Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #2742 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#385 | C2 |
| prime radical | C24×C3 | C23 |
| hypercenter | #385 | D3 |
| hypocenter | C3 | #2742 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #385 | |
416 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 141 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)(5,9)(8,11)(20,22,23,26)(21,25,24,27);(5,9)(8,11)(14,18)(16,19)(21,24)(25,27);(2,3)(4,6,7,10)(5,8,9,11)(12,14)(13,16)(15,18)(17,19)(21,24)(22,26);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13,15,17)(14,16,18,19)(20,21,23,24)(22,25,26,27)Polycyclic presentation:
2^7,3|a^2=e,b^2=ef,d^2=eg,b^a=beg,c^a=cfg,d^a=def,h^a=h^2,c^b=cg,d^b=deg,h^b=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10707743, #10707747, #10708227, … |
| maximal normal subgroup of | #10707743, #10707747, #10708227, … |
| maximal quotient of | #10722650, #10723119, #10723801, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 7120);