Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C4 |
| unambiguous semidirect product | #3027⋊C3 |
| Fitting, Sylow | #3027 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#120⋊C3 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C22×C3 |
| Frattini | C23 | C22×A4 |
| Fitting | #3027 | C3 |
| hypercenter | C4 | #120⋊C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2 ◁ C4 | |
172 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,5,6)(3,15,4,13,10,14,9,16,12,7,11,8);(1,10,5,11)(2,16,6,15)(3,4)(7,14)(8,13)(9,12)Polycyclic presentation:
2^2,3,2^5|b^2=h,d^2=h,b^a=bh,d^a=df,e^a=eg,d^c=defgh,e^c=dgh,f^c=fgh,g^c=f,g^d=gh,f^e=fh23 conjugacy classes — view the full character table.
| maximal subgroup of | #32894⋊C3, #32895⋊C3, #32947⋊C3, … |
| maximal normal subgroup of | #32894⋊C3, #32895⋊C3, #32947⋊C3, … |
| maximal quotient of | #20610⋊C3, #21482⋊C3, #23469⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5837);