Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×#535 ⋊ C2 |
| Sylow | C3 ⋊ #3551 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×#535 | C2 |
| hypercenter | #535 | D3 |
| hypocenter | C3 | #3551 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #535 | |
1050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(12,16,17,15)(13,21,19,20)(14,24,23,25)(18,26,22,27);(5,9)(8,10)(12,15,17,16)(13,20,19,21)(14,24,23,25)(18,26,22,27);(2,3)(4,6,7,11)(5,8,9,10)(13,19)(14,23)(20,21)(24,25);(2,3)(6,11)(8,10)(12,14)(13,18)(15,25)(16,24)(17,23)(19,22)(20,27)(21,26);(4,5)(6,10)(7,9)(8,11)(12,13)(14,22)(15,21)(16,20)(17,19)(18,23)(24,26)(25,27)Polycyclic presentation:
2^7,3|c^2=fg,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,h^b=h^2,h^c=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10711045, #10711049, #10711088, … |
| maximal normal subgroup of | #10711045, #10711049, #10711088, … |
| maximal quotient of | #10958045, #10958411, #10958664, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18903);