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Magma
magma: G := TransitiveGroup(24, 5003);
Group action invariants
| Degree $n$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $5003$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $\PGOPlus(4,3):C_4$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $4$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,22,7,17)(2,21,8,18)(3,23,5,19)(4,24,6,20)(9,15)(10,16)(11,13)(12,14), (1,5,12,3,8,9,2,6,11,4,7,10)(13,16,14,15)(17,23,18,24)(19,21,20,22) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $8$: $D_{4}$ x 2, $C_4\times C_2$ $16$: $C_2^2:C_4$ $72$: $C_3^2:D_4$ $144$: 12T79 $1152$: $S_4\wr C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 3: None
Degree 4: $C_2^2$
Degree 6: $C_3^2:D_4$
Degree 8: None
Degree 12: 12T34
Low degree siblings
24T5005, 32T205480, 32T205482, 36T3220 x 2, 36T3222, 36T3225Siblings 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 |
| $1^{24}$ | $1$ | $1$ | $()$ | |
| $2^{4},1^{16}$ | $6$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | |
| $2^{8},1^{8}$ | $9$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(17,18)(19,20)(21,22)(23,24)$ | |
| $3^{4},1^{12}$ | $16$ | $3$ | $( 1,11, 7)( 2,12, 8)( 3,10, 5)( 4, 9, 6)$ | |
| $3^{4},2^{4},1^{4}$ | $48$ | $6$ | $( 1,11, 7)( 2,12, 8)( 3,10, 5)( 4, 9, 6)(17,18)(19,20)(21,22)(23,24)$ | |
| $3^{8}$ | $64$ | $3$ | $( 1,11, 7)( 2,12, 8)( 3,10, 5)( 4, 9, 6)(13,17,22)(14,18,21)(15,20,24) (16,19,23)$ | |
| $2^{8},1^{8}$ | $36$ | $2$ | $( 5,10)( 6, 9)( 7,11)( 8,12)(13,22)(14,21)(15,24)(16,23)$ | |
| $4^{2},2^{6},1^{4}$ | $72$ | $4$ | $( 1, 2)( 3, 4)( 5,10, 6, 9)( 7,11, 8,12)(13,22)(14,21)(15,24)(16,23)$ | |
| $4^{4},2^{4}$ | $36$ | $4$ | $( 1, 2)( 3, 4)( 5,10, 6, 9)( 7,11, 8,12)(13,21,14,22)(15,23,16,24)(17,18) (19,20)$ | |
| $4^{4},2^{4}$ | $144$ | $4$ | $( 1,22, 7,17)( 2,21, 8,18)( 3,23, 5,19)( 4,24, 6,20)( 9,15)(10,16)(11,13) (12,14)$ | |
| $8^{2},4^{2}$ | $144$ | $8$ | $( 1,22, 8,18, 2,21, 7,17)( 3,23, 6,20, 4,24, 5,19)( 9,15,10,16)(11,13,12,14)$ | |
| $4^{4},2^{4}$ | $144$ | $4$ | $( 1,13, 7,17)( 2,14, 8,18)( 3,16, 5,19)( 4,15, 6,20)( 9,24)(10,23)(11,22) (12,21)$ | |
| $8^{2},4^{2}$ | $144$ | $8$ | $( 1,13, 7,18, 2,14, 8,17)( 3,16, 5,20, 4,15, 6,19)( 9,23,10,24)(11,21,12,22)$ | |
| $2^{4},1^{16}$ | $9$ | $2$ | $( 9,10)(11,12)(21,22)(23,24)$ | |
| $2^{8},1^{8}$ | $6$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(21,22)(23,24)$ | |
| $2^{12}$ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)$ | |
| $6^{2},2^{2},1^{8}$ | $48$ | $6$ | $( 1,11, 8, 2,12, 7)( 3,10, 6, 4, 9, 5)(21,22)(23,24)$ | |
| $6^{2},2^{6}$ | $16$ | $6$ | $( 1,11, 8, 2,12, 7)( 3,10, 6, 4, 9, 5)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)$ | |
| $6^{4}$ | $64$ | $6$ | $( 1,11, 8, 2,12, 7)( 3,10, 6, 4, 9, 5)(13,17,22,14,18,21)(15,20,24,16,19,23)$ | |
| $4^{4},1^{8}$ | $36$ | $4$ | $( 5,10, 6, 9)( 7,11, 8,12)(13,22,14,21)(15,24,16,23)$ | |
| $4^{2},2^{6},1^{4}$ | $72$ | $4$ | $( 1, 2)( 3, 4)( 5,10)( 6, 9)( 7,11)( 8,12)(13,22,14,21)(15,24,16,23)$ | |
| $2^{12}$ | $36$ | $2$ | $( 1, 2)( 3, 4)( 5,10)( 6, 9)( 7,11)( 8,12)(13,21)(14,22)(15,23)(16,24)(17,18) (19,20)$ | |
| $12,4^{3}$ | $96$ | $12$ | $( 1, 5,12, 3, 8, 9, 2, 6,11, 4, 7,10)(13,16,14,15)(17,23,18,24)(19,21,20,22)$ | |
| $12,4,2^{4}$ | $96$ | $12$ | $( 1, 5,12, 3, 8, 9, 2, 6,11, 4, 7,10)(13,15,14,16)(17,24)(18,23)(19,22)(20,21)$ | |
| $4^{6}$ | $36$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,16,14,15)(17,23,18,24)(19,21,20,22)$ | |
| $4^{6}$ | $12$ | $4$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,12,10,11)(13,16,14,15)(17,23,18,24)(19,21,20,22)$ | |
| $4^{4},2^{4}$ | $36$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,15,14,16)(17,24)(18,23)(19,22) (20,21)$ | |
| $4^{4},2^{4}$ | $12$ | $4$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)(17,24)(18,23)(19,22) (20,21)$ | |
| $12,4,2^{4}$ | $96$ | $12$ | $( 1,10)( 2, 9)( 3,12)( 4,11)( 5, 8, 6, 7)(13,23,18,15,22,19,14,24,17,16,21,20)$ | |
| $12,4^{3}$ | $96$ | $12$ | $( 1,10, 2, 9)( 3,12, 4,11)( 5, 7, 6, 8)(13,23,18,15,22,19,14,24,17,16,21,20)$ | |
| $4^{4},2^{4}$ | $12$ | $4$ | $( 1,10)( 2, 9)( 3,12)( 4,11)( 5, 8, 6, 7)(13,16,14,15)(17,19,18,20) (21,24,22,23)$ | |
| $4^{6}$ | $12$ | $4$ | $( 1,10, 2, 9)( 3,12, 4,11)( 5, 7, 6, 8)(13,16,14,15)(17,19,18,20)(21,24,22,23)$ | |
| $4^{4},2^{4}$ | $36$ | $4$ | $( 1,10)( 2, 9)( 3,12)( 4,11)( 5, 8, 6, 7)(13,16,14,15)(17,20,18,19) (21,23,22,24)$ | |
| $4^{6}$ | $36$ | $4$ | $( 1,10, 2, 9)( 3,12, 4,11)( 5, 7, 6, 8)(13,16,14,15)(17,20,18,19)(21,23,22,24)$ | |
| $6^{4}$ | $192$ | $6$ | $( 1,19, 8,15,11,24)( 2,20, 7,16,12,23)( 3,18, 6,13,10,22)( 4,17, 5,14, 9,21)$ | |
| $4^{4},2^{4}$ | $72$ | $4$ | $( 1,19, 2,20)( 3,18, 4,17)( 5,14, 6,13)( 7,16, 8,15)( 9,21)(10,22)(11,24) (12,23)$ | |
| $2^{12}$ | $24$ | $2$ | $( 1,19)( 2,20)( 3,18)( 4,17)( 5,14)( 6,13)( 7,16)( 8,15)( 9,21)(10,22)(11,24) (12,23)$ | |
| $6^{4}$ | $192$ | $6$ | $( 1,19,12,16, 8,24)( 2,20,11,15, 7,23)( 3,18, 9,14, 6,22)( 4,17,10,13, 5,21)$ | |
| $4^{4},2^{4}$ | $72$ | $4$ | $( 1,19, 2,20)( 3,18, 4,17)( 5,21, 6,22)( 7,23, 8,24)( 9,14)(10,13)(11,15) (12,16)$ | |
| $2^{12}$ | $24$ | $2$ | $( 1,19)( 2,20)( 3,18)( 4,17)( 5,21)( 6,22)( 7,23)( 8,24)( 9,14)(10,13)(11,15) (12,16)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $2304=2^{8} \cdot 3^{2}$ | 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 | ||
| Label: | 2304.fx | magma: IdentifyGroup(G);
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| Character table: | 40 x 40 character table |
magma: CharacterTable(G);