Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C3×#385 ⋊ C2 |
| Sylow | C3 ⋊ #2711 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#385 | C2 |
| prime radical | C2×#930 | C2 |
| hypercenter | #385 | D3 |
| hypocenter | C3 | #2711 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #385 | |
540 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)(12,14)(16,18,19,23)(17,21,20,22);(2,3)(5,7)(8,10,11,15)(9,12,13,14);(4,5)(6,7)(17,20)(18,23);(2,3)(8,9)(10,14)(11,13)(12,15)(16,17,19,20)(18,22,23,21)Polycyclic presentation:
2^7,3|a^2=e,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=def,h^a=h^2,c^b=cg,d^b=de,h^c=h^2,h^d=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #11002623, #11002625, #11002758, … |
| maximal normal subgroup of | #11002623, #11002625, #11002758, … |
| maximal quotient of | #10722754, #10722980, #10723253, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 7104);