Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#388 ⋊ C2 |
| Sylow | C3 ⋊ #2707 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#388 | C2 |
| prime radical | C22×#67 | C22 |
| hypercenter | #388 | D3 |
| hypocenter | C3 | #2707 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #388 | |
490 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)(9,13)(12,15)(16,19,20,18)(17,22,21,23);(2,3)(5,7)(9,13)(12,15)(16,18,20,19)(17,22,21,23);(4,5)(6,7)(8,10,11,14)(9,12,13,15)(17,21)(18,19);(8,9,11,13)(10,12,14,15)(16,17)(18,23)(19,22)(20,21)Polycyclic presentation:
2^7,3|a^2=e,b^2=e,c^2=ef,d^2=ef,b^a=bef,c^a=cf,d^a=de,c^b=cg,d^b=def,h^c=h^2,h^d=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #11002205, #11002215, #11002220, … |
| maximal normal subgroup of | #11002205, #11002215, #11002220, … |
| maximal quotient of | #10722506, #10722671, #10722746, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 8265);