Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×#537 ⋊ C2 |
| Sylow | C3 ⋊ #3560 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×#537 | C2 |
| hypercenter | #537 | D3 |
| hypocenter | C3 | #3560 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #537 | |
1056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(13,19)(17,20)(18,23)(26,27);(4,6,7,10)(5,9,8,11)(13,19)(14,22)(18,23)(21,24);(2,3)(4,5)(6,9)(7,8)(10,11)(12,15,16,25)(13,18,19,23)(14,21,22,24)(17,26,20,27);(2,3)(5,8)(9,11)(12,14)(13,17)(15,24)(16,22)(18,27)(19,20)(21,25)(23,26);(5,8)(6,10)(12,13)(14,20)(15,23)(16,19)(17,22)(18,25)(21,26)(24,27)Polycyclic presentation:
2^7,3|c^2=f,d^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,h^b=h^2,h^c=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10768608, #10768616, #10768623, … |
| maximal normal subgroup of | #10768608, #10768616, #10768623, … |
| maximal quotient of | #10958178, #10958280, #10958463, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18953);