Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #3529⋊C3 |
| Fitting, Sylow | #3529 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | C22⋊A4 |
| commutator, cosocle, hypocenter | H(3,𝔽4) | C2×C3 |
| Frattini, socle | C22 | C2×C22⋊A4 |
| Fitting | #3529 | C3 |
| LCS | G ▷ H(3,𝔽4) | |
| UCS | C1 ◁ C2×C4 | |
238 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 22 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,3)(6,8,7)(9,10,13,16)(11,21,20,22,17,15,18,23,12,14,19,24);(3,7)(5,8)(11,18)(14,22)(17,19)(21,23);(1,5)(2,3)(4,8)(6,7)(9,19,13,17)(10,23,16,21)(11,12,18,20)(14,15,22,24)Polycyclic presentation:
2,3,2^6|a^2=g,c^2=g,d^2=g,c^b=cd,d^b=c,e^b=fgh,f^b=ef,d^c=dg,e^c=eh,f^c=fgh,e^d=egh,f^d=fg34 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18233);