Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #3028⋊C3 |
| Fitting, Sylow | #3028 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#120⋊C3 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C22×C3 |
| Frattini | C23 | C22×A4 |
| Fitting | #3028 | C3 |
| prime radical | #5950 | C2 |
| hypercenter | C4 | #120⋊C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2 ◁ C4 | |
156 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 156 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,3,6,14)(2,7,10,11)(4,27,20,29,16,28,18,13,17,12,5,30)(8,26,24,25,19,31,15,22,23,21,9,32);(1,23,6,24)(2,20,10,17)(3,32,14,31)(4,19)(5,8)(7,28,11,30)(9,18)(12,22)(13,25)(15,16)(21,27)(26,29)Polycyclic presentation:
2^2,3,2^5|a^2=h,b^2=h,d^2=h,e^2=h,b^a=bh,d^a=df,e^a=eg,d^c=eg,e^c=defh,f^c=g,g^c=fgh,g^d=gh,f^e=fh23 conjugacy classes — view the full character table.
| maximal subgroup of | #32896⋊C3, #32897⋊C3, #32947⋊C3, … |
| maximal normal subgroup of | #32896⋊C3, #32897⋊C3, #32947⋊C3, … |
| maximal quotient of | #20610⋊C3, #21481⋊C3, #23470⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5838);