Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C28 |
| unambiguous semidirect product | C22⋊(C22⋊SL(2,𝔽3)) |
| Fitting, derived, Sylow | C24×Q8 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C22⋊(C22⋊A4) |
| commutator, Fitting, cosocle, hypocenter | C24×Q8 | C3 |
| socle | C25 | A4 |
| LCS | G ▷ C24×Q8 | |
| UCS | C1 ◁ C2 | |
1129 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,4,3)(6,7,8)(10,11,15)(13,14,16);(5,8)(6,7);(1,4)(2,3);(9,13,12,15)(10,16,14,11)Polycyclic presentation:
3,2^7|b^2=h,c^2=h,b^a=c,c^a=bch,d^a=de,e^a=d,f^a=dg,g^a=efg,c^b=ch32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20125);