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Properties

Label 21T15
Degree $21$
Order $252$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times F_7$

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Show commands: Magma

magma: G := TransitiveGroup(21, 15);
 

Group action invariants

Degree $n$:  $21$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $15$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3\times F_7$
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:  (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(19,21), (1,13,16)(2,15,17,3,14,18)(4,19,7)(5,21,8,6,20,9)(11,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $S_3$, $C_6$ x 3
$12$:  $D_{6}$, $C_6\times C_2$
$18$:  $S_3\times C_3$
$36$:  $C_6\times S_3$
$42$:  $F_7$
$84$:  $F_7 \times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 7: $F_7$

Low degree siblings

42T43, 42T44, 42T45, 42T52

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $252=2^{2} \cdot 3^{2} \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  252.26
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B1 3B-1 3C1 3C-1 6A1 6A-1 6B 6C1 6C-1 6D1 6D-1 6E1 6E-1 7A 14A 21A
Size 1 3 7 21 2 7 7 14 14 7 7 14 14 14 21 21 21 21 6 18 12
2 P 1A 1A 1A 1A 3A 3B-1 3B1 3C-1 3C1 3B1 3B-1 3A 3C1 3C-1 3B1 3B1 3B-1 3B-1 7A 7A 21A
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 2B 2B 2B 2B 2B 2C 2A 2A 2C 7A 14A 7A
7 P 1A 2A 2B 2C 3A 3B1 3B-1 3C1 3C-1 6A1 6A-1 6B 6C1 6C-1 6D1 6E1 6E-1 6D-1 1A 2A 3A
Type
252.26.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
252.26.1b R 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 1 1 −1 −1 1 −1 1
252.26.1c R 1 −1 1 −1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 −1 1
252.26.1d R 1 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 1 1 1 1 1
252.26.1e1 C 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 1 1 1
252.26.1e2 C 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 1
252.26.1f1 C 1 −1 −1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 −1 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 1 −1 1
252.26.1f2 C 1 −1 −1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 −1 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 1 −1 1
252.26.1g1 C 1 −1 1 −1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1 1 −1 1
252.26.1g2 C 1 −1 1 −1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 1 −1 1
252.26.1h1 C 1 1 −1 −1 1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 −1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3 ζ3−1 1 1 1
252.26.1h2 C 1 1 −1 −1 1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 −1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3−1 ζ3 1 1 1
252.26.2a R 2 0 2 0 −1 2 2 −1 −1 2 2 −1 −1 −1 0 0 0 0 2 0 −1
252.26.2b R 2 0 −2 0 −1 2 2 −1 −1 −2 −2 1 1 1 0 0 0 0 2 0 −1
252.26.2c1 C 2 0 2 0 −1 2ζ3−1 2ζ3 −ζ3 −ζ3−1 2ζ3 2ζ3−1 −1 −ζ3−1 −ζ3 0 0 0 0 2 0 −1
252.26.2c2 C 2 0 2 0 −1 2ζ3 2ζ3−1 −ζ3−1 −ζ3 2ζ3−1 2ζ3 −1 −ζ3 −ζ3−1 0 0 0 0 2 0 −1
252.26.2d1 C 2 0 −2 0 −1 2ζ3−1 2ζ3 −ζ3 −ζ3−1 −2ζ3 −2ζ3−1 1 ζ3−1 ζ3 0 0 0 0 2 0 −1
252.26.2d2 C 2 0 −2 0 −1 2ζ3 2ζ3−1 −ζ3−1 −ζ3 −2ζ3−1 −2ζ3 1 ζ3 ζ3−1 0 0 0 0 2 0 −1
252.26.6a R 6 6 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 −1 −1
252.26.6b R 6 −6 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 1 −1
252.26.12a R 12 0 0 0 −6 0 0 0 0 0 0 0 0 0 0 0 0 0 −2 0 1

magma: CharacterTable(G);