Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×#532 ⋊ C2 |
| Sylow | C3 ⋊ #3529 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C3×#532 | C2 |
| hypercenter | #532 | D3 |
| hypocenter | C3 | #3529 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #532 | |
1058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(12,15,16,23)(13,19,18,25)(14,22,21,27)(17,24,20,26);(5,9)(8,11)(13,18)(14,21)(19,25)(22,27);(2,3)(5,9)(6,10)(14,21)(17,20)(22,27)(24,26);(4,6)(5,8)(7,10)(9,11)(12,14)(13,17)(15,22)(16,21)(18,20)(19,24)(23,27)(25,26);(4,5)(6,8)(7,9)(10,11)(12,13)(14,20)(15,19)(16,18)(17,21)(22,26)(23,25)(24,27)Polycyclic presentation:
2^7,3|e^2=f,b^a=bf,c^a=cg,d^a=dfg,c^b=cfg,d^b=df,h^c=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | #10687619, #10687622, #10711024, … |
| maximal normal subgroup of | #10687619, #10687622, #10711024, … |
| maximal quotient of | #10957798, #10957800, #10957826, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18824);