Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×#385 ⋊ C2 |
| Sylow | C3 ⋊ D4×C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C3×#385 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C22×#67 | C22 |
| hypercenter | #385 | D3 |
| hypocenter | C3 | D4×C4⋊C4 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #385 | |
776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2,3)(9,11);(2,3)(12,13,14,15);(2,3)(8,9)(10,11);(4,5,6,7)(9,11)(13,15)Polycyclic presentation:
2^7,3|a^2=g,c^2=f,b^a=be,c^a=cf,d^b=de,h^b=h^2,h^c=h^278 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D4×C4⋊C4, C2×#74393, #10954500, … |
| maximal normal subgroup of | D3×D4×C4⋊C4, C2×#74393, #10954500, … |
| maximal quotient of | #10716599, #10716601, #10716625, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6969);