Finite Group DatabasebetaFinite Group Databasebeta| order | 22932 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊#11763985 |
| Fitting, socle, derived, Hall | C72×C13 ⋊ C32×C4 |
| Hall | ΓL(1,𝔽8)×#63270 ⋊ C4 |
| Hall | C7⋊(C7⋊#261) ⋊ C32 |
| Sylow | C72 ⋊ C3×AGL(1,𝔽13) |
| Sylow | C13 ⋊ ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C72×C13 | C32×C4 |
| prime radical | ΓL(1,𝔽8)×#10587716 | C2 |
| cosocle | C72×D13 | C2×C32 |
| LCS | G ▷ C72×C13 | |
| UCS | C1 | |
168 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 273 |
Polycyclic presentation:
2,3^2,2,7^2,13|a^2=d,e^a=e^6,f^a=f^6,g^a=g^5,f^b=f^4,g^b=g^9,e^c=e^2,g^d=g^1287 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22932, 850);