Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Sylow | C3 ⋊ #3458 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×#543 | C2 |
| prime radical | C24×C3 | C23 |
| hypercenter | #543 | D3 |
| hypocenter | C3 | #3458 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #543 | |
996 conjugacy classes of subgroups; only the 509 normal subgroups are shown.
This group has 996 conjugacy classes of subgroups — too many to draw. Only its 509 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 27 points:
(2,3)(12,15,16,23)(13,19,18,25)(14,22,21,27)(17,24,20,26);(1,2,3)(13,18)(17,20)(19,25)(24,26);(4,6,7,11)(5,8,9,10)(14,21)(17,20)(22,27)(24,26);(12,14)(13,17)(15,22)(16,21)(18,20)(19,24)(23,27)(25,26);(4,5,7,9)(6,10,11,8)(12,13)(14,20)(15,19)(16,18)(17,21)(22,26)(23,25)(24,27)Polycyclic presentation:
2^7,3|a^2=g,c^2=g,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=cf,h^e=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | #10767470, #10767475, #10767479, … |
| maximal normal subgroup of | #10767470, #10767475, #10767479, … |
| maximal quotient of | #10953810, #10954028, #10954338, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19147);