Finite Group DatabasebetaFinite Group Databasebeta| order | 2592 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | C2×#13049220 |
| Fitting | C23×H(3,𝔽3) ⋊ C2×D3 |
| derived | A4×H(3,𝔽3) ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #13049220 |
| commutator, cosocle, hypocenter | A4×H(3,𝔽3) | C23 |
| Frattini | C3 | C2×#89101 |
| Fitting | C23×H(3,𝔽3) | C2×D3 |
| socle | C23×C3 | #1113 |
| LCS | G ▷ A4×H(3,𝔽3) | |
| UCS | C1 ◁ C2 | |
1279 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 1279 conjugacy classes of subgroups — too many to draw. Only its 60 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:
(3,5,4)(8,14)(10,11)(12,15);(5,6)(7,8,10)(9,11,14,13,12,15);(1,2)64 conjugacy classes — view the full character table.
| maximal subgroup of | 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((3,5,4)(8,14)(10,11)(12,15), (5,6)(7,8,10)(9,11,14,13,12,15), (1,2));