Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| Fitting | C3×H(3,𝔽4) ⋊ C2 |
| Sylow | C3 ⋊ #3549 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×H(3,𝔽4) | C2 |
| hypercenter | H(3,𝔽4) | D3 |
| hypocenter | C3 | #3549 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ H(3,𝔽4) | |
1258 conjugacy classes of subgroups; only the 455 normal subgroups are shown.
This group has 1258 conjugacy classes of subgroups — too many to draw. Only its 455 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)(12,15,16,14)(13,18,17,19);(5,9)(8,10)(12,14,16,15)(13,18,17,19);(2,3)(4,6,7,11)(5,8,9,10)(12,14,16,15)(13,18,17,19);(6,11)(8,10)(12,13,16,17)(14,19,15,18);(4,5)(6,10)(7,9)(8,11)(13,17)(14,15)Polycyclic presentation:
2^7,3|b^2=f,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,h^c=h^2,h^e=h^257 conjugacy classes — view the full character table.
| maximal subgroup of | #10776998, #10777002, #10777032, … |
| maximal normal subgroup of | #10776998, #10777002, #10777032, … |
| maximal quotient of | #10958037, #10958703, #10958709, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19663);