Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #3025⋊C3 |
| Fitting, Sylow | #3025 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #120⋊C3 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C22×C3 |
| Frattini | C23 | C22×A4 |
| Fitting | #3025 | C3 |
| socle | C2 | C2×#120⋊C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C4 | |
171 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 171 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:
(1,2,5,8)(3,7,15,12,4,13,9,14,10,6,11,16);(1,10,5,15)(2,13,8,16)(3,11)(4,9)(6,14)(7,12)Polycyclic presentation:
2^2,3,2^5|b^2=h,e^2=h,d^a=dfh,e^a=egh,d^c=efh,e^c=defg,f^c=g,g^c=fg,g^d=gh,f^e=fh32 conjugacy classes — view the full character table.
| maximal subgroup of | #21483⋊C3, #21484⋊C3, C2×#3025⋊C3, … |
| maximal normal subgroup of | #21483⋊C3, #21484⋊C3, C2×#3025⋊C3, … |
| maximal quotient of | #20608⋊C3, C4×B(4,𝔽2)⋊C3, C4×#458⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5835);