Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#471 ⋊ C2 |
| Sylow | C3 ⋊ #3068 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×#121 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#471 | C2 |
| socle | C2×C3 | C2×#121 |
| hypercenter | #471 | D3 |
| hypocenter | C3 | #3068 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #471 | |
466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 19 points:
(2,3)(5,11)(9,17)(10,18)(16,19);(1,2,3)(4,6)(5,10)(7,13)(8,14)(9,16)(11,18)(12,15)(17,19);(4,5,7,9,8,11,15,17)(6,10,13,16,14,18,12,19);(5,9)(6,12)(7,15)(11,17)(13,14)(16,19)Polycyclic presentation:
2^7,3|b^2=e,e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,d^b=dg,f^c=fg,h^d=h^245 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3068, C2×#80930, #11662606, … |
| maximal normal subgroup of | D3×#3068, C2×#80930, #11662606, … |
| maximal quotient of | #10777948, #10777950, #10777981, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 13506);