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Properties

Label 21T18
Degree $21$
Order $294$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7^2:S_3$

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Show commands: Magma

magma: G := TransitiveGroup(21, 18);
 

Group invariants

Abstract group:  $C_7^2:S_3$
magma: IdentifyGroup(G);
 
Order:  $294=2 \cdot 3 \cdot 7^{2}$
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 $t$:  $18$
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:  $(1,17,3,15,5,20,7,18,2,16,4,21,6,19)(8,13,11,9,14,12,10)$, $(1,8,3,10,5,12,7,14,2,9,4,11,6,13)(15,17,19,21,16,18,20)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 7: None

Low degree siblings

14T15, 21T17, 42T56, 42T57, 42T62

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{21}$ $1$ $1$ $0$ $()$
2A $2^{7},1^{7}$ $21$ $2$ $7$ $( 8,21)( 9,20)(10,19)(11,18)(12,17)(13,16)(14,15)$
3A $3^{7}$ $98$ $3$ $14$ $( 1,14,18)( 2, 8,17)( 3, 9,16)( 4,10,15)( 5,11,21)( 6,12,20)( 7,13,19)$
7A1 $7^{3}$ $3$ $7$ $18$ $( 1, 4, 7, 3, 6, 2, 5)( 8,10,12,14, 9,11,13)(15,20,18,16,21,19,17)$
7A-1 $7^{3}$ $3$ $7$ $18$ $( 1, 7, 6, 5, 4, 3, 2)( 8,12, 9,13,10,14,11)(15,18,21,17,20,16,19)$
7A2 $7^{3}$ $3$ $7$ $18$ $( 1, 5, 2, 6, 3, 7, 4)( 8,13,11, 9,14,12,10)(15,17,19,21,16,18,20)$
7A-2 $7^{3}$ $3$ $7$ $18$ $( 1, 6, 4, 2, 7, 5, 3)( 8, 9,10,11,12,13,14)(15,21,20,19,18,17,16)$
7A3 $7^{3}$ $3$ $7$ $18$ $( 1, 3, 5, 7, 2, 4, 6)( 8,14,13,12,11,10, 9)(15,16,17,18,19,20,21)$
7A-3 $7^{3}$ $3$ $7$ $18$ $( 1, 2, 3, 4, 5, 6, 7)( 8,11,14,10,13, 9,12)(15,19,16,20,17,21,18)$
7B1 $7^{2},1^{7}$ $6$ $7$ $12$ $( 1, 5, 2, 6, 3, 7, 4)(15,19,16,20,17,21,18)$
7B-1 $7^{3}$ $6$ $7$ $18$ $( 1, 3, 5, 7, 2, 4, 6)( 8, 9,10,11,12,13,14)(15,18,21,17,20,16,19)$
7C1 $7^{2},1^{7}$ $6$ $7$ $12$ $( 1, 2, 3, 4, 5, 6, 7)(15,16,17,18,19,20,21)$
7C2 $7^{2},1^{7}$ $6$ $7$ $12$ $( 8,10,12,14, 9,11,13)(15,17,19,21,16,18,20)$
7C3 $7^{3}$ $6$ $7$ $18$ $( 1, 6, 4, 2, 7, 5, 3)( 8,11,14,10,13, 9,12)(15,16,17,18,19,20,21)$
14A1 $14,7$ $21$ $14$ $19$ $( 1, 7, 6, 5, 4, 3, 2)( 8,21, 9,20,10,19,11,18,12,17,13,16,14,15)$
14A-1 $14,7$ $21$ $14$ $19$ $( 1, 6, 4, 2, 7, 5, 3)( 8,21,10,19,12,17,14,15, 9,20,11,18,13,16)$
14A3 $14,7$ $21$ $14$ $19$ $( 1, 5, 2, 6, 3, 7, 4)( 8,21,11,18,14,15,10,19,13,16, 9,20,12,17)$
14A-3 $14,7$ $21$ $14$ $19$ $( 1, 2, 3, 4, 5, 6, 7)( 8,21,14,15,13,16,12,17,11,18,10,19, 9,20)$
14A5 $14,7$ $21$ $14$ $19$ $( 1, 3, 5, 7, 2, 4, 6)( 8,21,13,16,11,18, 9,20,14,15,12,17,10,19)$
14A-5 $14,7$ $21$ $14$ $19$ $( 1, 4, 7, 3, 6, 2, 5)( 8,21,12,17, 9,20,13,16,10,19,14,15,11,18)$

Malle's constant $a(G)$:     $1/7$

magma: ConjugacyClasses(G);
 

Character table

1A 2A 3A 7A1 7A-1 7A2 7A-2 7A3 7A-3 7B1 7B-1 7C1 7C2 7C3 14A1 14A-1 14A3 14A-3 14A5 14A-5
Size 1 21 98 3 3 3 3 3 3 6 6 6 6 6 21 21 21 21 21 21
2 P 1A 1A 3A 7A2 7A-3 7A-2 7A1 7A-1 7A3 7C1 7B1 7C2 7C3 7B-1 7A-3 7A1 7A-2 7A3 7A-1 7A2
3 P 1A 2A 1A 7A3 7A-1 7A-3 7A-2 7A2 7A1 7C2 7B-1 7C3 7C1 7B1 14A5 14A3 14A1 14A-5 14A-3 14A-1
7 P 1A 2A 3A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A
Type
294.7.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
294.7.1b R 1 −1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
294.7.2a R 2 0 −1 2 2 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0
294.7.3a1 C 3 1 0 ζ7−3+2ζ7−2 2ζ72+ζ73 ζ7+2ζ73 2ζ7−3+ζ7−1 ζ7−2+2ζ7 2ζ7−1+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−1+1+ζ7 ζ7−2+1+ζ72 ζ7−3+1+ζ73 ζ72 ζ7−2 ζ7−1 ζ7 ζ73 ζ7−3
294.7.3a2 C 3 1 0 2ζ72+ζ73 ζ7−3+2ζ7−2 2ζ7−3+ζ7−1 ζ7+2ζ73 2ζ7−1+ζ72 ζ7−2+2ζ7 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−1+1+ζ7 ζ7−2+1+ζ72 ζ7−3+1+ζ73 ζ7−2 ζ72 ζ7 ζ7−1 ζ7−3 ζ73
294.7.3a3 C 3 1 0 2ζ7−3+ζ7−1 ζ7+2ζ73 ζ7−2+2ζ7 2ζ7−1+ζ72 ζ7−3+2ζ7−2 2ζ72+ζ73 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−2+1+ζ72 ζ7−3+1+ζ73 ζ7−1+1+ζ7 ζ73 ζ7−3 ζ72 ζ7−2 ζ7 ζ7−1
294.7.3a4 C 3 1 0 ζ7+2ζ73 2ζ7−3+ζ7−1 2ζ7−1+ζ72 ζ7−2+2ζ7 2ζ72+ζ73 ζ7−3+2ζ7−2 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−2+1+ζ72 ζ7−3+1+ζ73 ζ7−1+1+ζ7 ζ7−3 ζ73 ζ7−2 ζ72 ζ7−1 ζ7
294.7.3a5 C 3 1 0 2ζ7−1+ζ72 ζ7−2+2ζ7 ζ7−3+2ζ7−2 2ζ72+ζ73 2ζ7−3+ζ7−1 ζ7+2ζ73 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+1+ζ73 ζ7−1+1+ζ7 ζ7−2+1+ζ72 ζ7 ζ7−1 ζ73 ζ7−3 ζ7−2 ζ72
294.7.3a6 C 3 1 0 ζ7−2+2ζ7 2ζ7−1+ζ72 2ζ72+ζ73 ζ7−3+2ζ7−2 ζ7+2ζ73 2ζ7−3+ζ7−1 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+1+ζ73 ζ7−1+1+ζ7 ζ7−2+1+ζ72 ζ7−1 ζ7 ζ7−3 ζ73 ζ72 ζ7−2
294.7.3b1 C 3 −1 0 ζ7−3+2ζ7−2 2ζ72+ζ73 ζ7+2ζ73 2ζ7−3+ζ7−1 ζ7−2+2ζ7 2ζ7−1+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−1+1+ζ7 ζ7−2+1+ζ72 ζ7−3+1+ζ73 −ζ72 −ζ7−2 −ζ7−1 −ζ7 −ζ73 −ζ7−3
294.7.3b2 C 3 −1 0 2ζ72+ζ73 ζ7−3+2ζ7−2 2ζ7−3+ζ7−1 ζ7+2ζ73 2ζ7−1+ζ72 ζ7−2+2ζ7 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−1+1+ζ7 ζ7−2+1+ζ72 ζ7−3+1+ζ73 −ζ7−2 −ζ72 −ζ7 −ζ7−1 −ζ7−3 −ζ73
294.7.3b3 C 3 −1 0 2ζ7−3+ζ7−1 ζ7+2ζ73 ζ7−2+2ζ7 2ζ7−1+ζ72 ζ7−3+2ζ7−2 2ζ72+ζ73 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−2+1+ζ72 ζ7−3+1+ζ73 ζ7−1+1+ζ7 −ζ73 −ζ7−3 −ζ72 −ζ7−2 −ζ7 −ζ7−1
294.7.3b4 C 3 −1 0 ζ7+2ζ73 2ζ7−3+ζ7−1 2ζ7−1+ζ72 ζ7−2+2ζ7 2ζ72+ζ73 ζ7−3+2ζ7−2 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−2+1+ζ72 ζ7−3+1+ζ73 ζ7−1+1+ζ7 −ζ7−3 −ζ73 −ζ7−2 −ζ72 −ζ7−1 −ζ7
294.7.3b5 C 3 −1 0 2ζ7−1+ζ72 ζ7−2+2ζ7 ζ7−3+2ζ7−2 2ζ72+ζ73 2ζ7−3+ζ7−1 ζ7+2ζ73 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+1+ζ73 ζ7−1+1+ζ7 ζ7−2+1+ζ72 −ζ7 −ζ7−1 −ζ73 −ζ7−3 −ζ7−2 −ζ72
294.7.3b6 C 3 −1 0 ζ7−2+2ζ7 2ζ7−1+ζ72 2ζ72+ζ73 ζ7−3+2ζ7−2 ζ7+2ζ73 2ζ7−3+ζ7−1 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+1+ζ73 ζ7−1+1+ζ7 ζ7−2+1+ζ72 −ζ7−1 −ζ7 −ζ7−3 −ζ73 −ζ72 −ζ7−2
294.7.6a1 C 6 0 0 −2ζ7−3−2−2ζ7−2ζ72 2ζ7−3+2ζ7+2ζ72 −2ζ7−3−2−2ζ7−2ζ72 2ζ7−3+2ζ7+2ζ72 2ζ7−3+2ζ7+2ζ72 −2ζ7−3−2−2ζ7−2ζ72 ζ7−3+3+ζ7+ζ72 −ζ7−3+2−ζ7−ζ72 −1 −1 −1 0 0 0 0 0 0
294.7.6a2 C 6 0 0 2ζ7−3+2ζ7+2ζ72 −2ζ7−3−2−2ζ7−2ζ72 2ζ7−3+2ζ7+2ζ72 −2ζ7−3−2−2ζ7−2ζ72 −2ζ7−3−2−2ζ7−2ζ72 2ζ7−3+2ζ7+2ζ72 −ζ7−3+2−ζ7−ζ72 ζ7−3+3+ζ7+ζ72 −1 −1 −1 0 0 0 0 0 0
294.7.6b1 R 6 0 0 2ζ7−3+2+2ζ73 2ζ7−3+2+2ζ73 2ζ7−1+2+2ζ7 2ζ7−1+2+2ζ7 2ζ7−2+2+2ζ72 2ζ7−2+2+2ζ72 −1 −1 ζ7−3−ζ7−2−1−ζ72+ζ73 −2ζ7−3−ζ7−2−2−ζ72−2ζ73 ζ7−3+2ζ7−2+2ζ72+ζ73 0 0 0 0 0 0
294.7.6b2 R 6 0 0 2ζ7−2+2+2ζ72 2ζ7−2+2+2ζ72 2ζ7−3+2+2ζ73 2ζ7−3+2+2ζ73 2ζ7−1+2+2ζ7 2ζ7−1+2+2ζ7 −1 −1 ζ7−3+2ζ7−2+2ζ72+ζ73 ζ7−3−ζ7−2−1−ζ72+ζ73 −2ζ7−3−ζ7−2−2−ζ72−2ζ73 0 0 0 0 0 0
294.7.6b3 R 6 0 0 2ζ7−1+2+2ζ7 2ζ7−1+2+2ζ7 2ζ7−2+2+2ζ72 2ζ7−2+2+2ζ72 2ζ7−3+2+2ζ73 2ζ7−3+2+2ζ73 −1 −1 −2ζ7−3−ζ7−2−2−ζ72−2ζ73 ζ7−3+2ζ7−2+2ζ72+ζ73 ζ7−3−ζ7−2−1−ζ72+ζ73 0 0 0 0 0 0

magma: CharacterTable(G);
 

Regular extensions

Data not computed