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Magma
magma: G := TransitiveGroup(28, 393);
Group action invariants
| Degree $n$: | $28$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $393$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $G(2,2)$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | yes | 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: | (1,7,12)(2,10,22,25,5,14)(3,23,13,15,9,28)(4,18,19,26,6,27)(8,16,20,17,11,24), (1,27,15,10,24,20)(3,22,17,26,11,7)(4,6,19,16,13,14)(5,9,18,23,12,25)(8,28,21) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: None
Degree 7: None
Degree 14: None
Low degree siblings
36T9590Siblings 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 |
| $1^{28}$ | $1$ | $1$ | $()$ | |
| $3^{9},1$ | $672$ | $3$ | $( 1,14,26)( 2,13,28)( 4,10, 8)( 5,21,19)( 6,16,23)( 7,18,17)( 9,20,22) (11,27,24)(12,25,15)$ | |
| $2^{12},1^{4}$ | $252$ | $2$ | $( 1,18)( 2,22)( 4, 6)( 7,26)( 8,23)( 9,13)(10,16)(11,15)(12,27)(14,17)(20,28) (24,25)$ | |
| $6^{4},3,1$ | $2016$ | $6$ | $( 1,17,26,18,14, 7)( 2, 9,28,22,13,20)( 4,16, 8, 6,10,23)( 5,21,19) (11,12,24,15,27,25)$ | |
| $2^{12},1^{4}$ | $63$ | $2$ | $( 1, 7)( 3,14)( 4,20)( 5,28)( 6,17)( 8,18)(10,13)(11,16)(12,21)(15,22)(19,27) (24,26)$ | |
| $4^{6},2^{2}$ | $378$ | $4$ | $( 1, 7)( 2,25, 9,23)( 3, 8,28,26)( 4,13,17,15)( 5,24,14,18)( 6,22,20,10) (11,16)(12,19,21,27)$ | |
| $8^{3},4$ | $1512$ | $8$ | $( 1,11, 7,16)( 2,21,23,19, 9,12,25,27)( 3, 6,26,10,28,20, 8,22) ( 4,24,15, 5,17,18,13,14)$ | |
| $3^{9},1$ | $56$ | $3$ | $( 1,10,19)( 2,16, 6)( 3,22,15)( 4, 7,25)( 5,26, 8)( 9,24,21)(11,12,13) (17,28,20)(18,23,27)$ | |
| $4^{6},1^{4}$ | $126$ | $4$ | $( 1, 3,20,25)( 2,18,12, 5)( 4,10,22,17)( 6,27,11, 8)( 7,19,15,28)(13,26,16,23)$ | |
| $6^{4},3,1$ | $504$ | $6$ | $( 1,28,10,20,19,17)( 2,11,16,12, 6,13)( 3, 7,22,25,15, 4)( 5,27,26,18, 8,23) ( 9,21,24)$ | |
| $12^{2},3,1$ | $1008$ | $12$ | $( 1, 4,28, 3,10, 7,20,22,19,25,17,15)( 2,26,11,18,16, 8,12,23, 6, 5,13,27) ( 9,24,21)$ | |
| $4^{6},1^{4}$ | $252$ | $4$ | $( 2,12, 9,21)( 3,14,28, 5)( 4,15,17,13)( 6,22,20,10)( 8,18,26,24)(19,23,27,25)$ | |
| $12^{2},3,1$ | $1008$ | $12$ | $( 1, 7,11)( 2, 3,20,12,14,10, 9,28, 6,21, 5,22)( 4,23,18,15,27,26,17,25,24,13, 19, 8)$ | |
| $12^{2},3,1$ | $1008$ | $12$ | $( 1,11, 7)( 2,10, 5,12, 6, 3, 9,22,14,21,20,28)( 4,26,19,15,24,23,17, 8,27,13, 18,25)$ | |
| $8^{3},2,1^{2}$ | $1512$ | $8$ | $( 1,26,11,14,16, 8, 7, 5)( 2,15,10,23,27, 4,20,12)( 6,25, 9,17,22,21,19,13) (18,24)$ | |
| $7^{4}$ | $1728$ | $7$ | $( 1, 9,26,10,16,25,17)( 2,24,14, 5,28,22,11)( 3,15,21, 8,13,12,18) ( 4, 7,23, 6,19,20,27)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $12096=2^{6} \cdot 3^{3} \cdot 7$ | 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: | no | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 12096.a | magma: IdentifyGroup(G);
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| Character table: |
| Size | |
| 2 P | |
| 3 P | |
| 7 P | |
| Type |
magma: CharacterTable(G);