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Properties

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

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Show commands: Magma

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

Group invariants

Abstract group:  $C_7^2:C_3$
magma: IdentifyGroup(G);
 
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
magma: NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $21$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $12$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(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

$\card{(G/N)}$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 TypeSizeOrderIndexRepresentative
1A $1^{21}$ $1$ $1$ $0$ $()$
3A1 $3^{7}$ $49$ $3$ $14$ $( 1,14,17)( 2, 8,16)( 3, 9,15)( 4,10,21)( 5,11,20)( 6,12,19)( 7,13,18)$
3A-1 $3^{7}$ $49$ $3$ $14$ $( 1,17,14)( 2,16, 8)( 3,15, 9)( 4,21,10)( 5,20,11)( 6,19,12)( 7,18,13)$
7A1 $7^{3}$ $3$ $7$ $18$ $( 1, 2, 3, 4, 5, 6, 7)( 8, 9,10,11,12,13,14)(15,17,19,21,16,18,20)$
7A-1 $7^{3}$ $3$ $7$ $18$ $( 1, 5, 2, 6, 3, 7, 4)( 8,14,13,12,11,10, 9)(15,18,21,17,20,16,19)$
7B1 $7^{2},1^{7}$ $3$ $7$ $12$ $( 1, 5, 2, 6, 3, 7, 4)(15,19,16,20,17,21,18)$
7B-1 $7^{3}$ $3$ $7$ $18$ $( 1, 3, 5, 7, 2, 4, 6)( 8,14,13,12,11,10, 9)(15,16,17,18,19,20,21)$
7C1 $7^{2},1^{7}$ $3$ $7$ $12$ $( 8,12, 9,13,10,14,11)(15,19,16,20,17,21,18)$
7C-1 $7^{3}$ $3$ $7$ $18$ $( 1, 6, 4, 2, 7, 5, 3)( 8,12, 9,13,10,14,11)(15,17,19,21,16,18,20)$
7C2 $7^{3}$ $3$ $7$ $18$ $( 1, 2, 3, 4, 5, 6, 7)( 8,12, 9,13,10,14,11)(15,20,18,16,21,19,17)$
7C-2 $7^{2},1^{7}$ $3$ $7$ $12$ $( 1, 7, 6, 5, 4, 3, 2)(15,21,20,19,18,17,16)$
7C3 $7^{2},1^{7}$ $3$ $7$ $12$ $( 8,13,11, 9,14,12,10)(15,20,18,16,21,19,17)$
7C-3 $7^{3}$ $3$ $7$ $18$ $( 1, 6, 4, 2, 7, 5, 3)( 8,14,13,12,11,10, 9)(15,19,16,20,17,21,18)$
7D1 $7^{2},1^{7}$ $3$ $7$ $12$ $( 1, 7, 6, 5, 4, 3, 2)( 8, 9,10,11,12,13,14)$
7D-1 $7^{3}$ $3$ $7$ $18$ $( 1, 4, 7, 3, 6, 2, 5)( 8,11,14,10,13, 9,12)(15,21,20,19,18,17,16)$
7D2 $7^{2},1^{7}$ $3$ $7$ $12$ $( 1, 3, 5, 7, 2, 4, 6)( 8,13,11, 9,14,12,10)$
7D-2 $7^{3}$ $3$ $7$ $18$ $( 1, 2, 3, 4, 5, 6, 7)( 8,10,12,14, 9,11,13)(15,18,21,17,20,16,19)$
7D3 $7^{3}$ $3$ $7$ $18$ $( 1, 6, 4, 2, 7, 5, 3)( 8,11,14,10,13, 9,12)(15,16,17,18,19,20,21)$
7D-3 $7^{3}$ $3$ $7$ $18$ $( 1, 4, 7, 3, 6, 2, 5)( 8,10,12,14, 9,11,13)(15,20,18,16,21,19,17)$

Malle's constant $a(G)$:     $1/12$

magma: ConjugacyClasses(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 7C1 7C-3 7D-1 7C-1 7D1 7C-2 7A-1 7D2 7D3 7A1 7D-2 7C3 7D-3 7B-1 7B1 7C2
7 P 1A 1A 1A 7C-2 7C-1 7D2 7C2 7D-2 7C-3 7A1 7D3 7D1 7A-1 7D-3 7C1 7D-1 7B1 7B-1 7C3
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);