Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#468 ⋊ C2 |
| Sylow | C3 ⋊ #3055 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#468 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #468 | D3 |
| hypocenter | C3 | #3055 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #468 | |
548 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)(9,13)(12,15);(2,3)(5,7);(8,9,10,12,11,13,14,15);(4,5)(6,7)(9,12)(10,14)(13,15)Polycyclic presentation:
2^7,3|b^2=e,e^2=g,b^a=be,c^a=cf,e^a=eg,d^b=dg,h^c=h^2,h^d=h^248 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3055, C2×#80805, #11662351, … |
| maximal normal subgroup of | D3×#3055, C2×#80805, #11662351, … |
| maximal quotient of | #10777950, #10777972, #10777994, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 13381);