Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C3×#568 ⋊ C2 |
| Sylow | C3 ⋊ #3585 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C23×D12 |
| commutator, cosocle | C3×C4 | C25 |
| Frattini | C4 | C24×D3 |
| Fitting | C3×#568 | C2 |
| socle | C2×C3 | C23×D4 |
| hypercenter | #568 | D3 |
| hypocenter | C3 | #3585 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ #568 | |
1464 conjugacy classes of subgroups; only the 545 normal subgroups are shown.
This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 545 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) = 48 |
A minimal faithful permutation representation — 5 generators on 19 points:
(1,2,3)(5,11)(9,12)(10,17)(16,18);(6,13)(10,17)(14,19)(16,18);(4,6)(5,10)(7,14)(8,13)(9,16)(11,17)(12,18)(15,19);(2,3)(5,9)(7,15)(10,16)(11,12)(14,19)(17,18);(2,3)(4,5)(6,10)(7,9)(8,11)(12,15)(13,17)(14,16)(18,19)Polycyclic presentation:
2^7,3|f^2=g,b^a=bf,e^a=eg,f^a=fg,h^a=h^2,f^b=fg,h^b=h^2,d^c=dg75 conjugacy classes — view the full character table.
| maximal subgroup of | #10826483, #10826484, #10826505, … |
| maximal normal subgroup of | #10826483, #10826484, #10826505, … |
| maximal quotient of | #11658759, #11658772, #11658872, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19743);