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Magma
magma: G := TransitiveGroup(42, 26);
Group invariants
Abstract group: | $F_8:C_3$ | magma: IdentifyGroup(G);
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Order: | $168=2^{3} \cdot 3 \cdot 7$ | 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 $n$: | $42$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $26$ | 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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$\card{\Aut(F/K)}$: | $6$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | $(1,25,39,2,26,40)(3,28,42)(4,27,41)(5,29,38)(6,30,37)(7,11,10,8,12,9)(13,34,24,14,33,23)(15,36,19,16,35,20)(17,32,21)(18,31,22)$, $(1,13,27,38,8,22,36)(2,14,28,37,7,21,35)(3,16,29,39,10,23,31)(4,15,30,40,9,24,32)(5,17,26,42,12,19,33)(6,18,25,41,11,20,34)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $3$: $C_3$ $21$: $C_7:C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $C_3$
Degree 6: None
Degree 7: $C_7:C_3$
Degree 14: 14T11
Degree 21: 21T2
Low degree siblings
8T36, 14T11, 24T283, 28T27Siblings 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 | $1^{42}$ | $1$ | $1$ | $0$ | $()$ |
2A | $2^{12},1^{18}$ | $7$ | $2$ | $12$ | $( 5, 6)( 7, 8)(11,12)(15,16)(21,22)(25,26)(27,28)(29,30)(31,32)(35,36)(39,40)(41,42)$ |
3A1 | $3^{14}$ | $28$ | $3$ | $28$ | $( 1,39,26)( 2,40,25)( 3,42,28)( 4,41,27)( 5,38,29)( 6,37,30)( 7,10,12)( 8, 9,11)(13,24,33)(14,23,34)(15,19,35)(16,20,36)(17,21,32)(18,22,31)$ |
3A-1 | $3^{14}$ | $28$ | $3$ | $28$ | $( 1,26,39)( 2,25,40)( 3,28,42)( 4,27,41)( 5,29,38)( 6,30,37)( 7,12,10)( 8,11, 9)(13,33,24)(14,34,23)(15,35,19)(16,36,20)(17,32,21)(18,31,22)$ |
6A1 | $6^{4},3^{6}$ | $28$ | $6$ | $32$ | $( 1, 6, 4, 2, 5, 3)( 7,26,16, 8,25,15)( 9,28,17,10,27,18)(11,29,14)(12,30,13)(19,31,38,20,32,37)(21,34,39)(22,33,40)(23,35,41)(24,36,42)$ |
6A-1 | $6^{4},3^{6}$ | $28$ | $6$ | $32$ | $( 1, 3, 5, 2, 4, 6)( 7,15,25, 8,16,26)( 9,18,27,10,17,28)(11,14,29)(12,13,30)(19,37,32,20,38,31)(21,39,34)(22,40,33)(23,41,35)(24,42,36)$ |
7A1 | $7^{6}$ | $24$ | $7$ | $36$ | $( 1,28, 8,35,14,37,22)( 2,27, 7,36,13,38,21)( 3,30, 9,31,15,39,23)( 4,29,10,32,16,40,24)( 5,25,12,33,17,41,20)( 6,26,11,34,18,42,19)$ |
7A-1 | $7^{6}$ | $24$ | $7$ | $36$ | $( 1,37,35,28,22,14, 8)( 2,38,36,27,21,13, 7)( 3,39,31,30,23,15, 9)( 4,40,32,29,24,16,10)( 5,41,33,25,20,17,12)( 6,42,34,26,19,18,11)$ |
Malle's constant $a(G)$: $1/12$
magma: ConjugacyClasses(G);
Character table
1A | 2A | 3A1 | 3A-1 | 6A1 | 6A-1 | 7A1 | 7A-1 | ||
Size | 1 | 7 | 28 | 28 | 28 | 28 | 24 | 24 | |
2 P | 1A | 1A | 3A-1 | 3A1 | 3A1 | 3A-1 | 7A1 | 7A-1 | |
3 P | 1A | 2A | 1A | 1A | 2A | 2A | 7A-1 | 7A1 | |
7 P | 1A | 2A | 3A1 | 3A-1 | 6A1 | 6A-1 | 1A | 1A | |
Type | |||||||||
168.43.1a | R | ||||||||
168.43.1b1 | C | ||||||||
168.43.1b2 | C | ||||||||
168.43.3a1 | C | ||||||||
168.43.3a2 | C | ||||||||
168.43.7a | R | ||||||||
168.43.7b1 | C | ||||||||
168.43.7b2 | C |
magma: CharacterTable(G);
Regular extensions
Data not computed