Finite Group DatabasebetaFinite Group Databasebeta| order | 5184 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2×C32 |
| direct product | C2×A4×PSU(3,𝔽2)⋊C3 |
| Fitting, socle | C23×C32 ⋊ C3×SL(2,𝔽3) |
| derived | C22×PSU(3,𝔽2) ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A4×PSU(3,𝔽2)⋊C3 |
| commutator, cosocle, hypocenter | C22×PSU(3,𝔽2) | C2×C32 |
| Frattini | C1 | G |
| Fitting, socle | C23×C32 | C3×SL(2,𝔽3) |
| LCS | G ▷ C22×PSU(3,𝔽2) | |
| UCS | C1 ◁ C2 | |
546 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 546 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,3,7,9)(2,5,4,8)(10,11)(13,15,14);(2,3,7)(5,8,6)(12,15)(13,14)80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×A4×AGL(2,𝔽3), C2×S4×PSU(3,𝔽2)⋊C3, C2×#72213290 |
| maximal normal subgroup of | C2×A4×AGL(2,𝔽3), C2×S4×PSU(3,𝔽2)⋊C3, C2×#72213290 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,3,7,9)(2,5,4,8)(10,11)(13,15,14), (2,3,7)(5,8,6)(12,15)(13,14));