Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#442 ⋊ C2 |
| Sylow | C3 ⋊ #3264 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#442 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #442 | D3 |
| hypocenter | C3 | #3264 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #442 | |
568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2,3)(9,13)(10,15)(12,14);(2,3)(5,7);(8,9)(10,13)(11,14)(12,15);(2,3)(4,5)(6,7)(9,12)(10,15)(13,14)Polycyclic presentation:
2^7,3|e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=h^2,d^b=de,e^b=eg,h^c=h^2,e^d=eg51 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3264, C2×#79878, #11670260, … |
| maximal normal subgroup of | D3×#3264, C2×#79878, #11670260, … |
| maximal quotient of | #10784536, #10784656, #10784776, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12454);