Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C3⋊#2104 |
| Sylow | C3 ⋊ #2104 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C3⋊#401 |
| commutator, prime radical, cosocle | C24×C3 | C23 |
| Frattini | C24 | C22×D3 |
| Fitting | C3×#342 | C2 |
| socle | C2×C3 | #401 |
| hypercenter | #342 | D3 |
| hypocenter | C3 | #2104 |
| LCS | G ▷ C24×C3 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ #342 | |
178 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 178 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) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,7)(5,10,11,19)(6,13)(8,16)(9,17,12,18)(14,15);(4,6)(5,9)(7,15,16,13)(8,14)(10,18,19,17)(11,12);(2,3)(4,5)(6,12,14,9)(7,10,16,19)(8,11)(13,18)(15,17)Polycyclic presentation:
2^7,3|a^2=d,b^2=df,c^2=de,b^a=bde,c^a=cef,e^a=eg,h^a=h^2,c^b=cf,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=eg45 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2104, C2×C3⋊#2104, #10826176, … |
| maximal normal subgroup of | D3×#2104, C2×C3⋊#2104, #10826176, … |
| maximal quotient of | #10700328, #10700329, #10700330, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1526);