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Magma
magma: G := TransitiveGroup(32, 404);
Group action invariants
| Degree $n$: | $32$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $404$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_2^3\times A_4$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $8$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,29,9,3,31,11)(2,30,10,4,32,12)(5,7)(6,8)(13,23,25,15,21,27)(14,24,26,16,22,28)(17,19)(18,20), (1,23)(2,24)(3,21)(4,22)(5,13,29,17,11,25)(6,14,30,18,12,26)(7,15,31,19,9,27)(8,16,32,20,10,28), (1,24)(2,23)(3,22)(4,21)(5,14,29,18,11,26)(6,13,30,17,12,25)(7,16,31,20,9,28)(8,15,32,19,10,27) | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 7 $3$: $C_3$ $4$: $C_2^2$ x 7 $6$: $C_6$ x 7 $8$: $C_2^3$ $12$: $A_4$, $C_6\times C_2$ x 7 $24$: $A_4\times C_2$ x 7, 24T3 $48$: $C_2^2 \times A_4$ x 7 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 7
Degree 8: $C_2^3$, $A_4\times C_2$ x 7
Degree 16: 16T58 x 7
Low degree siblings
24T135 x 7, 24T136 x 14Siblings 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 | $1^{32}$ | $1$ | $1$ | $0$ | $()$ |
| 2A | $2^{16}$ | $1$ | $2$ | $16$ | $( 1,23)( 2,24)( 3,21)( 4,22)( 5,17)( 6,18)( 7,19)( 8,20)( 9,15)(10,16)(11,13)(12,14)(25,29)(26,30)(27,31)(28,32)$ |
| 2B | $2^{16}$ | $1$ | $2$ | $16$ | $( 1,24)( 2,23)( 3,22)( 4,21)( 5,18)( 6,17)( 7,20)( 8,19)( 9,16)(10,15)(11,14)(12,13)(25,30)(26,29)(27,32)(28,31)$ |
| 2C | $2^{16}$ | $1$ | $2$ | $16$ | $( 1,22)( 2,21)( 3,24)( 4,23)( 5,20)( 6,19)( 7,18)( 8,17)( 9,14)(10,13)(11,16)(12,15)(25,32)(26,31)(27,30)(28,29)$ |
| 2D | $2^{16}$ | $1$ | $2$ | $16$ | $( 1,21)( 2,22)( 3,23)( 4,24)( 5,19)( 6,20)( 7,17)( 8,18)( 9,13)(10,14)(11,15)(12,16)(25,31)(26,32)(27,29)(28,30)$ |
| 2E | $2^{16}$ | $1$ | $2$ | $16$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$ |
| 2F | $2^{16}$ | $1$ | $2$ | $16$ | $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)$ |
| 2G | $2^{16}$ | $1$ | $2$ | $16$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)$ |
| 2H | $2^{16}$ | $3$ | $2$ | $16$ | $( 1,20)( 2,19)( 3,18)( 4,17)( 5,22)( 6,21)( 7,24)( 8,23)( 9,28)(10,27)(11,26)(12,25)(13,30)(14,29)(15,32)(16,31)$ |
| 2I | $2^{16}$ | $3$ | $2$ | $16$ | $( 1,17)( 2,18)( 3,19)( 4,20)( 5,23)( 6,24)( 7,21)( 8,22)( 9,25)(10,26)(11,27)(12,28)(13,31)(14,32)(15,29)(16,30)$ |
| 2J | $2^{16}$ | $3$ | $2$ | $16$ | $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,30)(10,29)(11,32)(12,31)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)$ |
| 2K | $2^{16}$ | $3$ | $2$ | $16$ | $( 1,19)( 2,20)( 3,17)( 4,18)( 5,21)( 6,22)( 7,23)( 8,24)( 9,27)(10,28)(11,25)(12,26)(13,29)(14,30)(15,31)(16,32)$ |
| 2L | $2^{16}$ | $3$ | $2$ | $16$ | $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,32)(10,31)(11,30)(12,29)(13,26)(14,25)(15,28)(16,27)(17,22)(18,21)(19,24)(20,23)$ |
| 2M | $2^{16}$ | $3$ | $2$ | $16$ | $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,29)(10,30)(11,31)(12,32)(13,27)(14,28)(15,25)(16,26)(17,23)(18,24)(19,21)(20,22)$ |
| 2N | $2^{16}$ | $3$ | $2$ | $16$ | $( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,31)(10,32)(11,29)(12,30)(13,25)(14,26)(15,27)(16,28)(17,21)(18,22)(19,23)(20,24)$ |
| 2O | $2^{16}$ | $3$ | $2$ | $16$ | $( 1,18)( 2,17)( 3,20)( 4,19)( 5,24)( 6,23)( 7,22)( 8,21)( 9,26)(10,25)(11,28)(12,27)(13,32)(14,31)(15,30)(16,29)$ |
| 3A1 | $3^{8},1^{8}$ | $4$ | $3$ | $16$ | $( 5,11,29)( 6,12,30)( 7, 9,31)( 8,10,32)(13,25,17)(14,26,18)(15,27,19)(16,28,20)$ |
| 3A-1 | $3^{8},1^{8}$ | $4$ | $3$ | $16$ | $( 5,29,11)( 6,30,12)( 7,31, 9)( 8,32,10)(13,17,25)(14,18,26)(15,19,27)(16,20,28)$ |
| 6A1 | $6^{4},2^{4}$ | $4$ | $6$ | $24$ | $( 1, 3)( 2, 4)( 5,31,11, 7,29, 9)( 6,32,12, 8,30,10)(13,19,25,15,17,27)(14,20,26,16,18,28)(21,23)(22,24)$ |
| 6A-1 | $6^{4},2^{4}$ | $4$ | $6$ | $24$ | $( 1,23)( 2,24)( 3,21)( 4,22)( 5,13,29,17,11,25)( 6,14,30,18,12,26)( 7,15,31,19, 9,27)( 8,16,32,20,10,28)$ |
| 6B1 | $6^{4},2^{4}$ | $4$ | $6$ | $24$ | $( 1,23)( 2,24)( 3,21)( 4,22)( 5,25,11,17,29,13)( 6,26,12,18,30,14)( 7,27, 9,19,31,15)( 8,28,10,20,32,16)$ |
| 6B-1 | $6^{4},2^{4}$ | $4$ | $6$ | $24$ | $( 1, 2)( 3, 4)( 5,12,29, 6,11,30)( 7,10,31, 8, 9,32)(13,26,17,14,25,18)(15,28,19,16,27,20)(21,22)(23,24)$ |
| 6C1 | $6^{4},2^{4}$ | $4$ | $6$ | $24$ | $( 1,24)( 2,23)( 3,22)( 4,21)( 5,26,11,18,29,14)( 6,25,12,17,30,13)( 7,28, 9,20,31,16)( 8,27,10,19,32,15)$ |
| 6C-1 | $6^{4},2^{4}$ | $4$ | $6$ | $24$ | $( 1,22)( 2,21)( 3,24)( 4,23)( 5,16,29,20,11,28)( 6,15,30,19,12,27)( 7,14,31,18, 9,26)( 8,13,32,17,10,25)$ |
| 6D1 | $6^{4},2^{4}$ | $4$ | $6$ | $24$ | $( 1,21)( 2,22)( 3,23)( 4,24)( 5,27,11,19,29,15)( 6,28,12,20,30,16)( 7,25, 9,17,31,13)( 8,26,10,18,32,14)$ |
| 6D-1 | $6^{4},2^{4}$ | $4$ | $6$ | $24$ | $( 1, 4)( 2, 3)( 5,32,11, 8,29,10)( 6,31,12, 7,30, 9)(13,20,25,16,17,28)(14,19,26,15,18,27)(21,24)(22,23)$ |
| 6E1 | $6^{4},2^{4}$ | $4$ | $6$ | $24$ | $( 1, 2)( 3, 4)( 5,30,11, 6,29,12)( 7,32, 9, 8,31,10)(13,18,25,14,17,26)(15,20,27,16,19,28)(21,22)(23,24)$ |
| 6E-1 | $6^{4},2^{4}$ | $4$ | $6$ | $24$ | $( 1, 3)( 2, 4)( 5, 9,29, 7,11,31)( 6,10,30, 8,12,32)(13,27,17,15,25,19)(14,28,18,16,26,20)(21,23)(22,24)$ |
| 6F1 | $6^{4},2^{4}$ | $4$ | $6$ | $24$ | $( 1, 4)( 2, 3)( 5,10,29, 8,11,32)( 6, 9,30, 7,12,31)(13,28,17,16,25,20)(14,27,18,15,26,19)(21,24)(22,23)$ |
| 6F-1 | $6^{4},2^{4}$ | $4$ | $6$ | $24$ | $( 1,24)( 2,23)( 3,22)( 4,21)( 5,14,29,18,11,26)( 6,13,30,17,12,25)( 7,16,31,20, 9,28)( 8,15,32,19,10,27)$ |
| 6G1 | $6^{4},2^{4}$ | $4$ | $6$ | $24$ | $( 1,21)( 2,22)( 3,23)( 4,24)( 5,15,29,19,11,27)( 6,16,30,20,12,28)( 7,13,31,17, 9,25)( 8,14,32,18,10,26)$ |
| 6G-1 | $6^{4},2^{4}$ | $4$ | $6$ | $24$ | $( 1,22)( 2,21)( 3,24)( 4,23)( 5,28,11,20,29,16)( 6,27,12,19,30,15)( 7,26, 9,18,31,14)( 8,25,10,17,32,13)$ |
Malle's constant $a(G)$: $1/16$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $96=2^{5} \cdot 3$ | 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: | 96.228 | magma: IdentifyGroup(G);
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| Character table: | 32 x 32 character table |
magma: CharacterTable(G);