Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | C4⋊(#120⋊C3) |
| unambiguous semidirect product | #2847⋊C3 |
| Fitting, Sylow | #2847 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#120⋊C3 |
| commutator, cosocle | C25 | C22×C3 |
| Frattini, socle | C23 | C22×A4 |
| Fitting | #2847 | C3 |
| hypercenter | C4 | #120⋊C3 |
| hypocenter | C24 | C3×D4 |
| LCS | G ▷ C25 ▷ C24 | |
| UCS | C1 ◁ C2 ◁ C4 | |
254 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 254 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) = 12 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,2,3,4)(7,8,9)(10,11,12);(2,4)(5,12,7,6)(8,11,9,10)Polycyclic presentation:
2^2,3,2^5|b^2=d,b^a=bd,e^a=eg,f^a=fh,e^c=fgh,f^c=efh,g^c=h,h^c=gh28 conjugacy classes — view the full character table.
| maximal subgroup of | #32706⋊C3, #32708⋊C3, #32878⋊C3, … |
| maximal normal subgroup of | #32706⋊C3, #32708⋊C3, #32878⋊C3, … |
| maximal quotient of | C4⋊(#379⋊C3), #14694⋊C3, C4⋊(#393⋊C3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5791);