Finite Group DatabasebetaFinite Group Databasebeta| order | 2592 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C2×A4×((C3⋊D3)⋊C3)⋊C2 |
| Fitting | C23×H(3,𝔽3) ⋊ C22×C3 |
| derived | C22×H(3,𝔽3) ⋊ C23×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A4×((C3⋊D3)⋊C3)⋊C2 |
| commutator, cosocle, hypocenter | C22×H(3,𝔽3) | C23×C3 |
| Frattini | C3 | C2×D32×A4 |
| Fitting | C23×H(3,𝔽3) | C22×C3 |
| socle | C23×C3 | C3×D32 |
| LCS | G ▷ C22×H(3,𝔽3) | |
| UCS | C1 ◁ C2 | |
1446 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 1446 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 3 generators on 15 points:
(1,2,9,7,6,8)(3,5,4)(13,14,15);(2,4)(5,9)(6,8)(12,15)(13,14);(10,11)88 conjugacy classes — view the full character table.
| maximal subgroup of | C2×A4×AGL(2,𝔽3), C2×S4×((C3⋊D3)⋊C3)⋊C2 |
| maximal normal subgroup of | C2×S4×((C3⋊D3)⋊C3)⋊C2 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,9,7,6,8)(3,5,4)(13,14,15), (2,4)(5,9)(6,8)(12,15)(13,14), (10,11));