Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #1791⋊C3 |
| Fitting, Sylow | #1791 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #120⋊C3 |
| commutator, hypocenter | H(5,𝔽2) | C3×C4 |
| Frattini | C22×C4 | C2×A4 |
| Fitting | #1791 | C3 |
| socle | C2 | #379⋊C3 |
| prime radical | #5950 | C2 |
| cosocle | #585 | C2×C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C4 | |
118 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 118 conjugacy classes of subgroups — too many to draw. Only its 12 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 — 2 generators on 32 points:
(1,2,7,20,6,11,3,8)(4,26,13,23,18,17,12,32,19,14,30,29,16,25,22,27,15,10,21,31,5,9,28,24);(1,15,6,18)(2,25,11,26)(3,28,7,30)(4,19,16,5)(8,31,20,32)(9,17,14,10)(12,22,21,13)(23,29,27,24)Polycyclic presentation:
2,3,2^6|a^2=ch,c^2=h,e^2=h,d^a=df,e^a=egh,d^b=defgh,e^b=dg,f^b=fg,g^b=fh,e^d=eh,g^d=gh,f^e=fh32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1791⋊C3, #20612⋊C3, #21483⋊C3, … |
| maximal normal subgroup of | C2×#1791⋊C3, #20612⋊C3, #21483⋊C3, … |
| maximal quotient of | #8316⋊C3, #8317⋊C3, #11391⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 603);