Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #3502⋊C3 |
| Fitting, Sylow | #3502 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C23×A4 |
| commutator, cosocle | C2×Q8 | C23×C3 |
| Fitting | #3502 | C3 |
| prime radical | C22×SL(2,𝔽3) | C22 |
| hypercenter | C22⋊Q8 | A4 |
| hypocenter | Q8 | C2×C3×Q8 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
254 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 254 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(10,14)(11,17,12)(15,24,16,21,23,22)(18,19,20);(1,3,4,8)(2,5,6,7)(9,19,13,17)(10,24,14,23)(11,12,18,20)(15,16,21,22);(1,2,4,6)(3,7,8,5)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)Polycyclic presentation:
2^3,3,2^4|a^2=e,b^2=e,f^2=h,g^2=h,b^a=be,c^a=ch,f^d=fg,g^d=f,g^f=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #17619⋊C3, #61818⋊C3, #61821⋊C3, … |
| maximal normal subgroup of | #17619⋊C3, #61818⋊C3, #61821⋊C3, … |
| maximal quotient of | #34182⋊C3, #34249⋊C3, #34335⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18189);