Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×D4⋊C8 ⋊ C2 |
| Sylow | C3 ⋊ #1619 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #882 |
| commutator, cosocle | C3×C42 | C23 |
| Frattini | C42 | C22×D3 |
| Fitting | C3×D4⋊C8 | C2 |
| socle | C22×C3 | #120 |
| hypercenter | D4⋊C8 | D3 |
| hypocenter | C3 | #1619 |
| LCS | G ▷ C3×C42 ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ D4⋊C8 | |
336 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(13,18)(14,15)(17,19);(2,3)(5,8)(6,11)(9,10)(13,17)(14,15)(18,19);(2,3)(4,5)(6,8)(7,10)(9,11)(12,13)(14,17)(15,18)(16,19)Polycyclic presentation:
2^7,3|d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^a=h^2,d^b=df,e^b=eg,h^b=h^2,d^c=dg,e^c=eg54 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1619, C2×#68194, #10770837, … |
| maximal normal subgroup of | D3×#1619, C2×#68194, #10770837, … |
| maximal quotient of | #10683886, #10683903, #10683904, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 770);