Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #548⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #2351⋊C3 |
| Fitting, Sylow | #2351 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C23 | C22×A4 |
| commutator | C2×Q8 | C2×C3×C4 |
| Fitting | #2351 | C3 |
| prime radical | C22×SL(2,𝔽3) | C22 |
| cosocle | C22×Q8 | C22×C3 |
| hypercenter | #95 | A4 |
| hypocenter | Q8 | C3×C4⋊C4 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C23 ◁ #95 | |
177 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 177 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:
(5,6,7,8)(10,14)(12,19,13)(15,24,16,21,23,22)(17,20,18);(1,2,3,4)(6,8)(9,24,11,23)(10,20,14,19)(12,16,17,22)(13,21,18,15)Polycyclic presentation:
2^2,3,2^5|a^2=eh,b^2=dh,f^2=h,g^2=h,b^a=bd,f^c=fg,g^c=f,g^f=gh70 conjugacy classes — view the full character table.
| maximal subgroup of | #17618⋊C3, #17619⋊C3, C2×#2351⋊C3, … |
| maximal normal subgroup of | #17618⋊C3, #17619⋊C3, C2×#2351⋊C3, … |
| maximal quotient of | SL(2,𝔽3)×#95, #22734⋊C3, #22738⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5753);