Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1842 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22×C4 | C22×D3 |
| commutator | C22×C3 | C22×C8 |
| Fitting | C3×C43 | C2 |
| socle, prime radical | C23×C3 | C22×C4 |
| cosocle | C22×C3×C4 | C23 |
| hypercenter | C43 | D3 |
| hypocenter | C3 | #1842 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C4 ◁ C43 | |
300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(8,9,10,11);(4,5,6,7);(2,3)(5,7)(9,11)(12,13,14,16,15,17,18,19)Polycyclic presentation:
2^7,3|a^2=f,b^2=d,c^2=e,f^2=g,b^a=bd,c^a=ce,h^a=h^2120 conjugacy classes — view the full character table.
| maximal subgroup of | #10681755, #10681756, #10681778, … |
| maximal normal subgroup of | #10681755, #10681756, #10681778, … |
| maximal quotient of | #10688925, #10688928, #10688932, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1970);