Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C3×#525 ⋊ C2 |
| Sylow | C3 ⋊ #3449 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C3×#525 | C2 |
| hypercenter | #525 | D3 |
| hypocenter | C3 | #3449 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #525 | |
1556 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2,3)(8,11,12,19)(9,15,14,21)(10,18,17,23)(13,20,16,22);(9,14)(13,16)(15,21)(20,22);(5,7)(10,17)(13,16)(18,23)(20,22);(2,3)(8,10)(9,13)(11,18)(12,17)(14,16)(15,20)(19,23)(21,22);(4,5)(6,7)(8,9)(10,16)(11,15)(12,14)(13,17)(18,22)(19,21)(20,23)Polycyclic presentation:
2^7,3|e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=cf,h^b=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10796369, #10796840, #10796955, … |
| maximal normal subgroup of | #10796369, #10796840, #10796955, … |
| maximal quotient of | #10953757, #10953760, #10953811, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18611);