Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#575 ⋊ C2 |
| Sylow | C3 ⋊ #2008 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #5223 |
| commutator, cosocle | C22×C3×C4 | C23 |
| Frattini | C22×C4 | C22×D3 |
| Fitting | C3×#575 | C2 |
| socle | C2×C3 | #392 |
| hypercenter | #575 | D3 |
| hypocenter | C3 | #2008 |
| LCS | G ▷ C22×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #575 | |
380 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 71 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:
(2,3)(5,11)(7,8)(10,14)(12,13)(15,17)(18,19);(1,2,3)(4,6)(5,10)(7,17)(8,15)(9,16)(11,19)(12,14)(13,18);(4,5)(6,14)(7,12)(8,11)(9,13)(10,15)(16,19)(17,18)Polycyclic presentation:
2^7,3|d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,c^b=cf,d^b=dg,d^c=dg,e^c=eg,h^c=h^245 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2008, C2×#72119, #10822193, … |
| maximal normal subgroup of | D3×#2008, C2×#72119, #10822193, … |
| maximal quotient of | #10695492, #10695506, #10695520, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4695);