Finite Group DatabasebetaFinite Group Databasebeta| order | 30492 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C2×C3×C11 |
| unambiguous semidirect product | (C22⋊ΓL(1,𝔽8))⋊(C112⋊C3) |
| unambiguous semidirect product | (C22⋊(C112⋊C3))⋊ΓL(1,𝔽8) |
| unambiguous semidirect product | (C7⋊(C112⋊C3))⋊A4 |
| Fitting, socle, derived, Hall | C22×C7×C112 ⋊ C32 |
| Hall | C7×C112 ⋊ C3×A4 |
| Hall | C22×C112 ⋊ C3×ΓL(1,𝔽8) |
| Sylow | C112 ⋊ A4×ΓL(1,𝔽8) |
| Hall | C22×C7 ⋊ C3×C112⋊C3 |
| Sylow | C7 ⋊ A4×C112⋊C3 |
| Sylow | C22 ⋊ ΓL(1,𝔽8)×C112⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C22×C7×C112 | C32 |
| LCS | G ▷ C22×C7×C112 | |
| UCS | C1 | |
96 conjugacy classes of subgroups.
| transitive degree | μt(G) = 924 |
Polycyclic presentation:
3^2,2^2,7,11^2|e^a=e^2,f^a=f^6g^3,g^a=f^4g^4,c^b=cd,d^b=c,e^b=e^2500 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30492, 362);