Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C3×#384 ⋊ C2 |
| Sylow | C3 ⋊ #2720 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#384 | C2 |
| prime radical | C23×D3 | C23 |
| hypercenter | #384 | D3 |
| hypocenter | C3 | #2720 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #384 | |
472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(8,11,12,10)(9,14,13,15)(17,22)(21,23);(2,3)(5,7)(8,10,12,11)(9,14,13,15)(16,19,20,18)(17,21,22,23);(4,5)(6,7)(9,13)(10,11)(16,18,20,19)(17,21,22,23);(8,9,12,13)(10,15,11,14)(16,17)(18,21)(19,23)(20,22)Polycyclic presentation:
2^7,3|a^2=e,b^2=ef,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,h^c=h^2,h^d=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #11002621, #11002628, #11002674, … |
| maximal normal subgroup of | #11002621, #11002628, #11002674, … |
| maximal quotient of | #10722881, #10723061, #10723126, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6923);