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Properties

Label 21T44
Degree $21$
Order $7560$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_3\times A_7$

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Show commands: Magma

magma: G := TransitiveGroup(21, 44);
 

Group invariants

Abstract group:  $C_3\times A_7$
magma: IdentifyGroup(G);
 
Order:  $7560=2^{3} \cdot 3^{3} \cdot 5 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
magma: NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $21$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $44$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $3$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,11,17,21,7,5,13,2,12,18,19,8,6,14,3,10,16,20,9,4,15)$, $(1,16,15,10,2,17,13,11,3,18,14,12)(4,8,6,7,5,9)(19,20,21)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$3$:  $C_3$
$2520$:  $A_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: $A_7$

Low degree siblings

45T442 x 2

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 $2^{6},1^{9}$ $105$ $2$ $6$ $( 7,11)( 8,12)( 9,10)(13,18)(14,16)(15,17)$
3A1 $3^{7}$ $1$ $3$ $14$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)$
3A-1 $3^{7}$ $1$ $3$ $14$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)$
3B $3^{3},1^{12}$ $70$ $3$ $6$ $(10,21,18)(11,19,16)(12,20,17)$
3C1 $3^{7}$ $70$ $3$ $14$ $( 1, 5,19)( 2, 6,20)( 3, 4,21)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)$
3C-1 $3^{7}$ $70$ $3$ $14$ $( 1,19, 5)( 2,20, 6)( 3,21, 4)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)$
3D $3^{6},1^{3}$ $280$ $3$ $12$ $( 1,13,10)( 2,14,11)( 3,15,12)( 4,16, 7)( 5,17, 8)( 6,18, 9)$
3E1 $3^{7}$ $280$ $3$ $14$ $( 1,12, 4)( 2,10, 5)( 3,11, 6)( 7,13,17)( 8,14,18)( 9,15,16)(19,21,20)$
3E-1 $3^{7}$ $280$ $3$ $14$ $( 1, 4,12)( 2, 5,10)( 3, 6,11)( 7,17,13)( 8,18,14)( 9,16,15)(19,20,21)$
4A $4^{3},2^{3},1^{3}$ $630$ $4$ $12$ $( 4,11)( 5,12)( 6,10)( 7,19,16,14)( 8,20,17,15)( 9,21,18,13)$
5A $5^{3},1^{6}$ $504$ $5$ $12$ $( 1,10,13, 6,21)( 2,11,14, 4,19)( 3,12,15, 5,20)$
6A1 $6^{2},3^{3}$ $105$ $6$ $16$ $( 1, 3, 2)( 4, 6, 5)( 7,18, 8,16, 9,17)(10,12,11)(13,20,14,21,15,19)$
6A-1 $6^{2},3^{3}$ $105$ $6$ $16$ $( 1, 2, 3)( 4, 5, 6)( 7,17, 9,16, 8,18)(10,11,12)(13,19,15,21,14,20)$
6B $3^{3},2^{6}$ $210$ $6$ $12$ $( 1,13)( 2,14)( 3,15)( 4, 7)( 5, 8)( 6, 9)(10,18,21)(11,16,19)(12,17,20)$
6C1 $6^{2},3^{3}$ $210$ $6$ $16$ $( 1,19, 5)( 2,20, 6)( 3,21, 4)( 7,12, 9,11, 8,10)(13,16,15,18,14,17)$
6C-1 $6^{2},3^{3}$ $210$ $6$ $16$ $( 1, 5,19)( 2, 6,20)( 3, 4,21)( 7,10, 8,11, 9,12)(13,17,14,18,15,16)$
7A1 $7^{3}$ $360$ $7$ $18$ $( 1, 9,21,18, 6,13,10)( 2, 7,19,16, 4,14,11)( 3, 8,20,17, 5,15,12)$
7A-1 $7^{3}$ $360$ $7$ $18$ $( 1,10,13, 6,18,21, 9)( 2,11,14, 4,16,19, 7)( 3,12,15, 5,17,20, 8)$
12A1 $12,6,3$ $630$ $12$ $18$ $( 1, 2, 3)( 4,12, 6,11, 5,10)( 7,15,18,19, 8,13,16,20, 9,14,17,21)$
12A-1 $12,6,3$ $630$ $12$ $18$ $( 1, 3, 2)( 4,10, 5,11, 6,12)( 7,21,17,14, 9,20,16,13, 8,19,18,15)$
15A1 $15,3^{2}$ $504$ $15$ $18$ $( 1,15,19,10, 5, 2,13,20,11, 6, 3,14,21,12, 4)( 7, 9, 8)(16,18,17)$
15A-1 $15,3^{2}$ $504$ $15$ $18$ $( 1, 4,12,21,14, 3, 6,11,20,13, 2, 5,10,19,15)( 7, 8, 9)(16,17,18)$
21A1 $21$ $360$ $21$ $20$ $( 1,15,16, 9,12, 4,21, 3,14,18, 8,11, 6,20, 2,13,17, 7,10, 5,19)$
21A-1 $21$ $360$ $21$ $20$ $( 1,19, 5,10, 7,17,13, 2,20, 6,11, 8,18,14, 3,21, 4,12, 9,16,15)$
21A2 $21$ $360$ $21$ $20$ $( 1,16,12,21,14, 8, 6, 2,17,10,19,15, 9, 4, 3,18,11,20,13, 7, 5)$
21A-2 $21$ $360$ $21$ $20$ $( 1, 5, 7,13,20,11,18, 3, 4, 9,15,19,10,17, 2, 6, 8,14,21,12,16)$

Malle's constant $a(G)$:     $1/6$

magma: ConjugacyClasses(G);
 

Character table

1A 2A 3A1 3A-1 3B 3C1 3C-1 3D 3E1 3E-1 4A 5A 6A1 6A-1 6B 6C1 6C-1 7A1 7A-1 12A1 12A-1 15A1 15A-1 21A1 21A-1 21A2 21A-2
Size 1 105 1 1 70 70 70 280 280 280 630 504 105 105 210 210 210 360 360 630 630 504 504 360 360 360 360
2 P 1A 1A 3A-1 3A1 3B 3C-1 3C1 3D 3E-1 3E1 2A 5A 3A-1 3A1 3B 3C1 3C-1 7A1 7A-1 6A1 6A-1 15A-1 15A1 21A2 21A-2 21A1 21A-1
3 P 1A 2A 1A 1A 1A 1A 1A 1A 1A 1A 4A 5A 2A 2A 2A 2A 2A 7A-1 7A1 4A 4A 5A 5A 7A1 7A-1 7A1 7A-1
5 P 1A 2A 3A-1 3A1 3B 3C-1 3C1 3D 3E-1 3E1 4A 1A 6A-1 6A1 6B 6C-1 6C1 7A-1 7A1 12A-1 12A1 3A-1 3A1 21A-1 21A1 21A-2 21A2
7 P 1A 2A 3A1 3A-1 3B 3C1 3C-1 3D 3E1 3E-1 4A 5A 6A1 6A-1 6B 6C1 6C-1 1A 1A 12A1 12A-1 15A1 15A-1 3A1 3A-1 3A-1 3A1
Type
7560.e.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
7560.e.1b1 C 1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3
7560.e.1b2 C 1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1
7560.e.6a R 6 2 6 6 3 3 3 0 0 0 0 1 2 2 −1 −1 −1 −1 −1 0 0 1 1 −1 −1 −1 −1
7560.e.6b1 C 6 2 6ζ3−1 6ζ3 3 3ζ3−1 3ζ3 0 0 0 0 1 2ζ3−1 2ζ3 −1 −ζ3 −ζ3−1 −1 −1 0 0 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3
7560.e.6b2 C 6 2 6ζ3 6ζ3−1 3 3ζ3 3ζ3−1 0 0 0 0 1 2ζ3 2ζ3−1 −1 −ζ3−1 −ζ3 −1 −1 0 0 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1
7560.e.10a1 C 10 −2 10 10 1 1 1 1 1 1 0 0 −2 −2 1 1 1 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 0 0 0 0 −ζ7−3−1−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+ζ7+ζ72
7560.e.10a2 C 10 −2 10 10 1 1 1 1 1 1 0 0 −2 −2 1 1 1 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 0 0 0 0 ζ7−3+ζ7+ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72
7560.e.10b1 C 10 −2 10ζ21−7 10ζ217 1 ζ21−7 ζ217 1 ζ21−7 ζ217 0 0 −2ζ21−7 −2ζ217 1 ζ217 ζ21−7 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 0 0 0 0 ζ21−10+ζ212+ζ218 ζ21−ζ212+ζ214−ζ219 −ζ21−10+1−ζ212+ζ217−ζ218 −ζ21+ζ212−ζ214−ζ217+ζ219
7560.e.10b2 C 10 −2 10ζ217 10ζ21−7 1 ζ217 ζ21−7 1 ζ217 ζ21−7 0 0 −2ζ217 −2ζ21−7 1 ζ21−7 ζ217 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 0 0 0 0 −ζ21+ζ212−ζ214−ζ217+ζ219 −ζ21−10+1−ζ212+ζ217−ζ218 ζ21−ζ212+ζ214−ζ219 ζ21−10+ζ212+ζ218
7560.e.10b3 C 10 −2 10ζ21−7 10ζ217 1 ζ21−7 ζ217 1 ζ21−7 ζ217 0 0 −2ζ21−7 −2ζ217 1 ζ217 ζ21−7 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 0 0 0 0 −ζ21−10+1−ζ212+ζ217−ζ218 −ζ21+ζ212−ζ214−ζ217+ζ219 ζ21−10+ζ212+ζ218 ζ21−ζ212+ζ214−ζ219
7560.e.10b4 C 10 −2 10ζ217 10ζ21−7 1 ζ217 ζ21−7 1 ζ217 ζ21−7 0 0 −2ζ217 −2ζ21−7 1 ζ21−7 ζ217 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 0 0 0 0 ζ21−ζ212+ζ214−ζ219 ζ21−10+ζ212+ζ218 −ζ21+ζ212−ζ214−ζ217+ζ219 −ζ21−10+1−ζ212+ζ217−ζ218
7560.e.14a R 14 2 14 14 −1 −1 −1 2 2 2 0 −1 2 2 −1 −1 −1 0 0 0 0 −1 −1 0 0 0 0
7560.e.14b R 14 2 14 14 2 2 2 −1 −1 −1 0 −1 2 2 2 2 2 0 0 0 0 −1 −1 0 0 0 0
7560.e.14c1 C 14 2 14ζ3−1 14ζ3 −1 −ζ3−1 −ζ3 2 2ζ3−1 2ζ3 0 −1 2ζ3−1 2ζ3 −1 −ζ3 −ζ3−1 0 0 0 0 −ζ3−1 −ζ3 0 0 0 0
7560.e.14c2 C 14 2 14ζ3 14ζ3−1 −1 −ζ3 −ζ3−1 2 2ζ3 2ζ3−1 0 −1 2ζ3 2ζ3−1 −1 −ζ3−1 −ζ3 0 0 0 0 −ζ3 −ζ3−1 0 0 0 0
7560.e.14d1 C 14 2 14ζ3−1 14ζ3 2 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 0 −1 2ζ3−1 2ζ3 2 2ζ3 2ζ3−1 0 0 0 0 −ζ3−1 −ζ3 0 0 0 0
7560.e.14d2 C 14 2 14ζ3 14ζ3−1 2 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 0 −1 2ζ3 2ζ3−1 2 2ζ3−1 2ζ3 0 0 0 0 −ζ3 −ζ3−1 0 0 0 0
7560.e.15a R 15 −1 15 15 3 3 3 0 0 0 −1 0 −1 −1 −1 −1 −1 1 1 −1 −1 0 0 1 1 1 1
7560.e.15b1 C 15 −1 15ζ3−1 15ζ3 3 3ζ3−1 3ζ3 0 0 0 −1 0 −ζ3−1 −ζ3 −1 −ζ3 −ζ3−1 1 1 −ζ3 −ζ3−1 0 0 ζ3−1 ζ3 ζ3−1 ζ3
7560.e.15b2 C 15 −1 15ζ3 15ζ3−1 3 3ζ3 3ζ3−1 0 0 0 −1 0 −ζ3 −ζ3−1 −1 −ζ3−1 −ζ3 1 1 −ζ3−1 −ζ3 0 0 ζ3 ζ3−1 ζ3 ζ3−1
7560.e.21a R 21 1 21 21 −3 −3 −3 0 0 0 −1 1 1 1 1 1 1 0 0 −1 −1 1 1 0 0 0 0
7560.e.21b1 C 21 1 21ζ3−1 21ζ3 −3 −3ζ3−1 −3ζ3 0 0 0 −1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 0 0 −ζ3 −ζ3−1 ζ3−1 ζ3 0 0 0 0
7560.e.21b2 C 21 1 21ζ3 21ζ3−1 −3 −3ζ3 −3ζ3−1 0 0 0 −1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 0 0 −ζ3−1 −ζ3 ζ3 ζ3−1 0 0 0 0
7560.e.35a R 35 −1 35 35 −1 −1 −1 −1 −1 −1 1 0 −1 −1 −1 −1 −1 0 0 1 1 0 0 0 0 0 0
7560.e.35b1 C 35 −1 35ζ3−1 35ζ3 −1 −ζ3−1 −ζ3 −1 −ζ3−1 −ζ3 1 0 −ζ3−1 −ζ3 −1 −ζ3 −ζ3−1 0 0 ζ3 ζ3−1 0 0 0 0 0 0
7560.e.35b2 C 35 −1 35ζ3 35ζ3−1 −1 −ζ3 −ζ3−1 −1 −ζ3 −ζ3−1 1 0 −ζ3 −ζ3−1 −1 −ζ3−1 −ζ3 0 0 ζ3−1 ζ3 0 0 0 0 0 0

magma: CharacterTable(G);
 

Regular extensions

Data not computed