Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#341 ⋊ C2 |
| Sylow | C3 ⋊ #2018 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #895 |
| commutator, cosocle | C22×C3×C4 | C23 |
| Frattini | C22×C4 | C22×D3 |
| Fitting | C3×#341 | C2 |
| socle | C22×C3 | #121 |
| hypercenter | #341 | D3 |
| hypocenter | C3 | #2018 |
| LCS | G ▷ C22×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #341 | |
382 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 382 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2)(9,15)(10,19)(11,12)(16,17)(18,22)(20,21);(2,3)(5,7)(8,10)(9,14)(11,20)(12,19)(13,21)(15,22)(16,18)(17,23);(2,3)(4,5)(6,7)(8,9)(10,18)(11,16)(12,15)(13,17)(14,19)(20,23)(21,22)Polycyclic presentation:
2^7,3|d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,h^a=h^2,c^b=cf,d^b=dg,f^b=fg,h^b=h^2,d^c=dg,e^c=eg,f^c=fg,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10707319, #10707324, #10707330, … |
| maximal normal subgroup of | #10707319, #10707324, #10707330, … |
| maximal quotient of | #10695788, #10695805, #10695856, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1518);