Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #3549⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C23×C4 |
| unambiguous semidirect product | #3549⋊C3 |
| Fitting, Sylow | #3549 ⋊ C3 |
| derived | H(3,𝔽4) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | H(3,𝔽4) | C2×C3 |
| Frattini, socle | C22 | C2×C22⋊A4 |
| Fitting | #3549 | C3 |
| LCS | G ▷ H(3,𝔽4) | |
| UCS | C1 | |
243 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 243 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,16,6,7,8)(3,22,19)(4,14,20,5,10,23)(9,21,15)(11,12,18)(13,17,24);(1,5,6,4)(2,14,7,10)(3,15,11,13)(9,12,17,22)(18,19)(21,24);(1,13,6,15)(2,12,7,22)(3,5,11,4)(8,20)(9,10,17,14)(16,23)(18,24)(19,21)Polycyclic presentation:
2,3,2^6|d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=cgh,d^a=dg,c^b=cdefgh,d^b=cfh,e^b=efh,f^b=e,g^b=gh,h^b=g,d^c=dg,e^c=egh,f^c=fh,e^d=eh,f^d=fg18 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18236);