Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #1839 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22×C4 | C22×D3 |
| commutator | C22×C3 | C22×C8 |
| Fitting | C3×C43 | C2 |
| socle, prime radical | C23×C3 | C22×C4 |
| cosocle | C22×C3×C4 | C23 |
| hypercenter | C43 | D3 |
| hypocenter | C3 | #1839 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C4 ◁ C43 | |
240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(13,17)(16,19)(20,22,23,27)(21,25,24,26);(12,14,15,18)(13,16,17,19)(21,24)(25,26);(2,3)(4,5,6,8,7,9,10,11)(12,13)(14,16)(15,17)(18,19)(20,21)(22,26)(23,24)(25,27)Polycyclic presentation:
2^7,3|a^2=f,b^2=de,c^2=d,f^2=g,b^a=bd,c^a=ce,h^a=h^2120 conjugacy classes — view the full character table.
| maximal subgroup of | #10681722, #10681723, #10681733, … |
| maximal normal subgroup of | #10681722, #10681723, #10681733, … |
| maximal quotient of | #10688907, #10688911, #10700792, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1969);