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Properties

Label 21T25
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, 25);
 

Group action invariants

Degree $n$:  $21$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $25$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_7^2:(C_3\times S_3)$
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:  (2,6,5,7,3,4)(8,20,9,15,13,16)(10,17)(11,19,14,18,12,21), (1,21,14)(2,16,10)(3,18,13)(4,20,9)(5,15,12)(6,17,8)(7,19,11)
magma: Generators(G);
 

Low degree resolvents

|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, 21T26, 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 TypeSizeOrderRepresentative
$1^{21}$ $1$ $1$ $()$
$7^{3}$ $6$ $7$ $( 1, 6, 4, 2, 7, 5, 3)( 8,12, 9,13,10,14,11)(15,21,20,19,18,17,16)$
$7^{3}$ $6$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,13,11, 9,14,12,10)(15,19,16,20,17,21,18)$
$7^{2},1^{7}$ $9$ $7$ $( 8,14,13,12,11,10, 9)(15,21,20,19,18,17,16)$
$7^{3}$ $18$ $7$ $( 1, 6, 4, 2, 7, 5, 3)( 8,11,14,10,13, 9,12)(15,20,18,16,21,19,17)$
$7^{2},1^{7}$ $9$ $7$ $( 8,12, 9,13,10,14,11)(15,19,16,20,17,21,18)$
$3^{6},1^{3}$ $49$ $3$ $( 2, 5, 3)( 4, 6, 7)( 9,12,10)(11,13,14)(16,19,17)(18,20,21)$
$3^{6},1^{3}$ $49$ $3$ $( 2, 3, 5)( 4, 7, 6)( 9,10,12)(11,14,13)(16,17,19)(18,21,20)$
$3^{7}$ $98$ $3$ $( 1,11,18)( 2,10,19)( 3, 9,20)( 4, 8,21)( 5,14,15)( 6,13,16)( 7,12,17)$
$21$ $42$ $21$ $( 1,13,19, 5,11,20, 2, 9,21, 6,14,15, 3,12,16, 7,10,17, 4, 8,18)$
$21$ $42$ $21$ $( 1,10,21, 6,11,17, 4,12,20, 2,13,16, 7,14,19, 5, 8,15, 3, 9,18)$
$3^{7}$ $14$ $3$ $( 1,11,18)( 2,14,20)( 3,10,15)( 4,13,17)( 5, 9,19)( 6,12,21)( 7, 8,16)$
$21$ $42$ $21$ $( 1,14,15, 2,12,19, 3,10,16, 4, 8,20, 5,13,17, 6,11,21, 7, 9,18)$
$21$ $42$ $21$ $( 1, 8,18, 4, 9,16, 7,10,21, 3,11,19, 6,12,17, 2,13,15, 5,14,20)$
$3^{7}$ $14$ $3$ $( 1,10,17)( 2, 8,21)( 3,13,18)( 4,11,15)( 5, 9,19)( 6,14,16)( 7,12,20)$
$6^{3},2,1$ $147$ $6$ $( 2, 6, 5, 7, 3, 4)( 8,20, 9,15,13,16)(10,17)(11,19,14,18,12,21)$
$14,2^{3},1$ $63$ $14$ $( 2, 7)( 3, 6)( 4, 5)( 8,21,13,19,11,17, 9,15,14,20,12,18,10,16)$
$14,2^{3},1$ $63$ $14$ $( 1, 6)( 2, 5)( 3, 4)( 8,20, 9,21,10,15,11,16,12,17,13,18,14,19)$
$2^{10},1$ $21$ $2$ $( 1, 2)( 3, 7)( 4, 6)( 8,18)( 9,19)(10,20)(11,21)(12,15)(13,16)(14,17)$
$6^{3},2,1$ $147$ $6$ $( 2, 4, 3, 7, 5, 6)( 8,18, 9,15,11,16)(10,19,13,17,12,20)(14,21)$

magma: ConjugacyClasses(G);
 

Group invariants

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
Label:  882.34
magma: IdentifyGroup(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);