Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C22×C3×#41 ⋊ C2 |
| Sylow | C3 ⋊ #3487 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C22×C3×#41 | C2 |
| hypercenter | C22×#41 | D3 |
| hypocenter | C3 | #3487 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×#41 | |
1536 conjugacy classes of subgroups; only the 549 normal subgroups are shown.
This group has 1536 conjugacy classes of subgroups — too many to draw. Only its 549 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2,3)(9,17)(14,18)(15,16)(22,23);(8,12,13,11)(9,16,17,15)(10,19,20,21)(14,22,18,23);(5,7)(8,11,13,12)(9,15,17,16)(10,19,20,21)(14,22,18,23);(8,10,13,20)(9,14,17,18)(11,19,12,21)(15,22,16,23);(2,3)(4,5)(6,7)(8,9)(10,18)(11,15)(12,16)(13,17)(14,20)(19,23)(21,22)Polycyclic presentation:
2^7,3|b^2=f,c^2=f,d^2=f,b^a=bf,c^a=cg,e^a=ef,h^a=h^2,d^b=df84 conjugacy classes — view the full character table.
| maximal subgroup of | #10687370, #10687372, #10710723, … |
| maximal normal subgroup of | #10687370, #10687372, #10710723, … |
| maximal quotient of | #10955217, #10955482, #10956125, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19850);