Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×#546 ⋊ C2 |
| Sylow | C3 ⋊ #3558 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×#546 | C2 |
| hypercenter | #546 | D3 |
| hypocenter | C3 | #3558 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #546 | |
1160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 19 points:
(2,3)(5,9)(10,11);(1,2,3)(4,7,8,6)(5,11,9,10)(12,14,15,18)(13,17,16,19);(4,6,8,7)(5,10,9,11)(12,13)(14,17)(15,16)(18,19);(5,9)(6,7)(13,16)(17,19);(4,5)(6,11)(7,10)(8,9)(13,16)(14,18)Polycyclic presentation:
2^7,3|c^2=f,d^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,h^e=h^263 conjugacy classes — view the full character table.
| maximal subgroup of | #11688693, #11688697, #11688750, … |
| maximal normal subgroup of | #11688693, #11688697, #11688750, … |
| maximal quotient of | #10958121, #10958189, #10958432, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19249);