Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Fitting | C3×#388 ⋊ C2 |
| Sylow | C3 ⋊ #2342 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C3×#388 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C24×D3 | C22 |
| hypercenter | #388 | D3 |
| hypocenter | C3 | #2342 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #388 | |
792 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2)(9,11);(2,3)(13,17)(16,19);(8,9)(10,11)(12,14,15,18)(13,16,17,19);(4,5,6,7)(9,11)(12,13)(14,16)(15,17)(18,19)Polycyclic presentation:
2^7,3|a^2=g,b^2=f,b^a=be,c^a=cf,d^b=de,h^c=h^2,h^d=h^278 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2342, C2×#75832, #10954458, … |
| maximal normal subgroup of | D3×#2342, C2×#75832, #10954458, … |
| maximal quotient of | #10716588, #10716591, #10716617, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 8408);