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Properties

Label 21T16
21T16 1 2 1->2 14 1->14 12 2->12 3 7 3->7 10 3->10 4 6 4->6 8 4->8 5 13 5->13 11 6->11 9 7->9 8->12 18 8->18 9->11 16 9->16 21 10->21 19 11->19 17 12->17 13->14 15 13->15 20 14->20 15->5 15->18 16->3 16->17 17->1 18->6 19->4 19->21 20->2 21->7
Degree $21$
Order $294$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7:F_7$

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

Group invariants

Abstract group:  $C_7:F_7$
Copy content magma:IdentifyGroup(G);
 
Order:  $294=2 \cdot 3 \cdot 7^{2}$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$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 TypeSizeOrderIndexRepresentative
1A $1^{21}$ $1$ $1$ $0$ $()$
2A $2^{9},1^{3}$ $49$ $2$ $9$ $( 1, 7)( 2, 6)( 3, 5)( 8,14)( 9,13)(10,12)(15,21)(16,20)(17,19)$
3A1 $3^{7}$ $7$ $3$ $14$ $( 1,15, 8)( 2,19,10)( 3,16,12)( 4,20,14)( 5,17, 9)( 6,21,11)( 7,18,13)$
3A-1 $3^{7}$ $7$ $3$ $14$ $( 1, 8,15)( 2,10,19)( 3,12,16)( 4,14,20)( 5, 9,17)( 6,11,21)( 7,13,18)$
6A1 $6^{3},3$ $49$ $6$ $17$ $( 1,10,20, 7,12,16)( 2, 8,17, 6,14,19)( 3,13,21, 5, 9,15)( 4,11,18)$
6A-1 $6^{3},3$ $49$ $6$ $17$ $( 1,16,12, 7,20,10)( 2,19,14, 6,17, 8)( 3,15, 9, 5,21,13)( 4,18,11)$
7A1 $7^{3}$ $2$ $7$ $18$ $( 1, 3, 5, 7, 2, 4, 6)( 8,12, 9,13,10,14,11)(15,16,17,18,19,20,21)$
7A2 $7^{3}$ $2$ $7$ $18$ $( 1, 5, 2, 6, 3, 7, 4)( 8, 9,10,11,12,13,14)(15,17,19,21,16,18,20)$
7A3 $7^{3}$ $2$ $7$ $18$ $( 1, 7, 6, 5, 4, 3, 2)( 8,13,11, 9,14,12,10)(15,18,21,17,20,16,19)$
7B $7^{3}$ $6$ $7$ $18$ $( 1, 2, 3, 4, 5, 6, 7)( 8, 9,10,11,12,13,14)(15,16,17,18,19,20,21)$
7C1 $7^{2},1^{7}$ $6$ $7$ $12$ $( 1, 5, 2, 6, 3, 7, 4)(15,21,20,19,18,17,16)$
7C2 $7^{2},1^{7}$ $6$ $7$ $12$ $( 8,14,13,12,11,10, 9)(15,19,16,20,17,21,18)$
7C3 $7^{2},1^{7}$ $6$ $7$ $12$ $( 1, 7, 6, 5, 4, 3, 2)( 8,12, 9,13,10,14,11)$
7D1 $7^{3}$ $6$ $7$ $18$ $( 1, 7, 6, 5, 4, 3, 2)( 8,10,12,14, 9,11,13)(15,16,17,18,19,20,21)$
7D2 $7^{3}$ $6$ $7$ $18$ $( 1, 3, 5, 7, 2, 4, 6)( 8, 9,10,11,12,13,14)(15,21,20,19,18,17,16)$
7D3 $7^{3}$ $6$ $7$ $18$ $( 1, 4, 7, 3, 6, 2, 5)( 8,13,11, 9,14,12,10)(15,17,19,21,16,18,20)$
21A1 $21$ $14$ $21$ $20$ $( 1,20,13, 3,21,10, 5,15,14, 7,16,11, 2,17, 8, 4,18,12, 6,19, 9)$
21A-1 $21$ $14$ $21$ $20$ $( 1,14,18, 3,11,19, 5, 8,20, 7,12,21, 2, 9,15, 4,13,16, 6,10,17)$
21A2 $21$ $14$ $21$ $20$ $( 1,13,21, 5,14,16, 2, 8,18, 6, 9,20, 3,10,15, 7,11,17, 4,12,19)$
21A-2 $21$ $14$ $21$ $20$ $( 1,18,11, 5,20,12, 2,15,13, 6,17,14, 3,19, 8, 7,21, 9, 4,16,10)$
21A4 $21$ $14$ $21$ $20$ $( 1,16, 9, 7,19,14, 6,15,12, 5,18,10, 4,21, 8, 3,17,13, 2,20,11)$
21A-4 $21$ $14$ $21$ $20$ $( 1,12,17, 7,10,20, 6, 8,16, 5,13,19, 4,11,15, 3, 9,18, 2,14,21)$

Malle's constant $a(G)$:     $1/9$

Copy content magma:ConjugacyClasses(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 7B 7C2 7C3 7C1 7D2 7D3 7D1 21A2 21A-2 21A4 21A-4 21A-1 21A1
3 P 1A 2A 1A 1A 2A 2A 7A3 7A1 7A2 7B 7C3 7C1 7C2 7D3 7D1 7D2 7A1 7A1 7A2 7A2 7A3 7A3
7 P 1A 2A 3A1 3A-1 6A1 6A-1 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 3A1 3A-1 3A-1 3A1 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 ζ3−1 ζ3−1 ζ3−1 ζ3
294.10.1c2 C 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1 1 1 1 1 1 1 ζ3−1 ζ3−1 ζ3 ζ3 ζ3 ζ3−1
294.10.1d1 C 1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3−1 ζ3
294.10.1d2 C 1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1 1 1 1 1 1 1 1 1 ζ3−1 ζ3−1 ζ3 ζ3 ζ3 ζ3−1
294.10.2a1 R 2 0 2 2 0 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−1+ζ7 2 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7
294.10.2a2 R 2 0 2 2 0 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−3+ζ73 2 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73
294.10.2a3 R 2 0 2 2 0 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−2+ζ72 2 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72
294.10.2b1 C 2 0 2ζ21−7 2ζ217 0 0 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−6+ζ216 ζ21−6+ζ216 ζ21−3+ζ213 ζ21−3+ζ213 2 ζ21−9+ζ219 ζ21−9+ζ219 ζ21−10−1+ζ21+ζ212−ζ213+ζ215−ζ216−ζ217+ζ218−ζ2110 −ζ21−10+1−ζ21−ζ212+ζ213−ζ214−ζ215+ζ216−ζ218 ζ21−10−ζ213−ζ2110 −ζ21−10+1−ζ212+ζ213−ζ215+ζ217+ζ2110 ζ212+ζ215 ζ214+ζ2110
294.10.2b2 C 2 0 2ζ217 2ζ21−7 0 0 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−6+ζ216 ζ21−6+ζ216 ζ21−3+ζ213 ζ21−3+ζ213 2 ζ21−9+ζ219 ζ21−9+ζ219 −ζ21−10+1−ζ212+ζ213−ζ215+ζ217+ζ2110 ζ212+ζ215 ζ214+ζ2110 ζ21−10−1+ζ21+ζ212−ζ213+ζ215−ζ216−ζ217+ζ218−ζ2110 −ζ21−10+1−ζ21−ζ212+ζ213−ζ214−ζ215+ζ216−ζ218 ζ21−10−ζ213−ζ2110
294.10.2b3 C 2 0 2ζ21−7 2ζ217 0 0 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−3+ζ213 ζ21−3+ζ213 ζ21−9+ζ219 ζ21−9+ζ219 2 ζ21−6+ζ216 ζ21−6+ζ216 ζ214+ζ2110 ζ21−10−1+ζ21+ζ212−ζ213+ζ215−ζ216−ζ217+ζ218−ζ2110 ζ212+ζ215 ζ21−10−ζ213−ζ2110 −ζ21−10+1−ζ212+ζ213−ζ215+ζ217+ζ2110 −ζ21−10+1−ζ21−ζ212+ζ213−ζ214−ζ215+ζ216−ζ218
294.10.2b4 C 2 0 2ζ217 2ζ21−7 0 0 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−3+ζ213 ζ21−3+ζ213 ζ21−9+ζ219 ζ21−9+ζ219 2 ζ21−6+ζ216 ζ21−6+ζ216 ζ21−10−ζ213−ζ2110 −ζ21−10+1−ζ212+ζ213−ζ215+ζ217+ζ2110 −ζ21−10+1−ζ21−ζ212+ζ213−ζ214−ζ215+ζ216−ζ218 ζ214+ζ2110 ζ21−10−1+ζ21+ζ212−ζ213+ζ215−ζ216−ζ217+ζ218−ζ2110 ζ212+ζ215
294.10.2b5 C 2 0 2ζ21−7 2ζ217 0 0 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−9+ζ219 ζ21−9+ζ219 ζ21−6+ζ216 ζ21−6+ζ216 2 ζ21−3+ζ213 ζ21−3+ζ213 −ζ21−10+1−ζ21−ζ212+ζ213−ζ214−ζ215+ζ216−ζ218 ζ214+ζ2110 −ζ21−10+1−ζ212+ζ213−ζ215+ζ217+ζ2110 ζ212+ζ215 ζ21−10−ζ213−ζ2110 ζ21−10−1+ζ21+ζ212−ζ213+ζ215−ζ216−ζ217+ζ218−ζ2110
294.10.2b6 C 2 0 2ζ217 2ζ21−7 0 0 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−9+ζ219 ζ21−9+ζ219 ζ21−6+ζ216 ζ21−6+ζ216 2 ζ21−3+ζ213 ζ21−3+ζ213 ζ212+ζ215 ζ21−10−ζ213−ζ2110 ζ21−10−1+ζ21+ζ212−ζ213+ζ215−ζ216−ζ217+ζ218−ζ2110 −ζ21−10+1−ζ21−ζ212+ζ213−ζ214−ζ215+ζ216−ζ218 ζ214+ζ2110 −ζ21−10+1−ζ212+ζ213−ζ215+ζ217+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 ζ7−3−ζ7−2−1−ζ72+ζ73 −ζ7−3+1−ζ73 −ζ7−2+1−ζ72 ζ7−3+2ζ7−2+2ζ72+ζ73 −1 −2ζ7−3−ζ7−2−2−ζ72−2ζ73 −ζ7−1+1−ζ7 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 ζ7−3+2ζ7−2+2ζ72+ζ73 −ζ7−2+1−ζ72 −ζ7−1+1−ζ7 −2ζ7−3−ζ7−2−2−ζ72−2ζ73 −1 ζ7−3−ζ7−2−1−ζ72+ζ73 −ζ7−3+1−ζ73 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 −2ζ7−3−ζ7−2−2−ζ72−2ζ73 −ζ7−1+1−ζ7 −ζ7−3+1−ζ73 ζ7−3−ζ7−2−1−ζ72+ζ73 −1 ζ7−3+2ζ7−2+2ζ72+ζ73 −ζ7−2+1−ζ72 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 −ζ7−3+1−ζ73 ζ7−3−ζ7−2−1−ζ72+ζ73 ζ7−3+2ζ7−2+2ζ72+ζ73 −ζ7−2+1−ζ72 −1 −ζ7−1+1−ζ7 −2ζ7−3−ζ7−2−2−ζ72−2ζ73 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 −ζ7−2+1−ζ72 ζ7−3+2ζ7−2+2ζ72+ζ73 −2ζ7−3−ζ7−2−2−ζ72−2ζ73 −ζ7−1+1−ζ7 −1 −ζ7−3+1−ζ73 ζ7−3−ζ7−2−1−ζ72+ζ73 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 −ζ7−1+1−ζ7 −2ζ7−3−ζ7−2−2−ζ72−2ζ73 ζ7−3−ζ7−2−1−ζ72+ζ73 −ζ7−3+1−ζ73 −1 −ζ7−2+1−ζ72 ζ7−3+2ζ7−2+2ζ72+ζ73 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed