Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#385 ⋊ C2 |
| Sylow | C3 ⋊ #2732 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#385 | C2 |
| prime radical | C22×#67 | C22 |
| hypercenter | #385 | D3 |
| hypocenter | C3 | #2732 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #385 | |
528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(10,15)(12,14)(16,19,20,18)(17,21,22,23);(5,7)(9,13)(10,15)(17,22)(21,23);(4,5)(6,7)(8,10)(9,12)(11,15)(13,14)(16,18,20,19)(17,21,22,23);(2,3)(8,9)(10,14)(11,13)(12,15)(16,17,20,22)(18,21,19,23)Polycyclic presentation:
2^7,3|a^2=e,b^2=e,d^2=e,b^a=bef,c^a=cf,d^a=de,h^a=h^2,c^b=cg,d^b=def60 conjugacy classes — view the full character table.
| maximal subgroup of | #11002777, #11002782, #11002790, … |
| maximal normal subgroup of | #11002777, #11002782, #11002790, … |
| maximal quotient of | #10723066, #10723069, #10723190, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 7117);