Finite Group DatabasebetaFinite Group Databasebeta| order | 5184 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×S4×((C3⋊D3)⋊C3)⋊C2 |
| Fitting | C23×H(3,𝔽3) ⋊ C22×D3 |
| derived | A4×H(3,𝔽3) ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | S4×((C3⋊D3)⋊C3)⋊C2 |
| commutator, cosocle, hypocenter | A4×H(3,𝔽3) | C24 |
| Frattini | C3 | C2×D32×S4 |
| Fitting | C23×H(3,𝔽3) | C22×D3 |
| socle | C23×C3 | D33 |
| LCS | G ▷ A4×H(3,𝔽3) | |
| UCS | C1 ◁ C2 | |
5497 conjugacy classes of subgroups; only the 169 normal subgroups are shown.
This group has 5497 conjugacy classes of subgroups — too many to draw. Only its 169 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 — 4 generators on 15 points:
(3,6,5)(7,8,15,13,12,14)(9,11,10);(8,10)(11,15)(12,14);(1,2);(4,6)110 conjugacy classes — view the full character table.
| maximal subgroup of | C2×S4×AGL(2,𝔽3) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,6,5)(7,8,15,13,12,14)(9,11,10), (8,10)(11,15)(12,14), (1,2), (4,6));