Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #3026⋊C3 |
| Fitting, Sylow | #3026 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#120⋊C3 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C22×C3 |
| Frattini | C23 | C22×A4 |
| Fitting | #3026 | C3 |
| hypercenter | C22 | #120⋊C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2 ◁ C22 | |
180 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 180 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:
(2,6)(3,10,13)(4,11,16)(7,15,12,9,14,8);(1,15,5,14)(2,16,6,10)(3,8)(4,7)(9,13)(11,12)Polycyclic presentation:
2^2,3,2^5|b^a=bh,d^a=dfh,e^a=egh,d^c=de,e^c=dh,f^c=fg,g^c=f,g^d=gh,f^e=fh23 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#3026⋊C3, #60398⋊C3, #60399⋊C3, … |
| maximal normal subgroup of | C2×#3026⋊C3, #60398⋊C3, #60399⋊C3, … |
| maximal quotient of | #20609⋊C3, #21481⋊C3, #21482⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5836);