Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C28 |
| unambiguous semidirect product | C22⋊#997 |
| Fitting, derived, Sylow | C22×H(5,𝔽2) ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C22⋊(C22⋊A4) |
| commutator, Fitting, cosocle, hypocenter | C22×H(5,𝔽2) | C3 |
| socle | C23 | C22⋊A4 |
| LCS | G ▷ C22×H(5,𝔽2) | |
| UCS | C1 ◁ C2 | |
1195 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 4 generators on 12 points:
(2,3,4)(6,7,9)(10,11,12);(1,4)(2,3);(6,10)(7,11);(5,9)(6,7)(8,12)(10,11)Polycyclic presentation:
3,2^7|b^a=c,c^a=bc,d^a=e,e^a=de,f^a=g,g^a=fg,e^b=eh,d^c=dh28 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20126);