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Properties

Label 21T26
Degree $21$
Order $882$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7^2:(C_3\times S_3)$

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Show commands: Magma

magma: G := TransitiveGroup(21, 26);
 

Group invariants

Abstract group:  $C_7^2:(C_3\times S_3)$
magma: IdentifyGroup(G);
 
Order:  $882=2 \cdot 3^{2} \cdot 7^{2}$
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 $t$:  $26$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,21,2,18,4,19)(3,15,6,20,5,16)(7,17)(8,12,14)(10,13,11)$, $(1,11,15)(2,10,16)(3,9,17)(4,8,18)(5,14,19)(6,13,20)(7,12,21)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$18$:  $S_3\times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 7: None

Low degree siblings

14T26, 21T25, 42T143, 42T144, 42T152, 42T153, 42T154, 42T155

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 $2^{7},1^{7}$ $21$ $2$ $7$ $( 1,15)( 2,16)( 3,17)( 4,18)( 5,19)( 6,20)( 7,21)$
3A1 $3^{7}$ $14$ $3$ $14$ $( 1,15, 8)( 2,17,11)( 3,19,14)( 4,21,10)( 5,16,13)( 6,18, 9)( 7,20,12)$
3A-1 $3^{7}$ $14$ $3$ $14$ $( 1,20,13)( 2,17,11)( 3,21, 9)( 4,18,14)( 5,15,12)( 6,19,10)( 7,16, 8)$
3B1 $3^{6},1^{3}$ $49$ $3$ $12$ $( 1, 7, 5)( 3, 4, 6)( 8,12,13)( 9,14,10)(15,20,16)(18,19,21)$
3B-1 $3^{6},1^{3}$ $49$ $3$ $12$ $( 1, 5, 7)( 3, 6, 4)( 8,13,12)( 9,10,14)(15,16,20)(18,21,19)$
3C $3^{7}$ $98$ $3$ $14$ $( 1,20,11)( 2,21,10)( 3,15, 9)( 4,16, 8)( 5,17,14)( 6,18,13)( 7,19,12)$
6A1 $6^{2},3^{2},2,1$ $147$ $6$ $15$ $( 1,15, 5,17, 6,21)( 2,19, 7,18, 3,16)( 4,20)( 8,12,14)(10,13,11)$
6A-1 $6^{2},3^{2},2,1$ $147$ $6$ $15$ $( 1,17, 6,20, 5,18)( 2,19, 3,21, 7,15)( 4,16)( 8,11,10)( 9,13,14)$
7A1 $7^{3}$ $6$ $7$ $18$ $( 1, 5, 2, 6, 3, 7, 4)( 8,13,11, 9,14,12,10)(15,16,17,18,19,20,21)$
7A-1 $7^{3}$ $6$ $7$ $18$ $( 1, 6, 4, 2, 7, 5, 3)( 8,12, 9,13,10,14,11)(15,21,20,19,18,17,16)$
7B1 $7^{3}$ $9$ $7$ $18$ $( 1, 5, 2, 6, 3, 7, 4)( 8,10,12,14, 9,11,13)(15,20,18,16,21,19,17)$
7B-1 $7^{3}$ $9$ $7$ $18$ $( 1, 5, 2, 6, 3, 7, 4)( 8, 9,10,11,12,13,14)(15,19,16,20,17,21,18)$
7C $7^{2},1^{7}$ $18$ $7$ $12$ $( 8,12, 9,13,10,14,11)(15,19,16,20,17,21,18)$
14A1 $14,7$ $63$ $14$ $19$ $( 1,18, 7,17, 6,16, 5,15, 4,21, 3,20, 2,19)( 8,14,13,12,11,10, 9)$
14A-1 $14,7$ $63$ $14$ $19$ $( 1,19, 2,20, 3,21, 4,15, 5,16, 6,17, 7,18)( 8, 9,10,11,12,13,14)$
21A1 $21$ $42$ $21$ $20$ $( 1,16,12, 5,17,10, 2,18, 8, 6,19,13, 3,20,11, 7,21, 9, 4,15,14)$
21A-1 $21$ $42$ $21$ $20$ $( 1,19,11, 4,17,12, 7,15,13, 3,20,14, 6,18, 8, 2,16, 9, 5,21,10)$
21A2 $21$ $42$ $21$ $20$ $( 1,15,10, 2,19, 8, 3,16,13, 4,20,11, 5,17, 9, 6,21,14, 7,18,12)$
21A-2 $21$ $42$ $21$ $20$ $( 1,18,13, 6,21,14, 4,17, 8, 2,20, 9, 7,16,10, 5,19,11, 3,15,12)$

Malle's constant $a(G)$:     $1/7$

magma: ConjugacyClasses(G);
 

Character table

1A 2A 3A1 3A-1 3B1 3B-1 3C 6A1 6A-1 7A1 7A-1 7B1 7B-1 7C 14A1 14A-1 21A1 21A-1 21A2 21A-2
Size 1 21 14 14 49 49 98 147 147 6 6 9 9 18 63 63 42 42 42 42
2 P 1A 1A 3A-1 3A1 3B-1 3B1 3C 3B1 3B-1 7A1 7A-1 7B1 7B-1 7C 7B1 7B-1 21A2 21A-2 21A1 21A-1
3 P 1A 2A 1A 1A 1A 1A 1A 2A 2A 7A-1 7A1 7B-1 7B1 7C 14A-1 14A1 7A1 7A-1 7A1 7A-1
7 P 1A 2A 3A1 3A-1 3B1 3B-1 3C 6A1 6A-1 1A 1A 1A 1A 1A 2A 2A 3A1 3A-1 3A-1 3A1
Type
882.34.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
882.34.1b R 1 −1 1 1 1 1 1 −1 −1 1 1 1 1 1 −1 −1 1 1 1 1
882.34.1c1 C 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 ζ3−1 ζ3 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1
882.34.1c2 C 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 ζ3 ζ3−1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3
882.34.1d1 C 1 −1 ζ3−1 ζ3 ζ3 ζ3−1 1 −ζ3−1 −ζ3 1 1 1 1 1 −1 −1 ζ3−1 ζ3 ζ3 ζ3−1
882.34.1d2 C 1 −1 ζ3 ζ3−1 ζ3−1 ζ3 1 −ζ3 −ζ3−1 1 1 1 1 1 −1 −1 ζ3 ζ3−1 ζ3−1 ζ3
882.34.2a R 2 0 −1 −1 2 2 −1 0 0 2 2 2 2 2 0 0 −1 −1 −1 −1
882.34.2b1 C 2 0 −ζ3 −ζ3−1 2ζ3−1 2ζ3 −1 0 0 2 2 2 2 2 0 0 −ζ3 −ζ3−1 −ζ3−1 −ζ3
882.34.2b2 C 2 0 −ζ3−1 −ζ3 2ζ3 2ζ3−1 −1 0 0 2 2 2 2 2 0 0 −ζ3−1 −ζ3 −ζ3 −ζ3−1
882.34.6a1 C 6 0 3 3 0 0 0 0 0 −ζ7−3+2−ζ7−ζ72 ζ7−3+3+ζ7+ζ72 2ζ7−3+2ζ7+2ζ72 −2ζ7−3−2−2ζ7−2ζ72 −1 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
882.34.6a2 C 6 0 3 3 0 0 0 0 0 ζ7−3+3+ζ7+ζ72 −ζ7−3+2−ζ7−ζ72 −2ζ7−3−2−2ζ7−2ζ72 2ζ7−3+2ζ7+2ζ72 −1 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
882.34.6b1 C 6 0 3ζ21−7 3ζ217 0 0 0 0 0 −ζ21−10+3−ζ21−ζ214−ζ218+ζ219 ζ21−10+2+ζ21+ζ214+ζ218−ζ219 2ζ21−10−2+2ζ21+2ζ214+2ζ218−2ζ219 −2ζ21−10−2ζ21−2ζ214−2ζ218+2ζ219 −1 0 0 −ζ21−10+1−ζ212+ζ217−ζ218 ζ21−ζ212+ζ214−ζ219 −ζ21+ζ212−ζ214−ζ217+ζ219 ζ21−10+ζ212+ζ218
882.34.6b2 C 6 0 3ζ217 3ζ21−7 0 0 0 0 0 ζ21−10+2+ζ21+ζ214+ζ218−ζ219 −ζ21−10+3−ζ21−ζ214−ζ218+ζ219 −2ζ21−10−2ζ21−2ζ214−2ζ218+2ζ219 2ζ21−10−2+2ζ21+2ζ214+2ζ218−2ζ219 −1 0 0 ζ21−ζ212+ζ214−ζ219 −ζ21−10+1−ζ212+ζ217−ζ218 ζ21−10+ζ212+ζ218 −ζ21+ζ212−ζ214−ζ217+ζ219
882.34.6b3 C 6 0 3ζ21−7 3ζ217 0 0 0 0 0 ζ21−10+2+ζ21+ζ214+ζ218−ζ219 −ζ21−10+3−ζ21−ζ214−ζ218+ζ219 −2ζ21−10−2ζ21−2ζ214−2ζ218+2ζ219 2ζ21−10−2+2ζ21+2ζ214+2ζ218−2ζ219 −1 0 0 ζ21−10+ζ212+ζ218 −ζ21+ζ212−ζ214−ζ217+ζ219 ζ21−ζ212+ζ214−ζ219 −ζ21−10+1−ζ212+ζ217−ζ218
882.34.6b4 C 6 0 3ζ217 3ζ21−7 0 0 0 0 0 −ζ21−10+3−ζ21−ζ214−ζ218+ζ219 ζ21−10+2+ζ21+ζ214+ζ218−ζ219 2ζ21−10−2+2ζ21+2ζ214+2ζ218−2ζ219 −2ζ21−10−2ζ21−2ζ214−2ζ218+2ζ219 −1 0 0 −ζ21+ζ212−ζ214−ζ217+ζ219 ζ21−10+ζ212+ζ218 −ζ21−10+1−ζ212+ζ217−ζ218 ζ21−ζ212+ζ214−ζ219
882.34.9a1 C 9 3 0 0 0 0 0 0 0 −3ζ7−3−3−3ζ7−3ζ72 3ζ7−3+3ζ7+3ζ72 −ζ7−3−2−ζ7−ζ72 ζ7−3−1+ζ7+ζ72 2 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 0 0 0 0
882.34.9a2 C 9 3 0 0 0 0 0 0 0 3ζ7−3+3ζ7+3ζ72 −3ζ7−3−3−3ζ7−3ζ72 ζ7−3−1+ζ7+ζ72 −ζ7−3−2−ζ7−ζ72 2 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 0 0 0 0
882.34.9b1 C 9 −3 0 0 0 0 0 0 0 −3ζ7−3−3−3ζ7−3ζ72 3ζ7−3+3ζ7+3ζ72 −ζ7−3−2−ζ7−ζ72 ζ7−3−1+ζ7+ζ72 2 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72 0 0 0 0
882.34.9b2 C 9 −3 0 0 0 0 0 0 0 3ζ7−3+3ζ7+3ζ72 −3ζ7−3−3−3ζ7−3ζ72 ζ7−3−1+ζ7+ζ72 −ζ7−3−2−ζ7−ζ72 2 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72 0 0 0 0
882.34.18a R 18 0 0 0 0 0 0 0 0 −3 −3 4 4 −3 0 0 0 0 0 0

magma: CharacterTable(G);
 

Regular extensions

Data not computed