Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×#525 ⋊ C2 |
| Sylow | C3 ⋊ #3459 |
| 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 | #3459 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #525 | |
1160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 19 points:
(1,2,3)(12,14,15,18)(13,17,16,19);(5,9)(8,10);(6,11)(8,10)(13,16)(17,19);(2,3)(4,6)(5,8)(7,11)(9,10);(4,5)(6,10)(7,9)(8,11)(12,13)(14,17)(15,16)(18,19)Polycyclic presentation:
2^7,3|e^2=g,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 | #10686967, #10686969, #10767289, … |
| maximal normal subgroup of | #10686967, #10686969, #10767289, … |
| maximal quotient of | #10953858, #10953862, #10954125, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18612);