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Properties

Label 21T16
Degree $21$
Order $294$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7:F_7$

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Show commands: Magma

magma: G := TransitiveGroup(21, 16);
 

Group action invariants

Degree $n$:  $21$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $16$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_7:F_7$
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:  (1,14,20,2,12,17)(3,10,21,7,9,16)(4,8,18,6,11,19)(5,13,15), (1,2)(3,7)(4,6)(8,12)(9,11)(13,14)(15,18)(16,17)(19,21)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$14$:  $D_{7}$
$42$:  $F_7$, 21T3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: None

Low degree siblings

21T16, 42T55 x 2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

magma: ConjugacyClasses(G);
 

Group invariants

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
Label:  294.10
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 6A1 6A-1 7A1 7A2 7A3 7B 7C1 7C2 7C3 7D1 7D2 7D3 21A1 21A-1 21A2 21A-2 21A4 21A-4
Size 1 49 7 7 49 49 2 2 2 6 6 6 6 6 6 6 14 14 14 14 14 14
2 P 1A 1A 3A-1 3A1 3A1 3A-1 7A2 7A3 7A1 7D3 7D2 7D1 7C3 7B 7C1 7C2 21A2 21A4 21A1 21A-1 21A-2 21A-4
3 P 1A 2A 1A 1A 2A 2A 7A3 7A1 7A2 7D1 7D3 7D2 7C1 7B 7C2 7C3 7A1 7A2 7A3 7A3 7A1 7A2
7 P 1A 2A 3A1 3A-1 6A1 6A-1 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 3A-1 3A1 3A1 3A-1 3A1 3A-1
Type
294.10.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
294.10.1b R 1 −1 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
294.10.1c1 C 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
294.10.1c2 C 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
294.10.1d1 C 1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
294.10.1d2 C 1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
294.10.2a1 R 2 0 2 2 0 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 2 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73
294.10.2a2 R 2 0 2 2 0 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 2 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72
294.10.2a3 R 2 0 2 2 0 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 2 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7
294.10.2b1 C 2 0 2ζ21−7 2ζ217 0 0 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−6+ζ216 2 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−3+ζ213 ζ214+ζ2110 ζ21−10−ζ213−ζ2110 −ζ21−10+1−ζ212+ζ213−ζ215+ζ217+ζ2110 ζ21−10−1+ζ21+ζ212−ζ213+ζ215−ζ216−ζ217+ζ218−ζ2110 −ζ21−10+1−ζ21−ζ212+ζ213−ζ214−ζ215+ζ216−ζ218 ζ212+ζ215
294.10.2b2 C 2 0 2ζ217 2ζ21−7 0 0 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−6+ζ216 2 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−10−ζ213−ζ2110 ζ214+ζ2110 ζ21−10−1+ζ21+ζ212−ζ213+ζ215−ζ216−ζ217+ζ218−ζ2110 −ζ21−10+1−ζ212+ζ213−ζ215+ζ217+ζ2110 ζ212+ζ215 −ζ21−10+1−ζ21−ζ212+ζ213−ζ214−ζ215+ζ216−ζ218
294.10.2b3 C 2 0 2ζ21−7 2ζ217 0 0 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−3+ζ213 2 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−9+ζ219 −ζ21−10+1−ζ21−ζ212+ζ213−ζ214−ζ215+ζ216−ζ218 ζ212+ζ215 ζ21−10−ζ213−ζ2110 ζ214+ζ2110 ζ21−10−1+ζ21+ζ212−ζ213+ζ215−ζ216−ζ217+ζ218−ζ2110 −ζ21−10+1−ζ212+ζ213−ζ215+ζ217+ζ2110
294.10.2b4 C 2 0 2ζ217 2ζ21−7 0 0 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−3+ζ213 2 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−9+ζ219 ζ212+ζ215 −ζ21−10+1−ζ21−ζ212+ζ213−ζ214−ζ215+ζ216−ζ218 ζ214+ζ2110 ζ21−10−ζ213−ζ2110 −ζ21−10+1−ζ212+ζ213−ζ215+ζ217+ζ2110 ζ21−10−1+ζ21+ζ212−ζ213+ζ215−ζ216−ζ217+ζ218−ζ2110
294.10.2b5 C 2 0 2ζ21−7 2ζ217 0 0 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−9+ζ219 2 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−10−1+ζ21+ζ212−ζ213+ζ215−ζ216−ζ217+ζ218−ζ2110 −ζ21−10+1−ζ212+ζ213−ζ215+ζ217+ζ2110 ζ212+ζ215 −ζ21−10+1−ζ21−ζ212+ζ213−ζ214−ζ215+ζ216−ζ218 ζ214+ζ2110 ζ21−10−ζ213−ζ2110
294.10.2b6 C 2 0 2ζ217 2ζ21−7 0 0 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−9+ζ219 2 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−6+ζ216 −ζ21−10+1−ζ212+ζ213−ζ215+ζ217+ζ2110 ζ21−10−1+ζ21+ζ212−ζ213+ζ215−ζ216−ζ217+ζ218−ζ2110 −ζ21−10+1−ζ21−ζ212+ζ213−ζ214−ζ215+ζ216−ζ218 ζ212+ζ215 ζ21−10−ζ213−ζ2110 ζ214+ζ2110
294.10.6a R 6 0 0 0 0 0 6 6 6 −1 −1 −1 −1 −1 −1 −1 0 0 0 0 0 0
294.10.6b1 R 6 0 0 0 0 0 3ζ7−3+3ζ73 3ζ7−1+3ζ7 3ζ7−2+3ζ72 −1 ζ7−3+2ζ7−2+2ζ72+ζ73 ζ7−3−ζ7−2−1−ζ72+ζ73 −2ζ7−3−ζ7−2−2−ζ72−2ζ73 −ζ7−3+1−ζ73 −ζ7−1+1−ζ7 −ζ7−2+1−ζ72 0 0 0 0 0 0
294.10.6b2 R 6 0 0 0 0 0 3ζ7−2+3ζ72 3ζ7−3+3ζ73 3ζ7−1+3ζ7 −1 −2ζ7−3−ζ7−2−2−ζ72−2ζ73 ζ7−3+2ζ7−2+2ζ72+ζ73 ζ7−3−ζ7−2−1−ζ72+ζ73 −ζ7−2+1−ζ72 −ζ7−3+1−ζ73 −ζ7−1+1−ζ7 0 0 0 0 0 0
294.10.6b3 R 6 0 0 0 0 0 3ζ7−1+3ζ7 3ζ7−2+3ζ72 3ζ7−3+3ζ73 −1 ζ7−3−ζ7−2−1−ζ72+ζ73 −2ζ7−3−ζ7−2−2−ζ72−2ζ73 ζ7−3+2ζ7−2+2ζ72+ζ73 −ζ7−1+1−ζ7 −ζ7−2+1−ζ72 −ζ7−3+1−ζ73 0 0 0 0 0 0
294.10.6c1 R 6 0 0 0 0 0 3ζ7−3+3ζ73 3ζ7−1+3ζ7 3ζ7−2+3ζ72 −1 −ζ7−2+1−ζ72 −ζ7−3+1−ζ73 −ζ7−1+1−ζ7 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
294.10.6c2 R 6 0 0 0 0 0 3ζ7−2+3ζ72 3ζ7−3+3ζ73 3ζ7−1+3ζ7 −1 −ζ7−1+1−ζ7 −ζ7−2+1−ζ72 −ζ7−3+1−ζ73 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
294.10.6c3 R 6 0 0 0 0 0 3ζ7−1+3ζ7 3ζ7−2+3ζ72 3ζ7−3+3ζ73 −1 −ζ7−3+1−ζ73 −ζ7−1+1−ζ7 −ζ7−2+1−ζ72 −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);