Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#385 ⋊ C2 |
| Sylow | C3 ⋊ #2463 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#385 | C2 |
| prime radical | #4960 | C2 |
| hypercenter | #385 | D3 |
| hypocenter | C3 | #2463 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #385 | |
668 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 668 conjugacy classes of subgroups — too many to draw. Only its 145 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:
(1,2,3)(5,7)(12,14,15,19)(13,17,16,18);(2,3)(5,7)(9,11);(5,7)(8,9)(10,11)(13,16)(17,18);(2,3)(4,5,6,7)(12,13,15,16)(14,18,19,17)Polycyclic presentation:
2^7,3|a^2=e,d^2=ef,b^a=be,c^a=cf,d^a=de,h^a=h^2,c^b=cg,h^c=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | #10707460, #10707464, #10708046, … |
| maximal normal subgroup of | #10707460, #10707464, #10708046, … |
| maximal quotient of | #10717277, #10718212, #10718744, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 7189);