Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#435 ⋊ C2 |
| Sylow | C3 ⋊ D4×M16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D3 |
| commutator, socle | C22×C3 | C23×C4 |
| Fitting | C3×#435 | C2 |
| prime radical | C2×D3×D4 | C4 |
| cosocle | C2×C3×C4 | C24 |
| hypercenter | #435 | D3 |
| hypocenter | C3 | D4×M16 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #435 | |
544 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2)(9,10)(14,15);(8,10,11,15,12,9,13,14);(5,7);(2,3)(4,5)(6,7)(8,9,11,14,12,10,13,15)Polycyclic presentation:
2^7,3|a^2=f,c^2=f,f^2=g,b^a=be,d^a=dg,h^a=h^2,d^c=dg,h^d=h^278 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D4×M16, C2×#79483, #11659109, … |
| maximal normal subgroup of | D3×D4×M16, C2×#79483, #11659109, … |
| maximal quotient of | #10766606, #10766614, #10766622, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12059);