Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C3×#532 ⋊ C2 |
| Sylow | C3 ⋊ #3497 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×#532 | C2 |
| hypercenter | #532 | D3 |
| hypocenter | C3 | #3497 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #532 | |
1492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2,3)(9,15)(13,16)(14,23)(21,22);(8,11,12,20)(9,14,15,23)(10,17,18,19)(13,21,16,22);(4,5,6,7);(2,3)(8,10,12,18)(9,13,15,16)(11,19,20,17)(14,22,23,21);(5,7)(8,9)(10,16)(11,14)(12,15)(13,18)(17,22)(19,21)(20,23)Polycyclic presentation:
2^7,3|b^2=f,c^2=g,d^2=f,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 | #10686738, #10686741, #10687157, … |
| maximal normal subgroup of | #10686738, #10686741, #10687157, … |
| maximal quotient of | #10955280, #10955543, #10955549, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18798);