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Magma
magma: G := TransitiveGroup(21, 25);
Group invariants
Abstract group: | $C_7^2:(C_3\times S_3)$ | magma: IdentifyGroup(G);
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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 | magma: NilpotencyClass(G);
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Group action invariants
Degree $n$: | $21$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $25$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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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);
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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, 21T26, 42T143, 42T144, 42T152, 42T153, 42T154, 42T155Siblings 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 | Index | Representative |
1A | $1^{21}$ | $1$ | $1$ | $0$ | $()$ |
2A | $2^{10},1$ | $21$ | $2$ | $10$ | $( 1, 3)( 4, 7)( 5, 6)( 8,21)( 9,15)(10,16)(11,17)(12,18)(13,19)(14,20)$ |
3A1 | $3^{7}$ | $14$ | $3$ | $14$ | $( 1,21,14)( 2,16,10)( 3,18,13)( 4,20, 9)( 5,15,12)( 6,17, 8)( 7,19,11)$ |
3A-1 | $3^{7}$ | $14$ | $3$ | $14$ | $( 1,14,21)( 2,10,16)( 3,13,18)( 4, 9,20)( 5,12,15)( 6, 8,17)( 7,11,19)$ |
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 | $3^{7}$ | $98$ | $3$ | $14$ | $( 1, 8,16)( 2,14,17)( 3,13,18)( 4,12,19)( 5,11,20)( 6,10,21)( 7, 9,15)$ |
6A1 | $6^{3},2,1$ | $147$ | $6$ | $16$ | $( 1, 6, 7, 3, 5, 4)( 8,19,12,21,13,18)( 9,16,14,15,10,20)(11,17)$ |
6A-1 | $6^{3},2,1$ | $147$ | $6$ | $16$ | $( 1, 4, 5, 3, 7, 6)( 8,18,13,21,12,19)( 9,20,10,15,14,16)(11,17)$ |
7A1 | $7^{3}$ | $6$ | $7$ | $18$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,11,14,10,13, 9,12)(15,17,19,21,16,18,20)$ |
7A-1 | $7^{3}$ | $6$ | $7$ | $18$ | $( 1, 7, 6, 5, 4, 3, 2)( 8,12, 9,13,10,14,11)(15,20,18,16,21,19,17)$ |
7B1 | $7^{2},1^{7}$ | $9$ | $7$ | $12$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,12, 9,13,10,14,11)$ |
7B-1 | $7^{2},1^{7}$ | $9$ | $7$ | $12$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,11,14,10,13, 9,12)$ |
7C | $7^{3}$ | $18$ | $7$ | $18$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,14,13,12,11,10, 9)(15,18,21,17,20,16,19)$ |
14A1 | $14,2^{3},1$ | $63$ | $14$ | $16$ | $( 1,12, 5, 9, 2,13, 6,10, 3,14, 7,11, 4, 8)(15,19)(16,18)(20,21)$ |
14A-1 | $14,2^{3},1$ | $63$ | $14$ | $16$ | $( 1, 8, 4,11, 7,14, 3,10, 6,13, 2, 9, 5,12)(15,19)(16,18)(20,21)$ |
21A1 | $21$ | $42$ | $21$ | $20$ | $( 1,17, 9, 2,19,12, 3,21, 8, 4,16,11, 5,18,14, 6,20,10, 7,15,13)$ |
21A-1 | $21$ | $42$ | $21$ | $20$ | $( 1,13,15, 7,10,20, 6,14,18, 5,11,16, 4, 8,21, 3,12,19, 2, 9,17)$ |
21A2 | $21$ | $42$ | $21$ | $20$ | $( 1, 9,19, 3, 8,16, 5,14,20, 7,13,17, 2,12,21, 4,11,18, 6,10,15)$ |
21A-2 | $21$ | $42$ | $21$ | $20$ | $( 1,15,10, 6,18,11, 4,21,12, 2,17,13, 7,20,14, 5,16, 8, 3,19, 9)$ |
Malle's constant $a(G)$: $1/10$
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 | ||||||||||||||||||||
882.34.1b | R | ||||||||||||||||||||
882.34.1c1 | C | ||||||||||||||||||||
882.34.1c2 | C | ||||||||||||||||||||
882.34.1d1 | C | ||||||||||||||||||||
882.34.1d2 | C | ||||||||||||||||||||
882.34.2a | R | ||||||||||||||||||||
882.34.2b1 | C | ||||||||||||||||||||
882.34.2b2 | C | ||||||||||||||||||||
882.34.6a1 | C | ||||||||||||||||||||
882.34.6a2 | C | ||||||||||||||||||||
882.34.6b1 | C | ||||||||||||||||||||
882.34.6b2 | C | ||||||||||||||||||||
882.34.6b3 | C | ||||||||||||||||||||
882.34.6b4 | C | ||||||||||||||||||||
882.34.9a1 | C | ||||||||||||||||||||
882.34.9a2 | C | ||||||||||||||||||||
882.34.9b1 | C | ||||||||||||||||||||
882.34.9b2 | C | ||||||||||||||||||||
882.34.18a | R |
magma: CharacterTable(G);
Regular extensions
Data not computed