Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×Q82 ⋊ C2 |
| Sylow | C3 ⋊ #3483 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×Q82 | C2 |
| hypercenter | Q82 | D3 |
| hypocenter | C3 | #3483 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ Q82 | |
1202 conjugacy classes of subgroups; only the 491 normal subgroups are shown.
This group has 1202 conjugacy classes of subgroups — too many to draw. Only its 491 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 — 5 generators on 19 points:
(1,2)(4,6,7,10)(5,9,8,11)(12,15,16,14)(13,19,17,18);(12,14,16,15)(13,19,17,18);(2,3)(5,8)(9,11)(12,14,16,15)(13,18,17,19);(12,13,16,17)(14,18,15,19);(2,3)(4,5)(6,9)(7,8)(10,11)(13,17)(18,19)Polycyclic presentation:
2^7,3|b^2=f,c^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,h^a=h^2,d^b=df,h^c=h^2,h^e=h^275 conjugacy classes — view the full character table.
| maximal subgroup of | #10771081, #10771087, #10771125, … |
| maximal normal subgroup of | #10771081, #10771087, #10771125, … |
| maximal quotient of | #10954667, #10954967, #10955039, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19616);