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Magma
magma: G := TransitiveGroup(42, 31);
Group action invariants
Degree $n$: | $42$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $31$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $D_7:A_4$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,9,19)(2,10,20)(3,12,23)(4,11,24)(5,7,21)(6,8,22)(13,34,30)(14,33,29)(15,32,26)(16,31,25)(17,35,28)(18,36,27)(37,39,41)(38,40,42), (1,13,33,38,28,11)(2,14,34,37,27,12)(3,18,35,40,26,7)(4,17,36,39,25,8)(5,15,32,42,29,10)(6,16,31,41,30,9)(19,24,22)(20,23,21) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ $12$: $A_4$ $24$: $A_4\times C_2$ $42$: $F_7$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $C_3$
Degree 6: $A_4$
Degree 7: $F_7$
Degree 14: None
Degree 21: 21T4
Low degree siblings
28T28, 42T30Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $7$ | $2$ | $( 7,38)( 8,37)( 9,40)(10,39)(11,41)(12,42)(13,31)(14,32)(15,36)(16,35)(17,34) (18,33)(19,25)(20,26)(21,28)(22,27)(23,30)(24,29)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 3, 4)( 5, 6)( 9,10)(11,12)(13,14)(17,18)(19,20)(21,22)(25,26)(27,28)(31,32) (33,34)(39,40)(41,42)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $21$ | $2$ | $( 3, 4)( 5, 6)( 7,38)( 8,37)( 9,39)(10,40)(11,42)(12,41)(13,32)(14,31)(15,36) (16,35)(17,33)(18,34)(19,26)(20,25)(21,27)(22,28)(23,30)(24,29)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $28$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7,13,25)( 8,14,26)( 9,17,30)(10,18,29)(11,15,27) (12,16,28)(19,38,31)(20,37,32)(21,42,35)(22,41,36)(23,40,34)(24,39,33)$ | |
$ 6, 6, 6, 6, 6, 6, 3, 3 $ | $28$ | $6$ | $( 1, 3, 5)( 2, 4, 6)( 7,31,25,38,13,19)( 8,32,26,37,14,20)( 9,34,30,40,17,23) (10,33,29,39,18,24)(11,36,27,41,15,22)(12,35,28,42,16,21)$ | |
$ 6, 6, 6, 6, 6, 6, 3, 3 $ | $28$ | $6$ | $( 1, 5, 3)( 2, 6, 4)( 7,19,13,38,25,31)( 8,20,14,37,26,32)( 9,23,17,40,30,34) (10,24,18,39,29,33)(11,22,15,41,27,36)(12,21,16,42,28,35)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $28$ | $3$ | $( 1, 5, 3)( 2, 6, 4)( 7,25,13)( 8,26,14)( 9,30,17)(10,29,18)(11,27,15) (12,28,16)(19,31,38)(20,32,37)(21,35,42)(22,36,41)(23,34,40)(24,33,39)$ | |
$ 7, 7, 7, 7, 7, 7 $ | $6$ | $7$ | $( 1, 7,16,23,30,35,38)( 2, 8,15,24,29,36,37)( 3, 9,13,21,28,31,40) ( 4,10,14,22,27,32,39)( 5,12,17,19,25,34,42)( 6,11,18,20,26,33,41)$ | |
$ 14, 14, 7, 7 $ | $6$ | $14$ | $( 1, 7,16,23,30,35,38)( 2, 8,15,24,29,36,37)( 3,10,13,22,28,32,40, 4, 9,14,21, 27,31,39)( 5,11,17,20,25,33,42, 6,12,18,19,26,34,41)$ | |
$ 14, 14, 7, 7 $ | $6$ | $14$ | $( 1, 8,16,24,30,36,38, 2, 7,15,23,29,35,37)( 3, 9,13,21,28,31,40) ( 4,10,14,22,27,32,39)( 5,11,17,20,25,33,42, 6,12,18,19,26,34,41)$ | |
$ 14, 14, 7, 7 $ | $6$ | $14$ | $( 1, 8,16,24,30,36,38, 2, 7,15,23,29,35,37)( 3,10,13,22,28,32,40, 4, 9,14,21, 27,31,39)( 5,12,17,19,25,34,42)( 6,11,18,20,26,33,41)$ |
magma: ConjugacyClasses(G);
Group invariants
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 | ||
Label: | 168.49 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 3A1 | 3A-1 | 6A1 | 6A-1 | 7A | 14A1 | 14A3 | 14A5 | ||
Size | 1 | 3 | 7 | 21 | 28 | 28 | 28 | 28 | 6 | 6 | 6 | 6 | |
2 P | 1A | 1A | 1A | 1A | 3A-1 | 3A1 | 3A1 | 3A-1 | 7A | 7A | 7A | 7A | |
3 P | 1A | 2A | 2B | 2C | 1A | 1A | 2B | 2B | 7A | 14A3 | 14A5 | 14A1 | |
7 P | 1A | 2A | 2B | 2C | 3A1 | 3A-1 | 6A1 | 6A-1 | 1A | 2A | 2A | 2A | |
Type | |||||||||||||
168.49.1a | R | ||||||||||||
168.49.1b | R | ||||||||||||
168.49.1c1 | C | ||||||||||||
168.49.1c2 | C | ||||||||||||
168.49.1d1 | C | ||||||||||||
168.49.1d2 | C | ||||||||||||
168.49.3a | R | ||||||||||||
168.49.3b | R | ||||||||||||
168.49.6a | R | ||||||||||||
168.49.6b1 | R | ||||||||||||
168.49.6b2 | R | ||||||||||||
168.49.6b3 | R |
magma: CharacterTable(G);