Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×D4×Q8 ⋊ C2 |
| Sylow | C3 ⋊ #3501 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×D4×Q8 | C2 |
| prime radical | C2×D3×D4 | C22 |
| hypercenter | D4×Q8 | D3 |
| hypocenter | C3 | #3501 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ D4×Q8 | |
1100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 19 points:
(2,3)(5,9)(8,10);(1,2,3)(6,11)(8,10);(12,14,15,19)(13,16,17,18);(4,6)(5,8)(7,11)(9,10);(4,5)(6,10)(7,9)(8,11)(12,13,15,17)(14,18,19,16)Polycyclic presentation:
2^7,3|a^2=g,c^2=g,b^a=bf,c^a=cg,e^a=ef,d^b=df,h^e=h^269 conjugacy classes — view the full character table.
| maximal subgroup of | #10772065, #10772071, #10772076, … |
| maximal normal subgroup of | #10772065, #10772071, #10772076, … |
| maximal quotient of | #10955271, #10955605, #10955613, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19337);