−ζ214−ζ217+ζ219 ζ21−10+ζ212+ζ218 −ζ21−10+1−ζ212+ζ217−ζ218 ζ21−ζ212+ζ214−ζ219
441.9.9a1 C 9 0 0 0 0 0 0 0 0 −3ζ7−3−3−3ζ7−3ζ72 3ζ7−3+3ζ7+3ζ72 3ζ7−3+3ζ7+3ζ72 −3ζ7−3−3−3ζ7−3ζ72 ζ7−3−1+ζ7+ζ72 −ζ7−3−2−ζ7−ζ72 2 2 0 0 0 0 0 0 0 0
441.9.9a2 C 9 0 0 0 0 0 0 0 0 3ζ7−3+3ζ7+3ζ72 −3ζ7−3−3−3ζ7−3ζ72 −3ζ7−3−3−3ζ7−3ζ72 3ζ7−3+3ζ7+3ζ72 −ζ7−3−2−ζ7−ζ72 ζ7−3−1+ζ7+ζ72 2 2 0 0 0 0 0 0 0 0
441.9.9b1 C 9 0 0 0 0 0 0 0 0 −3ζ7−3−3−3ζ7−3ζ72 3ζ7−3+3ζ7+3ζ72 −3ζ7−3−3−3ζ7−3ζ72 3ζ7−3+3ζ7+3ζ72 2 2 ζ7−3−1+ζ7+ζ72 −ζ7−3−2−ζ7−ζ72 0 0 0 0 0 0 0 0
441.9.9b2 C 9 0 0 0 0 0 0 0 0 3ζ7−3+3ζ7+3ζ72 −3ζ7−3−3−3ζ7−3ζ72 3ζ7−3+3ζ7+3ζ72 −3ζ7−3−3−3ζ7−3ζ72 2 2 −ζ7−3−2−ζ7−ζ72 ζ7−3−1+ζ7+ζ72 0 0 0 0 0 0 0 0

magma: CharacterTable(G);