Finite Group DatabasebetaFinite Group Databasebeta| order | 9492 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊#90673 |
| unambiguous semidirect product | (C7⋊#90673)⋊C3 |
| Fitting, socle, derived, Hall | C7×C113 ⋊ C3×C4 |
| Hall | C7⋊#90673 ⋊ C3 |
| Hall | ΓL(1,𝔽8)×C113 ⋊ C4 |
| Sylow | C113 ⋊ ΓL(1,𝔽8)⋊C4 |
| Hall | ΓL(1,𝔽8) ⋊ #90673 |
| Sylow | C7 ⋊ C3×#90673 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C7×C113 | C3×C4 |
| prime radical | ΓL(1,𝔽8)×D113 | C2 |
| cosocle | C7×D113 | C2×C3 |
| LCS | G ▷ C7×C113 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,2,7,113|a^2=c,d^a=d^6,e^a=e^15,d^b=d^2,e^c=e^112154 conjugacy classes — view the full character table.
| maximal subgroup of | C3×ΓL(1,𝔽8)⋊#90673, C3⋊(ΓL(1,𝔽8)⋊#90673) |
| maximal normal subgroup of | C3×ΓL(1,𝔽8)⋊#90673 |
| maximal quotient of | (C7⋊C9)⋊#90673, C3×ΓL(1,𝔽8)⋊#90673, C3⋊(ΓL(1,𝔽8)⋊#90673) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(9492, 25);