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Properties

Label 21T1
Degree $21$
Order $21$
Cyclic yes
Abelian yes
Solvable yes
Primitive no
$p$-group no
Group: $C_{21}$

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Show commands: Magma

magma: G := TransitiveGroup(21, 1);
 

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$
$7$:  $C_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: $C_7$

Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $21=3 \cdot 7$
magma: Order(G);
 
Cyclic:  yes
magma: IsCyclic(G);
 
Abelian:  yes
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $1$
Label:  21.2
magma: IdentifyGroup(G);
 
Character table:

1A 3A1 3A-1 7A1 7A-1 7A2 7A-2 7A3 7A-3 21A1 21A-1 21A2 21A-2 21A4 21A-4 21A5 21A-5 21A8 21A-8 21A10 21A-10
Size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
3 P 1A 3A-1 3A1 7A2 7A-3 7A-2 7A1 7A-1 7A3 21A-1 21A-8 21A8 21A-5 21A-10 21A5 21A10 21A1 21A2 21A-4 21A4 21A-2
7 P 1A 1A 1A 7A3 7A-1 7A-3 7A-2 7A2 7A1 7A3 7A3 7A-3 7A1 7A2 7A-1 7A-2 7A-3 7A1 7A-2 7A2 7A-1
Type
21.2.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
21.2.1b1 C 1 ζ3−1 ζ3 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3
21.2.1b2 C 1 ζ3 ζ3−1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1
21.2.1c1 C 1 1 1 ζ7−3 ζ73 ζ7 ζ7−1 ζ7−2 ζ72 ζ7−1 ζ7 ζ7−2 ζ72 ζ73 ζ7−3 ζ72 ζ7−2 ζ7−1 ζ7 ζ7−3 ζ73
21.2.1c2 C 1 1 1 ζ73 ζ7−3 ζ7−1 ζ7 ζ72 ζ7−2 ζ7 ζ7−1 ζ72 ζ7−2 ζ7−3 ζ73 ζ7−2 ζ72 ζ7 ζ7−1 ζ73 ζ7−3
21.2.1c3 C 1 1 1 ζ7−2 ζ72 ζ73 ζ7−3 ζ7 ζ7−1 ζ7−3 ζ73 ζ7 ζ7−1 ζ72 ζ7−2 ζ7−1 ζ7 ζ7−3 ζ73 ζ7−2 ζ72
21.2.1c4 C 1 1 1 ζ72 ζ7−2 ζ7−3 ζ73 ζ7−1 ζ7 ζ73 ζ7−3 ζ7−1 ζ7 ζ7−2 ζ72 ζ7 ζ7−1 ζ73 ζ7−3 ζ72 ζ7−2
21.2.1c5 C 1 1 1 ζ7−1 ζ7 ζ7−2 ζ72 ζ7−3 ζ73 ζ72 ζ7−2 ζ7−3 ζ73 ζ7 ζ7−1 ζ73 ζ7−3 ζ72 ζ7−2 ζ7−1 ζ7
21.2.1c6 C 1 1 1 ζ7 ζ7−1 ζ72 ζ7−2 ζ73 ζ7−3 ζ7−2 ζ72 ζ73 ζ7−3 ζ7−1 ζ7 ζ7−3 ζ73 ζ7−2 ζ72 ζ7 ζ7−1
21.2.1d1 C 1 ζ21−7 ζ217 ζ21−9 ζ219 ζ213 ζ21−3 ζ21−6 ζ216 ζ21−10 ζ2110 ζ21 ζ21−1 ζ212 ζ21−2 ζ21−8 ζ218 ζ214 ζ21−4 ζ215 ζ21−5
21.2.1d2 C 1 ζ217 ζ21−7 ζ219 ζ21−9 ζ21−3 ζ213 ζ216 ζ21−6 ζ2110 ζ21−10 ζ21−1 ζ21 ζ21−2 ζ212 ζ218 ζ21−8 ζ21−4 ζ214 ζ21−5 ζ215
21.2.1d3 C 1 ζ21−7 ζ217 ζ219 ζ21−9 ζ21−3 ζ213 ζ216 ζ21−6 ζ21−4 ζ214 ζ21−8 ζ218 ζ215 ζ21−5 ζ21 ζ21−1 ζ2110 ζ21−10 ζ212 ζ21−2
21.2.1d4 C 1 ζ217 ζ21−7 ζ21−9 ζ219 ζ213 ζ21−3 ζ21−6 ζ216 ζ214 ζ21−4 ζ218 ζ21−8 ζ21−5 ζ215 ζ21−1 ζ21 ζ21−10 ζ2110 ζ21−2 ζ212
21.2.1d5 C 1 ζ21−7 ζ217 ζ21−6 ζ216 ζ219 ζ21−9 ζ213 ζ21−3 ζ215 ζ21−5 ζ2110 ζ21−10 ζ21−1 ζ21 ζ214 ζ21−4 ζ21−2 ζ212 ζ218 ζ21−8
21.2.1d6 C 1 ζ217 ζ21−7 ζ216 ζ21−6 ζ21−9 ζ219 ζ21−3 ζ213 ζ21−5 ζ215 ζ21−10 ζ2110 ζ21 ζ21−1 ζ21−4 ζ214 ζ212 ζ21−2 ζ21−8 ζ218
21.2.1d7 C 1 ζ21−7 ζ217 ζ216 ζ21−6 ζ21−9 ζ219 ζ21−3 ζ213 ζ212 ζ21−2 ζ214 ζ21−4 ζ218 ζ21−8 ζ2110 ζ21−10 ζ21−5 ζ215 ζ21−1 ζ21
21.2.1d8 C 1 ζ217 ζ21−7 ζ21−6 ζ216 ζ219 ζ21−9 ζ213 ζ21−3 ζ21−2 ζ212 ζ21−4 ζ214 ζ21−8 ζ218 ζ21−10 ζ2110 ζ215 ζ21−5 ζ21 ζ21−1
21.2.1d9 C 1 ζ21−7 ζ217 ζ21−3 ζ213 ζ21−6 ζ216 ζ21−9 ζ219 ζ21−1 ζ21 ζ21−2 ζ212 ζ21−4 ζ214 ζ21−5 ζ215 ζ21−8 ζ218 ζ21−10 ζ2110
21.2.1d10 C 1 ζ217 ζ21−7 ζ213 ζ21−3 ζ216 ζ21−6 ζ219 ζ21−9 ζ21 ζ21−1 ζ212 ζ21−2 ζ214 ζ21−4 ζ215 ζ21−5 ζ218 ζ21−8 ζ2110 ζ21−10
21.2.1d11 C 1 ζ21−7 ζ217 ζ213 ζ21−3 ζ216 ζ21−6 ζ219 ζ21−9 ζ218 ζ21−8 ζ21−5 ζ215 ζ21−10 ζ2110 ζ21−2 ζ212 ζ21 ζ21−1 ζ21−4 ζ214
21.2.1d12 C 1 ζ217 ζ21−7 ζ21−3 ζ213 ζ21−6 ζ216 ζ21−9 ζ219 ζ21−8 ζ218 ζ215 ζ21−5 ζ2110 ζ21−10 ζ212 ζ21−2 ζ21−1 ζ21 ζ214 ζ21−4

magma: CharacterTable(G);