Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C2×C3×#121 ⋊ C2 |
| Sylow | C3 ⋊ #3542 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C2×C3×#121 | C2 |
| hypercenter | C2×#121 | D3 |
| hypocenter | C3 | #3542 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×#121 | |
1160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 19 points:
(1,2,3)(12,15,16,19)(13,17,14,18);(5,9)(8,11)(12,14,16,13)(15,18,19,17);(5,9)(6,10);(2,3)(4,6)(5,8)(7,10)(9,11)(13,14)(17,18);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13,16,14)(15,18,19,17)Polycyclic presentation:
2^7,3|a^2=f,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,h^a=h^2,c^b=cg,d^b=df,h^b=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | #11680728, #11680732, #11680790, … |
| maximal normal subgroup of | #11680728, #11680732, #11680790, … |
| maximal quotient of | #10957986, #10957997, #10958515, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18532);