Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #3459⋊C3 |
| Fitting, Sylow | #3459 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | C22⋊A4 |
| commutator, cosocle, hypocenter | #543 | C2×C3 |
| Frattini, socle | C22 | C2×C22⋊A4 |
| Fitting | #3459 | C3 |
| LCS | G ▷ #543 | |
| UCS | C1 ◁ C2×C4 | |
250 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,3)(6,7,8)(9,10,13,16)(11,21,20,23,17,15,19,22,12,14,18,24);(3,8)(5,7);(1,7)(2,3)(4,5)(6,8)(9,17,13,18)(10,21,16,22)(11,12,19,20)(14,15,23,24)Polycyclic presentation:
2,3,2^6|a^2=h,c^2=h,e^2=h,c^b=cegh,d^b=fg,e^b=c,f^b=df,d^c=dg,e^c=eh,f^c=fg,e^d=eg36 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18227);