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Magma
magma: G := TransitiveGroup(21, 24);
Group action invariants
Degree $n$: | $21$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_7:(C_3\times F_7)$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,5,6)(2,7,3)(9,10,12)(11,14,13)(15,18,17)(16,20,21), (1,8,18,2,9,19)(3,10,20,7,14,17)(4,11,21,6,13,16)(5,12,15) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ x 4 $6$: $C_6$ x 4 $9$: $C_3^2$ $18$: $C_6 \times C_3$ $42$: $F_7$ x 2 $126$: 21T9 x 2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 7: None
Low degree siblings
21T24, 42T142 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Label | Cycle Type | Size | Order | Representative |
$1^{21}$ | $1$ | $1$ | $()$ | |
$7^{3}$ | $18$ | $7$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,12, 9,13,10,14,11)(15,16,17,18,19,20,21)$ | |
$7^{2},1^{7}$ | $18$ | $7$ | $( 8,10,12,14, 9,11,13)(15,17,19,21,16,18,20)$ | |
$7^{3}$ | $6$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,13,11, 9,14,12,10)(15,16,17,18,19,20,21)$ | |
$7^{3}$ | $6$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,12, 9,13,10,14,11)(15,21,20,19,18,17,16)$ | |
$3^{6},1^{3}$ | $49$ | $3$ | $( 2, 3, 5)( 4, 7, 6)( 9,10,12)(11,14,13)(16,17,19)(18,21,20)$ | |
$3^{6},1^{3}$ | $49$ | $3$ | $( 2, 5, 3)( 4, 6, 7)( 9,12,10)(11,13,14)(16,19,17)(18,20,21)$ | |
$3^{7}$ | $49$ | $3$ | $( 1,18, 9)( 2,19, 8)( 3,20,14)( 4,21,13)( 5,15,12)( 6,16,11)( 7,17,10)$ | |
$21$ | $42$ | $21$ | $( 1,21,11, 4,20,13, 7,19, 8, 3,18,10, 6,17,12, 2,16,14, 5,15, 9)$ | |
$3^{7}$ | $7$ | $3$ | $( 1,17,10)( 2,19,13)( 3,21, 9)( 4,16,12)( 5,18, 8)( 6,20,11)( 7,15,14)$ | |
$21$ | $42$ | $21$ | $( 1,20,11, 7,16,13, 6,19, 8, 5,15,10, 4,18,12, 3,21,14, 2,17, 9)$ | |
$3^{7}$ | $7$ | $3$ | $( 1,18,11)( 2,15, 9)( 3,19,14)( 4,16,12)( 5,20,10)( 6,17, 8)( 7,21,13)$ | |
$3^{7}$ | $49$ | $3$ | $( 1, 9,18)( 2, 8,19)( 3,14,20)( 4,13,21)( 5,12,15)( 6,11,16)( 7,10,17)$ | |
$21$ | $42$ | $21$ | $( 1,10,19, 3,13,20, 5, 9,21, 7,12,15, 2, 8,16, 4,11,17, 6,14,18)$ | |
$3^{7}$ | $7$ | $3$ | $( 1,12,19)( 2,10,16)( 3, 8,20)( 4,13,17)( 5,11,21)( 6, 9,18)( 7,14,15)$ | |
$21$ | $42$ | $21$ | $( 1,12,15, 3,11,19, 5,10,16, 7, 9,20, 2, 8,17, 4,14,21, 6,13,18)$ | |
$3^{7}$ | $7$ | $3$ | $( 1,13,20)( 2, 9,15)( 3,12,17)( 4, 8,19)( 5,11,21)( 6,14,16)( 7,10,18)$ | |
$2^{9},1^{3}$ | $49$ | $2$ | $( 2, 7)( 3, 6)( 4, 5)( 9,14)(10,13)(11,12)(16,21)(17,20)(18,19)$ | |
$6^{3},1^{3}$ | $49$ | $6$ | $( 2, 6, 5, 7, 3, 4)( 9,13,12,14,10,11)(16,20,19,21,17,18)$ | |
$6^{3},1^{3}$ | $49$ | $6$ | $( 2, 4, 3, 7, 5, 6)( 9,11,10,14,12,13)(16,18,17,21,19,20)$ | |
$6^{3},3$ | $49$ | $6$ | $( 1,18, 8, 2,17,14)( 3,16,13, 7,19, 9)( 4,15,12, 6,20,10)( 5,21,11)$ | |
$6^{3},3$ | $49$ | $6$ | $( 1,21,14)( 2,19,10, 7,16,11)( 3,17,13, 6,18, 8)( 4,15, 9, 5,20,12)$ | |
$6^{3},3$ | $49$ | $6$ | $( 1,20, 9, 2,16,14)( 3,19,12, 7,17,11)( 4,15,10, 6,21,13)( 5,18, 8)$ | |
$6^{3},3$ | $49$ | $6$ | $( 1, 9,20, 7, 8,19)( 2,10,21, 6,14,18)( 3,11,15, 5,13,17)( 4,12,16)$ | |
$6^{3},3$ | $49$ | $6$ | $( 1,10,20, 6,13,19)( 2,12,17, 5,11,15)( 3,14,21, 4, 9,18)( 7, 8,16)$ | |
$6^{3},3$ | $49$ | $6$ | $( 1,12,19)( 2, 9,21, 7, 8,17)( 3,13,16, 6,11,15)( 4,10,18, 5,14,20)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $882=2 \cdot 3^{2} \cdot 7^{2}$ | magma: Order(G);
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Cyclic: | no | magma: IsCyclic(G);
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Abelian: | no | magma: IsAbelian(G);
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Solvable: | yes | magma: IsSolvable(G);
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Nilpotency class: | not nilpotent | ||
Label: | 882.36 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 3A1 | 3A-1 | 3B1 | 3B-1 | 3C1 | 3C-1 | 3D1 | 3D-1 | 6A1 | 6A-1 | 6B1 | 6B-1 | 6C1 | 6C-1 | 6D1 | 6D-1 | 7A | 7B | 7C | 7D | 21A1 | 21A-1 | 21B1 | 21B-1 | ||
Size | 1 | 49 | 7 | 7 | 7 | 7 | 49 | 49 | 49 | 49 | 49 | 49 | 49 | 49 | 49 | 49 | 49 | 49 | 6 | 6 | 18 | 18 | 42 | 42 | 42 | 42 | |
2 P | 1A | 1A | 3A1 | 3A-1 | 3B1 | 3B-1 | 3C-1 | 3C1 | 3D-1 | 3D1 | 3A1 | 3C-1 | 3A-1 | 3D1 | 3D-1 | 3C1 | 3B1 | 3B-1 | 7A | 7B | 7C | 7D | 21B-1 | 21B1 | 21A-1 | 21A1 | |
3 P | 1A | 2A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 2A | 2A | 2A | 2A | 2A | 2A | 2A | 2A | 7A | 7B | 7C | 7D | 7B | 7B | 7A | 7A | |
7 P | 1A | 2A | 3A-1 | 3A1 | 3B-1 | 3B1 | 3C1 | 3C-1 | 3D1 | 3D-1 | 6B1 | 6A-1 | 6B-1 | 6C1 | 6C-1 | 6A1 | 6D1 | 6D-1 | 1A | 1A | 1A | 1A | 3A-1 | 3A1 | 3B1 | 3B-1 | |
Type | |||||||||||||||||||||||||||
882.36.1a | R | ||||||||||||||||||||||||||
882.36.1b | R | ||||||||||||||||||||||||||
882.36.1c1 | C | ||||||||||||||||||||||||||
882.36.1c2 | C | ||||||||||||||||||||||||||
882.36.1d1 | C | ||||||||||||||||||||||||||
882.36.1d2 | C | ||||||||||||||||||||||||||
882.36.1e1 | C | ||||||||||||||||||||||||||
882.36.1e2 | C | ||||||||||||||||||||||||||
882.36.1f1 | C | ||||||||||||||||||||||||||
882.36.1f2 | C | ||||||||||||||||||||||||||
882.36.1g1 | C | ||||||||||||||||||||||||||
882.36.1g2 | C | ||||||||||||||||||||||||||
882.36.1h1 | C | ||||||||||||||||||||||||||
882.36.1h2 | C | ||||||||||||||||||||||||||
882.36.1i1 | C | ||||||||||||||||||||||||||
882.36.1i2 | C | ||||||||||||||||||||||||||
882.36.1j1 | C | ||||||||||||||||||||||||||
882.36.1j2 | C | ||||||||||||||||||||||||||
882.36.6a | R | ||||||||||||||||||||||||||
882.36.6b | R | ||||||||||||||||||||||||||
882.36.6c1 | C | ||||||||||||||||||||||||||
882.36.6c2 | C | ||||||||||||||||||||||||||
882.36.6d1 | C | ||||||||||||||||||||||||||
882.36.6d2 | C | ||||||||||||||||||||||||||
882.36.18a | R | ||||||||||||||||||||||||||
882.36.18b | R |
magma: CharacterTable(G);