Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #3522⋊C3 |
| Fitting, Sylow | #3522 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C23×A4 |
| commutator, cosocle | C2×Q8 | C23×C3 |
| Fitting | #3522 | C3 |
| hypercenter | C22⋊Q8 | A4 |
| hypocenter | Q8 | C3×#41 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
279 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 279 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)(12,19,13)(15,24,16,21,23,22)(17,20,18);(2,5)(6,8)(9,19,11,20)(10,24,14,23)(12,13,17,18)(15,22,21,16);(1,2)(3,6)(4,5)(7,8)(9,10,11,14)(12,15,17,21)(13,16,18,22)(19,23,20,24)Polycyclic presentation:
2^3,3,2^4|a^2=h,c^2=eh,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 | #17550⋊C3, #61823⋊C3, #61843⋊C3, … |
| maximal normal subgroup of | #17550⋊C3, #61823⋊C3, #61843⋊C3, … |
| maximal quotient of | #34196⋊C3, #34263⋊C3, #34294⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18195);