Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #548⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #2341⋊C3 |
| Fitting, Sylow | #2341 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C23 | C22×A4 |
| commutator | C2×Q8 | C2×C3×C4 |
| Fitting | #2341 | C3 |
| prime radical | C22×#41⋊C3 | C2 |
| cosocle | C22×Q8 | C22×C3 |
| hypercenter | #95 | A4 |
| hypocenter | Q8 | C3×C22⋊C4 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C23 ◁ #95 | |
237 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 237 conjugacy classes of subgroups — too many to draw. Only its 48 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 — 2 generators on 24 points:
(6,8)(9,10,11,14)(12,22,19,21,18,24,17,16,20,15,13,23);(1,2,3,4)(5,6)(7,8)(9,20,11,19)(10,24,14,23)(12,13,17,18)(15,16,21,22)Polycyclic presentation:
2^2,3,2^5|a^2=eh,b^2=h,f^2=h,g^2=h,b^a=bd,f^c=gh,g^c=fg,g^f=gh70 conjugacy classes — view the full character table.
| maximal subgroup of | #17606⋊C3, #17611⋊C3, C2×#2341⋊C3, … |
| maximal normal subgroup of | #17606⋊C3, #17611⋊C3, C2×#2341⋊C3, … |
| maximal quotient of | SL(2,𝔽3)×#95, #22530⋊C3, #22534⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5751);