Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #1446⋊C3 |
| Fitting, Sylow | #1446 ⋊ C3 |
| derived | C22×Q8 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #120⋊C3 |
| commutator, hypocenter | C22×Q8 | C3×C4 |
| Frattini, socle | C24 | C2×A4 |
| Fitting | #1446 | C3 |
| prime radical | C2×C22⋊SL(2,𝔽3) | C2 |
| cosocle | C23×Q8 | C2×C3 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C22 | |
161 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 161 conjugacy classes of subgroups — too many to draw. Only its 19 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) = 48 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,3,4)(5,6)(7,11,12,9,8,10)(14,15,17)(18,20,19);(6,10)(9,11)(13,17,16,19)(14,20,18,15)Polycyclic presentation:
2,3,2^6|a^2=c,d^2=h,e^2=h,d^a=df,e^a=eg,d^b=ef,e^b=defg,f^b=g,g^b=fg,e^d=eh36 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1446⋊C3, #13851⋊C3, C4×#393⋊C3, … |
| maximal normal subgroup of | C2×#1446⋊C3, #13851⋊C3, C4×#393⋊C3, … |
| maximal quotient of | #11380⋊C3, #11390⋊C3, #11391⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 587);