Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #1705⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C8 |
| unambiguous semidirect product | #1705⋊C3 |
| Fitting, Sylow | #1705 ⋊ C3 |
| derived | C82 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C82 | C2×C3 |
| Frattini | C42 | C2×A4 |
| Fitting | #1705 | C3 |
| socle | C22 | #127⋊C3 |
| LCS | G ▷ C82 | |
| UCS | C1 | |
49 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,6,13,14,15)(3,8,16)(4,7,18,5,9,19)(10,11,17,21,22,23)(12,20,24);(1,21,13,12,5,10,4,3)(2,11,14,8,9,22,7,20)Polycyclic presentation:
2,3,2^6|c^2=e,d^2=f,e^2=g,f^2=h,b^a=beh,c^a=ce,d^a=df,e^a=eg,f^a=fh,c^b=cdegh,d^b=ce,e^b=efg,f^b=eg,g^b=gh,h^b=g18 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 593);