Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #2852⋊C3 |
| Fitting, Sylow | #2852 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#126⋊C3 |
| commutator, cosocle | C2×C42 | C22×C3 |
| Frattini, socle | C23 | C22×A4 |
| Fitting | #2852 | C3 |
| hypercenter | C22 | #126⋊C3 |
| hypocenter | C42 | C3×D4 |
| LCS | G ▷ C2×C42 ▷ C42 | |
| UCS | C1 ◁ C2 ◁ C22 | |
137 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 137 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 20 points:
(2,4)(5,10,7)(8,20,12)(9,13,15)(11,19,14)(16,18,17);(1,2)(3,4)(5,18,6,20)(7,15,10,19)(8,12,11,9)(13,17,16,14)Polycyclic presentation:
2^2,3,2^5|e^2=gh,f^2=g,b^a=bd,c^a=ch,e^a=eg,f^a=fh,c^b=cg,e^c=ef,f^c=eg,g^c=gh,h^c=g28 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#126⋊C3, #36166⋊C3, C2×#2852⋊C3, … |
| maximal normal subgroup of | D4×#126⋊C3, #36166⋊C3, C2×#2852⋊C3, … |
| maximal quotient of | #14683⋊C3, #14710⋊C3, #14711⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5793);