Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| Fitting | C3×#493 ⋊ C2 |
| Sylow | C3 ⋊ #1721 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #940 |
| commutator, cosocle | C3×C42 | C23 |
| Frattini | C42 | C22×D3 |
| Fitting | C3×#493 | C2 |
| socle | C22×C3 | #127 |
| hypercenter | #493 | D3 |
| hypocenter | C3 | #1721 |
| LCS | G ▷ C3×C42 ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #493 | |
242 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 65 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)(5,9)(8,11)(12,13,17,19,15,18,14,16);(4,5,6,8,7,9,10,11);(2,3)(5,8)(6,10)(9,11)(13,16)(14,17)(18,19)Polycyclic presentation:
2^7,3|b^2=d,c^2=eg,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^a=h^2,c^b=cf,h^c=h^248 conjugacy classes — view the full character table.
| maximal subgroup of | #10708599, D3×#1721, C2×#71280, … |
| maximal normal subgroup of | #10708599, D3×#1721, C2×#71280, … |
| maximal quotient of | #10688977, #10694988, #10694996, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3856);