Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | C3×#388 ⋊ C2 |
| Sylow | C3 ⋊ #2735 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#388 | C2 |
| prime radical | C24×D3 | C22 |
| hypercenter | #388 | D3 |
| hypocenter | C3 | #2735 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #388 | |
552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(5,9)(6,11)(13,17)(16,19);(2,3)(13,17)(16,19)(22,26)(24,27);(4,6)(5,8)(7,11)(9,10)(12,14,15,18)(13,16,17,19)(20,22)(21,24)(23,26)(25,27);(4,5)(6,10)(7,9)(8,11)(12,13)(14,16)(15,17)(18,19)(20,21,23,25)(22,24,26,27)Polycyclic presentation:
2^7,3|a^2=g,b^2=f,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,h^c=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #11002282, #11002285, #11002286, … |
| maximal normal subgroup of | #11002282, #11002285, #11002286, … |
| maximal quotient of | #10722577, #10722578, #10722589, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 8509);