↑

Properties

Label 21T12
Degree $21$
Order $147$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7^2:C_3$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(21, 12);
 

Group action invariants

Degree $n$:  $21$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $12$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_7^2:C_3$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $7$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,17,14)(2,16,8)(3,15,9)(4,21,10)(5,20,11)(6,19,12)(7,18,13), (1,20,10)(2,19,11)(3,18,12)(4,17,13)(5,16,14)(6,15,8)(7,21,9)
magma: Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: None

Low degree siblings

21T12

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $147=3 \cdot 7^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  147.5
magma: IdentifyGroup(G);
 
Character table:

1A 3A1 3A-1 7A1 7A-1 7B1 7B-1 7C1 7C-1 7C2 7C-2 7C3 7C-3 7D1 7D-1 7D2 7D-2 7D3 7D-3
Size 1 49 49 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
3 P 1A 3A-1 3A1 7D3 7B1 7D-1 7C1 7D-3 7A-1 7D1 7C-2 7C-3 7C2 7C-1 7A1 7D-2 7D2 7C3 7B-1
7 P 1A 1A 1A 7D1 7B-1 7D2 7C-2 7D-1 7A1 7D-2 7C-3 7C-1 7C3 7C2 7A-1 7D-3 7D3 7C1 7B1
Type
147.5.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
147.5.1b1 C 1 ζ3−1 ζ3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
147.5.1b2 C 1 ζ3 ζ3−1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
147.5.3a1 C 3 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 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
147.5.3a2 C 3 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 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
147.5.3b1 C 3 0 0 3 3 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −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 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
147.5.3b2 C 3 0 0 3 3 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 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+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
147.5.3c1 C 3 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 2ζ72+ζ73 ζ7−3+2ζ7−2 2ζ7−3+ζ7−1 ζ7+2ζ73 2ζ7−1+ζ72 ζ7−2+2ζ7 ζ7−1+1+ζ7 ζ7−1+1+ζ7 ζ7−2+1+ζ72 ζ7−2+1+ζ72 ζ7−3+1+ζ73 ζ7−3+1+ζ73
147.5.3c2 C 3 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+2ζ7−2 2ζ72+ζ73 ζ7+2ζ73 2ζ7−3+ζ7−1 ζ7−2+2ζ7 2ζ7−1+ζ72 ζ7−1+1+ζ7 ζ7−1+1+ζ7 ζ7−2+1+ζ72 ζ7−2+1+ζ72 ζ7−3+1+ζ73 ζ7−3+1+ζ73
147.5.3c3 C 3 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 2ζ7−3+ζ7−1 ζ7+2ζ73 ζ7−2+2ζ7 2ζ7−1+ζ72 ζ7−3+2ζ7−2 2ζ72+ζ73 ζ7−2+1+ζ72 ζ7−2+1+ζ72 ζ7−3+1+ζ73 ζ7−3+1+ζ73 ζ7−1+1+ζ7 ζ7−1+1+ζ7
147.5.3c4 C 3 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7+2ζ73 2ζ7−3+ζ7−1 2ζ7−1+ζ72 ζ7−2+2ζ7 2ζ72+ζ73 ζ7−3+2ζ7−2 ζ7−2+1+ζ72 ζ7−2+1+ζ72 ζ7−3+1+ζ73 ζ7−3+1+ζ73 ζ7−1+1+ζ7 ζ7−1+1+ζ7
147.5.3c5 C 3 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−2+2ζ7 2ζ7−1+ζ72 2ζ72+ζ73 ζ7−3+2ζ7−2 ζ7+2ζ73 2ζ7−3+ζ7−1 ζ7−3+1+ζ73 ζ7−3+1+ζ73 ζ7−1+1+ζ7 ζ7−1+1+ζ7 ζ7−2+1+ζ72 ζ7−2+1+ζ72
147.5.3c6 C 3 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 2ζ7−1+ζ72 ζ7−2+2ζ7 ζ7−3+2ζ7−2 2ζ72+ζ73 2ζ7−3+ζ7−1 ζ7+2ζ73 ζ7−3+1+ζ73 ζ7−3+1+ζ73 ζ7−1+1+ζ7 ζ7−1+1+ζ7 ζ7−2+1+ζ72 ζ7−2+1+ζ72
147.5.3d1 C 3 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+1+ζ73 ζ7−3+1+ζ73 ζ7−1+1+ζ7 ζ7−1+1+ζ7 ζ7−2+1+ζ72 ζ7−2+1+ζ72 ζ7+2ζ73 2ζ7−3+ζ7−1 2ζ7−1+ζ72 ζ7−2+2ζ7 2ζ72+ζ73 ζ7−3+2ζ7−2
147.5.3d2 C 3 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+1+ζ73 ζ7−3+1+ζ73 ζ7−1+1+ζ7 ζ7−1+1+ζ7 ζ7−2+1+ζ72 ζ7−2+1+ζ72 2ζ7−3+ζ7−1 ζ7+2ζ73 ζ7−2+2ζ7 2ζ7−1+ζ72 ζ7−3+2ζ7−2 2ζ72+ζ73
147.5.3d3 C 3 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−2+1+ζ72 ζ7−2+1+ζ72 ζ7−3+1+ζ73 ζ7−3+1+ζ73 ζ7−1+1+ζ7 ζ7−1+1+ζ7 ζ7−3+2ζ7−2 2ζ72+ζ73 ζ7+2ζ73 2ζ7−3+ζ7−1 ζ7−2+2ζ7 2ζ7−1+ζ72
147.5.3d4 C 3 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−2+1+ζ72 ζ7−2+1+ζ72 ζ7−3+1+ζ73 ζ7−3+1+ζ73 ζ7−1+1+ζ7 ζ7−1+1+ζ7 2ζ72+ζ73 ζ7−3+2ζ7−2 2ζ7−3+ζ7−1 ζ7+2ζ73 2ζ7−1+ζ72 ζ7−2+2ζ7
147.5.3d5 C 3 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−1+1+ζ7 ζ7−1+1+ζ7 ζ7−2+1+ζ72 ζ7−2+1+ζ72 ζ7−3+1+ζ73 ζ7−3+1+ζ73 2ζ7−1+ζ72 ζ7−2+2ζ7 ζ7−3+2ζ7−2 2ζ72+ζ73 2ζ7−3+ζ7−1 ζ7+2ζ73
147.5.3d6 C 3 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−1+1+ζ7 ζ7−1+1+ζ7 ζ7−2+1+ζ72 ζ7−2+1+ζ72 ζ7−3+1+ζ73 ζ7−3+1+ζ73 ζ7−2+2ζ7 2ζ7−1+ζ72 2ζ72+ζ73 ζ7−3+2ζ7−2 ζ7+2ζ73 2ζ7−3+ζ7−1

magma: CharacterTable(G);