Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting, Sylow | #3483 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | C22⋊A4 |
| commutator, cosocle, hypocenter | Q82 | C2×C3 |
| Frattini, socle | C22 | C2×C22⋊A4 |
| Fitting | #3483 | C3 |
| LCS | G ▷ Q82 | |
| UCS | C1 ◁ C2×C4 | |
252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,6,8)(3,7,4)(9,10,12,15)(11,21,19,22,17,16,18,24,13,14,20,23);(1,7,5,6)(2,8,3,4)(9,12)(10,15)(11,18)(13,19)(14,22)(16,23)(17,20)(21,24);(1,8,5,4)(2,7,3,6)(9,20,12,17)(10,24,15,21)(11,13,18,19)(14,16,22,23)Polycyclic presentation:
2,3,2^6|a^2=gh,c^2=gh,d^2=g,e^2=h,f^2=g,c^b=cdefh,d^b=dfg,e^b=cf,f^b=dh,d^c=dg,e^c=eh,f^d=fg38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18230);