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 ⋊ #2363 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C3×#388 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C2×#884 | C2 |
| hypercenter | #388 | D3 |
| hypocenter | C3 | #2363 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #388 | |
688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(2,3)(12,15,16,14)(13,18,17,19);(1,2,3)(9,11)(12,14,16,15)(13,18,17,19);(4,6,7,5)(12,13,16,17)(14,18,15,19);(4,5,7,6)(8,9)(10,11)(13,17)(18,19)Polycyclic presentation:
2^7,3|a^2=g,b^2=eg,c^2=e,d^2=e,b^a=be,c^a=cf,d^b=de,h^d=h^278 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2363, C2×#75480, #10954747, … |
| maximal normal subgroup of | D3×#2363, C2×#75480, #10954747, … |
| maximal quotient of | #10716607, #10716716, #10716720, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 8056);