Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C3×C4×#41 ⋊ C2 |
| Sylow | C3 ⋊ #3499 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×C4×#41 | C2 |
| hypercenter | C4×#41 | D3 |
| hypocenter | C3 | #3499 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C4×#41 | |
1476 conjugacy classes of subgroups; only the 557 normal subgroups are shown.
This group has 1476 conjugacy classes of subgroups — too many to draw. Only its 557 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) = 192 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2,3)(8,11,12,20)(9,14,15,19)(10,18,17,23)(13,21,16,22);(10,17)(13,16)(18,23)(21,22);(4,5,6,7);(8,10)(9,13)(11,18)(12,17)(14,21)(15,16)(19,22)(20,23);(2,3)(5,7)(8,9,12,15)(10,16,17,13)(11,19,20,14)(18,21,23,22)Polycyclic presentation:
2^7,3|a^2=f,c^2=g,e^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 | #10687102, #10687106, #10687415, … |
| maximal normal subgroup of | #10687102, #10687106, #10687415, … |
| maximal quotient of | #10955291, #10955564, #10955583, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18403);