Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #3586⋊C3 |
| Fitting, Sylow | #3586 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×D4×A4 |
| commutator, cosocle | #41 | C23×C3 |
| Frattini | C4 | C23×A4 |
| Fitting | #3586 | C3 |
| hypercenter | AGL(1,ℤ/8) | A4 |
| hypocenter | Q8 | C2×C3×D4 |
| LCS | G ▷ #41 ▷ Q8 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8) | |
337 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 337 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,8)(4,15,18)(5,16,17)(7,9)(10,26,22,21,25,11)(13,29,28)(14,27,30)(19,32,24,23,31,20);(1,15,6,17)(2,31,8,32)(3,30,12,29)(4,18,16,5)(7,25,9,26)(10,24,21,20)(11,19,22,23)(13,14,27,28);(1,2,6,8)(3,7,12,9)(4,10,16,21)(5,11,18,22)(13,19,27,23)(14,20,28,24)(15,25,17,26)(29,31,30,32)Polycyclic presentation:
2^3,3,2^4|a^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=ch,e^a=eh,e^b=eh,f^d=fg,g^d=fh,g^f=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #60389⋊C3, #60391⋊C3, #60470⋊C3, … |
| maximal normal subgroup of | #60389⋊C3, #60391⋊C3, #60470⋊C3, … |
| maximal quotient of | #60088⋊C3, #60089⋊C3, #60109⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18217);