Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #3523⋊C3 |
| Fitting, Sylow | #3523 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C23×A4 |
| commutator, cosocle | C2×Q8 | C23×C3 |
| Fitting | #3523 | C3 |
| hypercenter | #121 | A4 |
| hypocenter | Q8 | C3×#41 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ #121 | |
277 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 277 conjugacy classes of subgroups — too many to draw. Only its 58 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:
(1,3,4,7)(2,6,5,8)(10,14)(12,18,20)(13,19,17)(15,16,24,21,22,23);(2,5)(6,8)(9,20,11,19)(10,23,14,24)(12,13,17,18)(15,22,21,16);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)Polycyclic presentation:
2^3,3,2^4|c^2=eh,f^2=h,g^2=h,b^a=be,c^a=ch,f^d=gh,g^d=fg,g^f=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #17549⋊C3, #61824⋊C3, #61837⋊C3, … |
| maximal normal subgroup of | #17549⋊C3, #61824⋊C3, #61837⋊C3, … |
| maximal quotient of | #34196⋊C3, #34262⋊C3, #34291⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18196);