Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #2812⋊C3 |
| Fitting, Sylow | #2812 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #5450 |
| commutator, cosocle | C25 | C22×C3 |
| Frattini, socle | C23 | C22×A4 |
| Fitting | #2812 | C3 |
| hypocenter | C24 | C3×D4 |
| LCS | G ▷ C25 ▷ C24 | |
| UCS | C1 ◁ C2 | |
162 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 20 points:
(2,4)(5,7)(6,11)(8,14,9,17,10,15)(12,18,13,19,16,20);(1,2)(3,4)(5,13,10,6)(7,19,14,18)(8,12,9,16)(11,17,20,15)Polycyclic presentation:
2^2,3,2^5|b^a=bd,e^a=eg,f^a=fgh,e^b=eh,f^b=fg,e^c=fgh,f^c=efg,g^c=gh,h^c=g22 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2812⋊C3, #11710041, #45439⋊C3, … |
| maximal normal subgroup of | C2×#2812⋊C3, #11710041, #45439⋊C3, … |
| maximal quotient of | #14657⋊C3, #14673⋊C3, #17216⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5780);