Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C2×C3×#121 ⋊ C2 |
| Sylow | C3 ⋊ #3505 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C2×C3×#121 | C2 |
| hypercenter | C2×#121 | D3 |
| hypocenter | C3 | #3505 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×#121 | |
1152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 19 points:
(1,2,3)(5,9)(8,10);(6,11)(8,10);(12,15,16,14)(13,17,18,19);(2,3)(4,6)(5,8)(7,11)(9,10)(12,14,16,15)(13,17,18,19);(4,5)(6,10)(7,9)(8,11)(12,13)(14,17)(15,19)(16,18)Polycyclic presentation:
2^7,3|b^2=g,c^2=g,b^a=bf,c^a=cg,e^a=ef,d^b=df,h^b=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10686726, #10686730, #10687774, … |
| maximal normal subgroup of | #10686726, #10686730, #10687774, … |
| maximal quotient of | #10955304, #10955541, #10955612, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18513);