Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#382 ⋊ C2 |
| Sylow | C3 ⋊ #2349 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C3×#382 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C22×D12 | C22 |
| hypercenter | #382 | D3 |
| hypocenter | C3 | #2349 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #382 | |
704 conjugacy classes of subgroups; only the 233 normal subgroups are shown.
This group has 704 conjugacy classes of subgroups — too many to draw. Only its 233 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) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2)(14,19)(16,18);(8,9,10,11)(13,17)(16,18);(2,3)(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=ef,b^a=be,c^a=cf,d^b=de,h^b=h^2,h^d=h^278 conjugacy classes — view the full character table.
| maximal subgroup of | #10688751, #10688756, #10691091, … |
| maximal normal subgroup of | #10688751, #10688756, #10691091, … |
| maximal quotient of | #10716615, #10716635, #10716656, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6804);