Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C22×C3×C42 ⋊ C2 |
| Sylow | C3 ⋊ #3444 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C22×C3×C42 | C2 |
| hypercenter | C22×C42 | D3 |
| hypocenter | C3 | #3444 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C42 ◁ C22×C42 | |
1212 conjugacy classes of subgroups; only the 509 normal subgroups are shown.
This group has 1212 conjugacy classes of subgroups — too many to draw. Only its 509 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 — 5 generators on 19 points:
(1,2,3)(12,14,15,18)(13,17,16,19);(4,6,7,10)(5,9,8,11);(5,8)(9,11);(13,16)(17,19);(2,3)(4,5)(6,9)(7,8)(10,11)(12,13)(14,17)(15,16)(18,19)Polycyclic presentation:
2^7,3|d^2=g,e^2=f,b^a=bf,c^a=cg,h^a=h^2120 conjugacy classes — view the full character table.
| maximal subgroup of | #10761698, #10761701, #10761721, … |
| maximal normal subgroup of | #10761698, #10761701, #10761721, … |
| maximal quotient of | #10953489, #10953595, C4×#5791, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18256);