Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#385 ⋊ C2 |
| Sylow | C3 ⋊ #2315 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C3×#385 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C24×D3 | C22 |
| hypercenter | #385 | D3 |
| hypocenter | C3 | #2315 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #385 | |
816 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 19 points:
(2,3)(13,17)(16,18);(1,2,3)(8,9,10,11)(14,19)(16,18);(12,14)(13,16)(15,19)(17,18);(4,5,6,7)(9,11)(12,13)(14,18)(15,17)(16,19)Polycyclic presentation:
2^7,3|a^2=g,c^2=f,b^a=be,c^a=cf,d^a=de,c^b=ce,h^d=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10704363, #10704609, #10796277, … |
| maximal normal subgroup of | #10704363, #10704609, #10796277, … |
| maximal quotient of | #10716440, #10716467, #10716468, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 7167);