Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #3503⋊C3 |
| Fitting, Sylow | #3503 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C23×A4 |
| commutator, cosocle | C2×Q8 | C23×C3 |
| Fitting | #3503 | C3 |
| prime radical | C2×#41⋊C3 | C22 |
| hypercenter | C4⋊Q8 | A4 |
| hypocenter | Q8 | C2×C3×Q8 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
265 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 265 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(9,11,12,10)(13,24,20,15,21,18,19,23,14,17,22,16);(1,3,4,8)(2,5,6,7)(9,23)(10,22)(11,21)(12,24)(13,16)(14,17)(15,20)(18,19);(1,2,4,6)(3,7,8,5)(10,11)(15,17)(16,18)(23,24)Polycyclic presentation:
2^3,3,2^4|a^2=eh,b^2=e,c^2=h,f^2=h,g^2=h,b^a=be,c^a=ch,f^d=fgh,g^d=f,g^f=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #17722⋊C3, #17723⋊C3, #17777⋊C3, … |
| maximal normal subgroup of | #17722⋊C3, #17723⋊C3, #17777⋊C3, … |
| maximal quotient of | #34183⋊C3, #34250⋊C3, #34335⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18190);