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Magma
magma: G := TransitiveGroup(24, 5000);
Group action invariants
| Degree $n$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $5000$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_2^6:S_3^2$ | ||
| 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: | (1,14,9,21,7,20)(2,13,10,22,8,19)(3,15,11,24,5,17)(4,16,12,23,6,18), (1,14,2,13)(3,16,4,15)(5,18)(6,17)(7,20)(8,19)(9,21,10,22)(11,23,12,24), (1,4,2,3)(5,10)(6,9)(7,11)(8,12)(13,23)(14,24)(15,22)(16,21)(17,20,18,19) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_2^2$ x 7 $6$: $S_3$ x 2 $8$: $D_{4}$ x 2, $C_2^3$ $12$: $D_{6}$ x 6 $16$: $D_4\times C_2$ $24$: $S_3 \times C_2^2$ x 2, $(C_6\times C_2):C_2$ x 2 $36$: $S_3^2$ $48$: 12T28, 24T25 $72$: 12T37 $144$: 24T204 $576$: $(A_4\wr C_2):C_2$ $1152$: 12T195 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 3: None
Degree 4: $C_2^2$
Degree 6: $S_3^2$
Degree 8: None
Degree 12: $S_3^2$
Low degree siblings
24T5014, 24T5082 x 2, 32T205462, 32T205497, 36T3109 x 2, 36T3165 x 2, 36T3437 x 4Siblings 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 |
| $1^{24}$ | $1$ | $1$ | $()$ | |
| $2^{8},1^{8}$ | $9$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(13,14)(15,16)(17,18)(19,20)$ | |
| $2^{4},1^{16}$ | $6$ | $2$ | $(13,14)(15,16)(21,22)(23,24)$ | |
| $3^{8}$ | $32$ | $3$ | $( 1, 9, 7)( 2,10, 8)( 3,11, 5)( 4,12, 6)(13,22,19)(14,21,20)(15,24,17) (16,23,18)$ | |
| $3^{4},1^{12}$ | $16$ | $3$ | $(13,20,21)(14,19,22)(15,18,23)(16,17,24)$ | |
| $3^{4},2^{4},1^{4}$ | $48$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(13,19,22)(14,20,21)(15,17,24)(16,18,23)$ | |
| $3^{8}$ | $32$ | $3$ | $( 1, 7, 9)( 2, 8,10)( 3, 5,11)( 4, 6,12)(13,21,19)(14,22,20)(15,23,17) (16,24,18)$ | |
| $6^{4}$ | $96$ | $6$ | $( 1,14, 9,21, 7,20)( 2,13,10,22, 8,19)( 3,15,11,24, 5,17)( 4,16,12,23, 6,18)$ | |
| $2^{12}$ | $12$ | $2$ | $( 1,21)( 2,22)( 3,24)( 4,23)( 5,15)( 6,16)( 7,14)( 8,13)( 9,20)(10,19)(11,17) (12,18)$ | |
| $4^{4},2^{4}$ | $36$ | $4$ | $( 1,21, 2,22)( 3,24, 4,23)( 5,16)( 6,15)( 7,13)( 8,14)( 9,19,10,20) (11,18,12,17)$ | |
| $2^{6},1^{12}$ | $18$ | $2$ | $( 3, 4)( 7, 8)( 9,10)(15,16)(19,20)(21,22)$ | |
| $2^{6},1^{12}$ | $12$ | $2$ | $( 3, 4)( 5, 6)(11,12)(13,14)(19,20)(23,24)$ | |
| $2^{6},1^{12}$ | $2$ | $2$ | $( 3, 4)( 5, 6)(11,12)(15,16)(17,18)(23,24)$ | |
| $6^{2},3^{4}$ | $32$ | $6$ | $( 1, 9, 8)( 2,10, 7)( 3,12, 6, 4,11, 5)(13,22,20)(14,21,19)(15,23,18,16,24,17)$ | |
| $6^{2},3^{4}$ | $32$ | $6$ | $( 1, 7,10)( 2, 8, 9)( 3, 6,12, 4, 5,11)(13,19,21)(14,20,22)(15,18,23,16,17,24)$ | |
| $6,3^{2},2^{3},1^{6}$ | $48$ | $6$ | $( 3, 4)( 7, 8)( 9,10)(13,20,22)(14,19,21)(15,17,24,16,18,23)$ | |
| $6,3^{2},2^{3},1^{6}$ | $16$ | $6$ | $( 3, 4)( 5, 6)(11,12)(13,20,21)(14,19,22)(15,17,23,16,18,24)$ | |
| $6,3^{2},2^{3},1^{6}$ | $48$ | $6$ | $( 1, 9, 8)( 2,10, 7)( 3,12, 6, 4,11, 5)(13,14)(19,20)(23,24)$ | |
| $6,3^{2},2^{3},1^{6}$ | $16$ | $6$ | $( 1, 9, 7)( 2,10, 8)( 3,12, 5, 4,11, 6)(15,16)(17,18)(23,24)$ | |
| $6^{2},3^{4}$ | $64$ | $6$ | $( 1, 7,10)( 2, 8, 9)( 3, 6,12, 4, 5,11)(13,21,20)(14,22,19)(15,24,18,16,23,17)$ | |
| $6^{4}$ | $96$ | $6$ | $( 1,14, 9,22, 7,19)( 2,13,10,21, 8,20)( 3,16,11,24, 5,17)( 4,15,12,23, 6,18)$ | |
| $4^{4},2^{4}$ | $72$ | $4$ | $( 1,21, 2,22)( 3,23, 4,24)( 5,15, 6,16)( 7,13, 8,14)( 9,19)(10,20)(11,17) (12,18)$ | |
| $2^{12}$ | $24$ | $2$ | $( 1,22)( 2,21)( 3,24)( 4,23)( 5,16)( 6,15)( 7,14)( 8,13)( 9,19)(10,20)(11,17) (12,18)$ | |
| $6^{4}$ | $96$ | $6$ | $( 1,20, 8,21,10,14)( 2,19, 7,22, 9,13)( 3,18, 6,23,12,16)( 4,17, 5,24,11,15)$ | |
| $2^{4},1^{16}$ | $9$ | $2$ | $( 9,10)(11,12)(21,22)(23,24)$ | |
| $2^{12}$ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)$ | |
| $2^{8},1^{8}$ | $6$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(17,18)(19,20)$ | |
| $6^{4}$ | $32$ | $6$ | $( 1, 9, 8, 2,10, 7)( 3,11, 6, 4,12, 5)(13,22,20,14,21,19)(15,24,18,16,23,17)$ | |
| $6^{2},2^{2},1^{8}$ | $48$ | $6$ | $( 9,10)(11,12)(13,20,21,14,19,22)(15,18,23,16,17,24)$ | |
| $6^{2},2^{6}$ | $16$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,19,22,14,20,21)(15,17,24,16,18, 23)$ | |
| $6^{4}$ | $32$ | $6$ | $( 1, 7, 9, 2, 8,10)( 3, 5,11, 4, 6,12)(13,21,20,14,22,19)(15,23,18,16,24,17)$ | |
| $6^{4}$ | $96$ | $6$ | $( 1,14, 9,22, 7,20)( 2,13,10,21, 8,19)( 3,15,11,23, 5,17)( 4,16,12,24, 6,18)$ | |
| $4^{4},2^{4}$ | $36$ | $4$ | $( 1,21, 2,22)( 3,24, 4,23)( 5,15)( 6,16)( 7,14)( 8,13)( 9,19,10,20) (11,18,12,17)$ | |
| $2^{12}$ | $12$ | $2$ | $( 1,21)( 2,22)( 3,24)( 4,23)( 5,16)( 6,15)( 7,13)( 8,14)( 9,20)(10,19)(11,17) (12,18)$ | |
| $4^{2},2^{8}$ | $72$ | $4$ | $( 1, 4, 2, 3)( 5,10)( 6, 9)( 7,11)( 8,12)(13,23)(14,24)(15,22)(16,21) (17,20,18,19)$ | |
| $4^{6}$ | $72$ | $4$ | $( 1, 3, 2, 4)( 5,10, 6, 9)( 7,11, 8,12)(13,23,14,24)(15,22,16,21)(17,19,18,20)$ | |
| $4^{4},2^{4}$ | $144$ | $4$ | $( 1, 4, 2, 3)( 5,10)( 6, 9)( 7,11)( 8,12)(13,24,14,23)(15,21,16,22) (17,19,18,20)$ | |
| $12^{2}$ | $192$ | $12$ | $( 1,16, 7,24, 9,18, 2,15, 8,23,10,17)( 3,14, 5,22,11,20, 4,13, 6,21,12,19)$ | |
| $4^{2},2^{8}$ | $72$ | $4$ | $( 1,17)( 2,18)( 3,19)( 4,20)( 5,14)( 6,13)( 7,16)( 8,15)( 9,23,10,24) (11,21,12,22)$ | |
| $4^{6}$ | $24$ | $4$ | $( 1,18, 2,17)( 3,20, 4,19)( 5,13, 6,14)( 7,15, 8,16)( 9,23,10,24)(11,21,12,22)$ | |
| $4^{2},2^{8}$ | $144$ | $4$ | $( 1, 3)( 2, 4)( 5, 9, 6,10)( 7,11, 8,12)(13,23)(14,24)(15,21)(16,22)(17,19) (18,20)$ | |
| $4^{4},2^{4}$ | $72$ | $4$ | $( 1, 3)( 2, 4)( 5, 9, 6,10)( 7,11, 8,12)(13,24,14,23)(15,22,16,21)(17,20) (18,19)$ | |
| $2^{12}$ | $72$ | $2$ | $( 1, 4)( 2, 3)( 5, 9)( 6,10)( 7,11)( 8,12)(13,24)(14,23)(15,22)(16,21)(17,19) (18,20)$ | |
| $6^{4}$ | $192$ | $6$ | $( 1,15, 7,24, 9,18)( 2,16, 8,23,10,17)( 3,14, 5,21,11,19)( 4,13, 6,22,12,20)$ | |
| $4^{4},2^{4}$ | $72$ | $4$ | $( 1,18)( 2,17)( 3,19)( 4,20)( 5,13, 6,14)( 7,16, 8,15)( 9,23,10,24) (11,22,12,21)$ | |
| $2^{12}$ | $24$ | $2$ | $( 1,18)( 2,17)( 3,19)( 4,20)( 5,14)( 6,13)( 7,15)( 8,16)( 9,24)(10,23)(11,21) (12,22)$ |
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.fu | magma: IdentifyGroup(G);
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| Character table: | 46 x 46 character table |
magma: CharacterTable(G);