Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C3×#333 ⋊ C2 |
| Sylow | C3 ⋊ #1624 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #882 |
| commutator, cosocle | C3×C42 | C23 |
| Frattini | C42 | C22×D3 |
| Fitting | C3×#333 | C2 |
| socle | C22×C3 | #120 |
| hypercenter | #333 | D3 |
| hypocenter | C3 | #1624 |
| LCS | G ▷ C3×C42 ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #333 | |
240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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) = 192 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2)(4,6,8,7)(5,9,10,11)(12,13,15,17)(14,19,18,16);(4,5,8,10)(6,9,7,11)(14,18)(16,19);(2,3)(5,9)(6,7)(10,11)(13,16)(14,18)(17,19)Polycyclic presentation:
2^7,3|b^2=f,c^2=fg,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,d^c=dg,e^c=eg,h^c=h^245 conjugacy classes — view the full character table.
| maximal subgroup of | #10715947, D3×#1624, C2×#68491, … |
| maximal normal subgroup of | #10715947, D3×#1624, C2×#68491, … |
| maximal quotient of | #10683933, #10683939, #10683954, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1067);