Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C2×C3×C4×D4 ⋊ C2 |
| Sylow | C3 ⋊ #3520 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C2×C3×C4×D4 | C2 |
| hypercenter | C2×C4×D4 | D3 |
| hypocenter | C3 | #3520 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 | |
1124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 19 points:
(1,2,3)(5,9)(8,10)(12,14,15,18)(13,17,16,19);(6,11)(8,10);(5,9)(8,10)(13,16)(17,19);(4,6)(5,8)(7,11)(9,10);(2,3)(4,5,7,9)(6,10,11,8)(12,13)(14,17)(15,16)(18,19)Polycyclic presentation:
2^7,3|a^2=f,e^2=fg,b^a=bf,c^a=cg,e^a=ef,h^a=h^2,d^b=df84 conjugacy classes — view the full character table.
| maximal subgroup of | #10687081, #10687085, #10762392, … |
| maximal normal subgroup of | #10687081, #10687085, #10762392, … |
| maximal quotient of | #10955403, #10955907, #10955936, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18343);