Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×#556 ⋊ C2 |
| Sylow | C3 ⋊ #3540 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×#556 | C2 |
| hypercenter | #556 | D3 |
| hypocenter | C3 | #3540 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #556 | |
1058 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)(12,16,17,15)(13,19,20,25)(14,24,23,22)(18,26,21,27);(2,3)(5,9)(8,11)(14,23)(18,21)(22,24)(26,27);(5,9)(6,10)(12,15,17,16)(13,19,20,25)(14,22,23,24)(18,26,21,27);(4,6)(5,8)(7,10)(9,11)(12,14,17,23)(13,18,20,21)(15,22,16,24)(19,26,25,27);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13)(14,21)(15,19)(16,25)(17,20)(18,23)(22,27)(24,26)Polycyclic presentation:
2^7,3|b^2=f,c^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^d=h^263 conjugacy classes — view the full character table.
| maximal subgroup of | #11681145, #11681147, #11681149, … |
| maximal normal subgroup of | #11681145, #11681147, #11681149, … |
| maximal quotient of | #10957963, #10958323, #10958324, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19584);