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 ⋊ #3516 |
| 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 | #3516 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ D4×Q8 | |
1012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2)(4,6,7,10)(5,9,8,11)(13,19)(17,20)(18,27)(25,26);(12,15,16,24)(13,18,19,27)(14,21,22,23)(17,25,20,26);(5,8)(9,11);(12,14,16,22)(13,17,19,20)(15,23,24,21)(18,26,27,25);(2,3)(4,5)(6,9)(7,8)(10,11)(12,13)(14,20)(15,18)(16,19)(17,22)(21,26)(23,25)(24,27)Polycyclic presentation:
2^7,3|b^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,e^a=ef,h^a=h^2,d^b=df,h^e=h^269 conjugacy classes — view the full character table.
| maximal subgroup of | #11677578, #11677582, #11677609, … |
| maximal normal subgroup of | #11677578, #11677582, #11677609, … |
| maximal quotient of | #10955373, #10955851, #10956010, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19325);