Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #548⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #3482⋊C3 |
| Fitting, Sylow | #3482 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×A4 |
| commutator, cosocle | C2×Q8 | C23×C3 |
| Frattini, socle | C22 | C23×A4 |
| Fitting | #3482 | C3 |
| hypercenter | C4×D4 | A4 |
| hypocenter | Q8 | C3×#41 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C2×C4 ◁ C4×D4 | |
294 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 66 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)(12,18,19)(13,20,17)(15,22,23)(16,24,21);(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,f^d=gh,g^d=fgh,g^f=gh70 conjugacy classes — view the full character table.
| maximal subgroup of | #17518⋊C3, #60331⋊C3, #60335⋊C3, … |
| maximal normal subgroup of | #17518⋊C3, #60331⋊C3, #60335⋊C3, … |
| maximal quotient of | #34128⋊C3, #34129⋊C3, C4×#548⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18179);