Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#398 ⋊ C2 |
| Sylow | C3 ⋊ #2468 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#398 | C2 |
| prime radical | C22×D12 | C22 |
| hypercenter | #398 | D3 |
| hypocenter | C3 | #2468 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #398 | |
532 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)(8,10,11,14)(9,13,12,15)(16,19,20,18)(17,22,23,21);(5,7)(9,12)(13,15)(16,18,20,19)(17,22,23,21);(4,5)(6,7)(16,18,20,19)(17,21,23,22);(2,3)(8,9)(10,13)(11,12)(14,15)(16,17)(18,22)(19,21)(20,23)Polycyclic presentation:
2^7,3|b^2=e,c^2=e,d^2=ef,b^a=be,c^a=cf,d^a=de,h^a=h^2,c^b=cg,h^d=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2468, C2×#77754, #10963359, … |
| maximal normal subgroup of | D3×#2468, C2×#77754, #10963359, … |
| maximal quotient of | #10717302, #10717475, #10718130, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 10330);