Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×C4×#41 ⋊ C2 |
| Sylow | C3 ⋊ #3508 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×C4×#41 | C2 |
| hypercenter | C4×#41 | D3 |
| hypocenter | C3 | #3508 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C4×#41 | |
1232 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)(12,14,15,18)(13,17,16,19);(13,16)(17,19);(4,7,8,6)(5,9,10,11);(4,6,8,7)(5,9,10,11)(12,13)(14,17)(15,16)(18,19);(2,3)(4,5)(6,9)(7,11)(8,10)(13,16)(14,18)Polycyclic presentation:
2^7,3|b^2=g,c^2=g,e^2=f,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 | #10687137, #10687141, #10687420, … |
| maximal normal subgroup of | #10687137, #10687141, #10687420, … |
| maximal quotient of | #10955333, #10955592, #10955638, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18424);