Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #2388⋊C3 |
| Fitting, Sylow | #2388 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #120⋊C3 |
| commutator, cosocle, hypocenter | C22×Q8 | C22×C3 |
| Frattini, socle | C23 | C22×A4 |
| Fitting | #2388 | C3 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C4 | |
199 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 199 conjugacy classes of subgroups — too many to draw. Only its 21 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 — 2 generators on 24 points:
(1,6,20,8,3,16,5,10,9,2,12,21)(4,17,19,15,14,18,13,11,24,7,22,23);(1,15)(2,4)(3,22)(5,7)(6,11)(8,13)(9,16,20,21)(10,17)(12,14)(18,19,23,24)Polycyclic presentation:
2^2,3,2^5|b^2=h,d^2=h,e^2=h,d^a=df,e^a=eg,d^c=deh,e^c=dh,f^c=fg,g^c=f,e^d=eh36 conjugacy classes — view the full character table.
| maximal subgroup of | #21442⋊C3, #21444⋊C3, #23311⋊C3, … |
| maximal normal subgroup of | #21442⋊C3, #21444⋊C3, #23311⋊C3, … |
| maximal quotient of | #13851⋊C3, C4×#393⋊C3, C4×#395⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5760);