Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C2×C3×#117 ⋊ C2 |
| Sylow | C3 ⋊ #3493 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C2×C3×#117 | C2 |
| hypercenter | C2×#117 | D3 |
| hypocenter | C3 | #3493 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×#117 | |
1040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(13,21)(18,22)(19,20)(26,27);(12,16,17,15)(13,20,21,19)(14,23,24,25)(18,26,22,27);(5,9)(8,11)(12,15,17,16)(13,19,21,20)(14,23,24,25)(18,26,22,27);(4,6,7,10)(5,8,9,11)(12,14)(13,18)(15,23)(16,25)(17,24)(19,26)(20,27)(21,22);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13)(14,22)(15,19)(16,20)(17,21)(18,24)(23,27)(25,26)Polycyclic presentation:
2^7,3|b^2=g,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 | #10686712, #10686715, #10687365, … |
| maximal normal subgroup of | #10686712, #10686715, #10687365, … |
| maximal quotient of | #10955256, #10955482, #10955696, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18322);