Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #2062 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #937 |
| commutator, prime radical, cosocle | C24×C3 | C23 |
| Frattini | C24 | C22×D3 |
| Fitting | C3×#385 | C2 |
| socle | C22×C3 | #120 |
| hypercenter | #385 | D3 |
| hypocenter | C3 | #2062 |
| LCS | G ▷ C24×C3 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ #385 | |
314 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 314 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(8,11)(12,15,16,14)(13,17,18,19);(4,6)(5,8)(7,10)(9,11)(12,14,16,15)(13,17,18,19);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13,16,18)(14,19,15,17)Polycyclic presentation:
2^7,3|a^2=d,b^2=d,c^2=d,b^a=bd,c^a=ce,f^a=fg,h^a=h^2,c^b=cf,e^b=eg54 conjugacy classes — view the full character table.
| maximal subgroup of | #10708041, #10708043, #10708048, … |
| maximal normal subgroup of | #10708041, #10708043, #10708048, … |
| maximal quotient of | #10699588, #10699590, #10699593, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2170);