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Magma
magma: G := TransitiveGroup(42, 47);
Group action invariants
| Degree $n$: | $42$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $47$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_3^2:C_{28}$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $7$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| Generators: | (1,2,3)(4,6,5)(7,9,8)(10,11,12)(13,15,14)(16,17,18)(19,21,20)(22,23,24)(25,27,26)(28,29,30)(31,33,32)(34,36,35)(37,39,38)(40,42,41), (1,17,31,6,19,35,7,24,38,10,27,40,14,28,2,16,32,5,21,36,8,22,37,12,25,42,13,30)(3,18,33,4,20,34,9,23,39,11,26,41,15,29) | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $7$: $C_7$ $14$: $C_{14}$ $28$: $C_{28}$ $36$: $C_3^2:C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 6: $C_3^2:C_4$
Degree 7: $C_7$
Degree 14: $C_{14}$
Degree 21: None
Low degree siblings
42T47Siblings 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 | $2^{14},1^{14}$ | $9$ | $2$ | $14$ | $( 1, 2)( 5, 6)( 7, 8)(10,12)(13,14)(16,17)(19,21)(22,24)(25,27)(28,30)(31,32)(35,36)(37,38)(40,42)$ |
| 3A | $3^{14}$ | $4$ | $3$ | $28$ | $( 1, 2, 3)( 4, 6, 5)( 7, 9, 8)(10,11,12)(13,15,14)(16,17,18)(19,21,20)(22,23,24)(25,27,26)(28,29,30)(31,33,32)(34,36,35)(37,39,38)(40,42,41)$ |
| 3B | $3^{7},1^{21}$ | $4$ | $3$ | $14$ | $( 4, 5, 6)(10,12,11)(16,18,17)(22,24,23)(28,30,29)(34,35,36)(40,41,42)$ |
| 4A1 | $4^{7},2^{7}$ | $9$ | $4$ | $28$ | $( 1,24, 2,22)( 3,23)( 4,26)( 5,25, 6,27)( 7,28, 8,30)( 9,29)(10,32,12,31)(11,33)(13,35,14,36)(15,34)(16,37,17,38)(18,39)(19,40,21,42)(20,41)$ |
| 4A-1 | $4^{7},2^{7}$ | $9$ | $4$ | $28$ | $( 1,22, 2,24)( 3,23)( 4,26)( 5,27, 6,25)( 7,30, 8,28)( 9,29)(10,31,12,32)(11,33)(13,36,14,35)(15,34)(16,38,17,37)(18,39)(19,42,21,40)(20,41)$ |
| 7A1 | $7^{6}$ | $1$ | $7$ | $36$ | $( 1,19,38,14,32, 8,25)( 2,21,37,13,31, 7,27)( 3,20,39,15,33, 9,26)( 4,23,41,18,34,11,29)( 5,22,42,17,35,10,28)( 6,24,40,16,36,12,30)$ |
| 7A-1 | $7^{6}$ | $1$ | $7$ | $36$ | $( 1,14,25,38, 8,19,32)( 2,13,27,37, 7,21,31)( 3,15,26,39, 9,20,33)( 4,18,29,41,11,23,34)( 5,17,28,42,10,22,35)( 6,16,30,40,12,24,36)$ |
| 7A2 | $7^{6}$ | $1$ | $7$ | $36$ | $( 1,25, 8,32,14,38,19)( 2,27, 7,31,13,37,21)( 3,26, 9,33,15,39,20)( 4,29,11,34,18,41,23)( 5,28,10,35,17,42,22)( 6,30,12,36,16,40,24)$ |
| 7A-2 | $7^{6}$ | $1$ | $7$ | $36$ | $( 1,32,19, 8,38,25,14)( 2,31,21, 7,37,27,13)( 3,33,20, 9,39,26,15)( 4,34,23,11,41,29,18)( 5,35,22,10,42,28,17)( 6,36,24,12,40,30,16)$ |
| 7A3 | $7^{6}$ | $1$ | $7$ | $36$ | $( 1, 8,14,19,25,32,38)( 2, 7,13,21,27,31,37)( 3, 9,15,20,26,33,39)( 4,11,18,23,29,34,41)( 5,10,17,22,28,35,42)( 6,12,16,24,30,36,40)$ |
| 7A-3 | $7^{6}$ | $1$ | $7$ | $36$ | $( 1,38,32,25,19,14, 8)( 2,37,31,27,21,13, 7)( 3,39,33,26,20,15, 9)( 4,41,34,29,23,18,11)( 5,42,35,28,22,17,10)( 6,40,36,30,24,16,12)$ |
| 14A1 | $14^{2},7^{2}$ | $9$ | $14$ | $38$ | $( 1,27, 8,31,14,37,19, 2,25, 7,32,13,38,21)( 3,26, 9,33,15,39,20)( 4,29,11,34,18,41,23)( 5,30,10,36,17,40,22, 6,28,12,35,16,42,24)$ |
| 14A-1 | $14^{2},7^{2}$ | $9$ | $14$ | $38$ | $( 1,13,25,37, 8,21,32, 2,14,27,38, 7,19,31)( 3,15,26,39, 9,20,33)( 4,18,29,41,11,23,34)( 5,16,28,40,10,24,35, 6,17,30,42,12,22,36)$ |
| 14A3 | $14^{2},7^{2}$ | $9$ | $14$ | $38$ | $( 1,31,19, 7,38,27,14, 2,32,21, 8,37,25,13)( 3,33,20, 9,39,26,15)( 4,34,23,11,41,29,18)( 5,36,22,12,42,30,17, 6,35,24,10,40,28,16)$ |
| 14A-3 | $14^{2},7^{2}$ | $9$ | $14$ | $38$ | $( 1, 7,14,21,25,31,38, 2, 8,13,19,27,32,37)( 3, 9,15,20,26,33,39)( 4,11,18,23,29,34,41)( 5,12,17,24,28,36,42, 6,10,16,22,30,35,40)$ |
| 14A5 | $14^{2},7^{2}$ | $9$ | $14$ | $38$ | $( 1,21,38,13,32, 7,25, 2,19,37,14,31, 8,27)( 3,20,39,15,33, 9,26)( 4,23,41,18,34,11,29)( 5,24,42,16,35,12,28, 6,22,40,17,36,10,30)$ |
| 14A-5 | $14^{2},7^{2}$ | $9$ | $14$ | $38$ | $( 1,37,32,27,19,13, 8, 2,38,31,25,21,14, 7)( 3,39,33,26,20,15, 9)( 4,41,34,29,23,18,11)( 5,40,35,30,22,16,10, 6,42,36,28,24,17,12)$ |
| 21A1 | $21^{2}$ | $4$ | $21$ | $40$ | $( 1,27, 9,32,13,39,19, 2,26, 8,31,15,38,21, 3,25, 7,33,14,37,20)( 4,30,10,34,16,42,23, 6,28,11,36,17,41,24, 5,29,12,35,18,40,22)$ |
| 21A-1 | $21,7^{3}$ | $4$ | $21$ | $38$ | $( 1, 8,14,19,25,32,38)( 2, 7,13,21,27,31,37)( 3, 9,15,20,26,33,39)( 4,10,16,23,28,36,41, 5,12,18,22,30,34,42, 6,11,17,24,29,35,40)$ |
| 21A2 | $21,7^{3}$ | $4$ | $21$ | $38$ | $( 1,25, 8,32,14,38,19)( 2,27, 7,31,13,37,21)( 3,26, 9,33,15,39,20)( 4,28,12,34,17,40,23, 5,30,11,35,16,41,22, 6,29,10,36,18,42,24)$ |
| 21A-2 | $21,7^{3}$ | $4$ | $21$ | $38$ | $( 1,32,19, 8,38,25,14)( 2,31,21, 7,37,27,13)( 3,33,20, 9,39,26,15)( 4,35,24,11,42,30,18, 5,36,23,10,40,29,17, 6,34,22,12,41,28,16)$ |
| 21A4 | $21^{2}$ | $4$ | $21$ | $40$ | $( 1,31,20, 8,37,26,14, 2,33,19, 7,39,25,13, 3,32,21, 9,38,27,15)( 4,36,22,11,40,28,18, 6,35,23,12,42,29,16, 5,34,24,10,41,30,17)$ |
| 21A-4 | $21^{2}$ | $4$ | $21$ | $40$ | $( 1,13,26,38, 7,20,32, 2,15,25,37, 9,19,31, 3,14,27,39, 8,21,33)( 4,16,28,41,12,22,34, 6,17,29,40,10,23,36, 5,18,30,42,11,24,35)$ |
| 21B1 | $21^{2}$ | $4$ | $21$ | $40$ | $( 1,37,33,25,21,15, 8, 2,39,32,27,20,14, 7, 3,38,31,26,19,13, 9)( 4,40,35,29,24,17,11, 6,42,34,30,22,18,12, 5,41,36,28,23,16,10)$ |
| 21B-1 | $21,7^{3}$ | $4$ | $21$ | $38$ | $( 1,14,25,38, 8,19,32)( 2,13,27,37, 7,21,31)( 3,15,26,39, 9,20,33)( 4,17,30,41,10,24,34, 5,16,29,42,12,23,35, 6,18,28,40,11,22,36)$ |
| 21B2 | $21^{2}$ | $4$ | $21$ | $40$ | $( 1, 7,15,19,27,33,38, 2, 9,14,21,26,32,37, 3, 8,13,20,25,31,39)( 4,12,17,23,30,35,41, 6,10,18,24,28,34,40, 5,11,16,22,29,36,42)$ |
| 21B-2 | $21^{2}$ | $4$ | $21$ | $40$ | $( 1,21,39,14,31, 9,25, 2,20,38,13,33, 8,27, 3,19,37,15,32, 7,26)( 4,24,42,18,36,10,29, 6,22,41,16,35,11,30, 5,23,40,17,34,12,28)$ |
| 21B4 | $21,7^{3}$ | $4$ | $21$ | $38$ | $( 1,38,32,25,19,14, 8)( 2,37,31,27,21,13, 7)( 3,39,33,26,20,15, 9)( 4,42,36,29,22,16,11, 5,40,34,28,24,18,10, 6,41,35,30,23,17,12)$ |
| 21B-4 | $21,7^{3}$ | $4$ | $21$ | $38$ | $( 1,19,38,14,32, 8,25)( 2,21,37,13,31, 7,27)( 3,20,39,15,33, 9,26)( 4,22,40,18,35,12,29, 5,24,41,17,36,11,28, 6,23,42,16,34,10,30)$ |
| 28A1 | $28,14$ | $9$ | $28$ | $40$ | $( 1,35,27,16, 8,42,31,24,14, 5,37,30,19,10, 2,36,25,17, 7,40,32,22,13, 6,38,28,21,12)( 3,34,26,18, 9,41,33,23,15, 4,39,29,20,11)$ |
| 28A-1 | $28,14$ | $9$ | $28$ | $40$ | $( 1,12,21,28,38, 6,13,22,32,40, 7,17,25,36, 2,10,19,30,37, 5,14,24,31,42, 8,16,27,35)( 3,11,20,29,39, 4,15,23,33,41, 9,18,26,34)$ |
| 28A3 | $28,14$ | $9$ | $28$ | $40$ | $( 1,28,13,40,25,10,37,24, 8,35,21, 6,32,17, 2,30,14,42,27,12,38,22, 7,36,19, 5,31,16)( 3,29,15,41,26,11,39,23, 9,34,20, 4,33,18)$ |
| 28A-3 | $28,14$ | $9$ | $28$ | $40$ | $( 1, 5, 7,12,14,17,21,24,25,28,31,36,38,42, 2, 6, 8,10,13,16,19,22,27,30,32,35,37,40)( 3, 4, 9,11,15,18,20,23,26,29,33,34,39,41)$ |
| 28A5 | $28,14$ | $9$ | $28$ | $40$ | $( 1,30,13,42,25,12,37,22, 8,36,21, 5,32,16, 2,28,14,40,27,10,38,24, 7,35,19, 6,31,17)( 3,29,15,41,26,11,39,23, 9,34,20, 4,33,18)$ |
| 28A-5 | $28,14$ | $9$ | $28$ | $40$ | $( 1, 6, 7,10,14,16,21,22,25,30,31,35,38,40, 2, 5, 8,12,13,17,19,24,27,28,32,36,37,42)( 3, 4, 9,11,15,18,20,23,26,29,33,34,39,41)$ |
| 28A9 | $28,14$ | $9$ | $28$ | $40$ | $( 1,17,31, 6,19,35, 7,24,38,10,27,40,14,28, 2,16,32, 5,21,36, 8,22,37,12,25,42,13,30)( 3,18,33, 4,20,34, 9,23,39,11,26,41,15,29)$ |
| 28A-9 | $28,14$ | $9$ | $28$ | $40$ | $( 1,42,37,36,32,28,27,24,19,17,13,12, 8, 5, 2,40,38,35,31,30,25,22,21,16,14,10, 7, 6)( 3,41,39,34,33,29,26,23,20,18,15,11, 9, 4)$ |
| 28A11 | $28,14$ | $9$ | $28$ | $40$ | $( 1,10,21,30,38, 5,13,24,32,42, 7,16,25,35, 2,12,19,28,37, 6,14,22,31,40, 8,17,27,36)( 3,11,20,29,39, 4,15,23,33,41, 9,18,26,34)$ |
| 28A-11 | $28,14$ | $9$ | $28$ | $40$ | $( 1,36,27,17, 8,40,31,22,14, 6,37,28,19,12, 2,35,25,16, 7,42,32,24,13, 5,38,30,21,10)( 3,34,26,18, 9,41,33,23,15, 4,39,29,20,11)$ |
| 28A13 | $28,14$ | $9$ | $28$ | $40$ | $( 1,16,31, 5,19,36, 7,22,38,12,27,42,14,30, 2,17,32, 6,21,35, 8,24,37,10,25,40,13,28)( 3,18,33, 4,20,34, 9,23,39,11,26,41,15,29)$ |
| 28A-13 | $28,14$ | $9$ | $28$ | $40$ | $( 1,40,37,35,32,30,27,22,19,16,13,10, 8, 6, 2,42,38,36,31,28,25,24,21,17,14,12, 7, 5)( 3,41,39,34,33,29,26,23,20,18,15,11, 9, 4)$ |
Malle's constant $a(G)$: $1/14$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $252=2^{2} \cdot 3^{2} \cdot 7$ | 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: | 252.31 | magma: IdentifyGroup(G);
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| Character table: | 42 x 42 character table |
magma: CharacterTable(G);