Finite Group DatabasebetaFinite Group Databasebeta| order | 30492 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C3×C11 |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊((C11⋊Dic11)⋊C3) |
| unambiguous semidirect product | (C11⋊(C11⋊(ΓL(1,𝔽8)⋊C4)))⋊C3 |
| unambiguous semidirect product | #23778989⋊C3 |
| unambiguous semidirect product | #23778995⋊C3 |
| derived, Hall | C7×C112 ⋊ C32×C4 |
| Hall | ΓL(1,𝔽8)×C112⋊C3 ⋊ C4 |
| Hall | C7⋊(C11⋊Dic11) ⋊ C32 |
| Sylow | C112 ⋊ C3×ΓL(1,𝔽8)⋊C4 |
| Sylow | C7 ⋊ C3×(C11⋊Dic11)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | ΓL(1,𝔽8)⋊((C11⋊D11)⋊C3) |
| commutator, hypocenter | C7×C112 | C32×C4 |
| Fitting, socle, cosocle | C2×C7×C112 | C2×C32 |
| prime radical | C2×ΓL(1,𝔽8)×C112⋊C3 | C2 |
| LCS | G ▷ C7×C112 | |
| UCS | C1 ◁ C2 | |
120 conjugacy classes of subgroups.
| transitive degree | μt(G) = 924 |
Polycyclic presentation:
2,3^2,7,11^2,2|a^2=g,d^a=d^6,e^a=e^10,f^a=f^10,e^b=e^7f^5,f^b=e^4f^3,d^c=d^2242 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30492, 220);