Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C3×D4×Q8 ⋊ C2 |
| Sylow | C3 ⋊ #3499 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×D4×Q8 | C2 |
| hypercenter | D4×Q8 | D3 |
| hypocenter | C3 | #3499 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ D4×Q8 | |
1476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2)(8,11,12,20)(9,14,15,19)(10,18,17,23)(13,21,16,22);(2,3)(10,17)(13,16)(18,23)(21,22);(4,5,6,7);(2,3)(8,10)(9,13)(11,18)(12,17)(14,21)(15,16)(19,22)(20,23);(5,7)(8,9,12,15)(10,16,17,13)(11,19,20,14)(18,21,23,22)Polycyclic presentation:
2^7,3|a^2=f,c^2=g,e^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df,h^b=h^2,h^d=h^2,h^e=h^269 conjugacy classes — view the full character table.
| maximal subgroup of | #10771850, #10771856, #10771938, … |
| maximal normal subgroup of | #10771850, #10771856, #10771938, … |
| maximal quotient of | #10955290, #10955557, #10955576, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19322);