Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | C3×#384 ⋊ C2 |
| Sylow | C3 ⋊ #2795 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#384 | C2 |
| prime radical | C22×D12 | C22 |
| hypercenter | #384 | D3 |
| hypocenter | C3 | #2795 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #384 | |
420 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 420 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(12,14,15,19)(13,17,16,18)(20,23,24,22)(21,25,27,26);(6,10)(8,11)(20,22,24,23)(21,26,27,25);(2,3)(4,6)(5,8)(7,10)(9,11)(13,16)(14,19)(20,22,24,23)(21,25,27,26);(4,5,7,9)(6,8,10,11)(12,13,15,16)(14,18,19,17)(20,21,24,27)(22,25,23,26)Polycyclic presentation:
2^7,3|a^2=e,b^2=fg,c^2=fg,d^2=eg,b^a=bef,c^a=cfg,d^a=deg,c^b=cg,d^b=def,h^b=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10707537, #10707541, #10708123, … |
| maximal normal subgroup of | #10707537, #10707541, #10708123, … |
| maximal quotient of | #10723263, #10723739, #10724119, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6929);