Finite Group DatabasebetaFinite Group Databasebeta| order | 13440 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C22 |
| direct product | C2×C5×AGL(3,𝔽2) |
| Fitting, socle | C24×C5 ⋊ GL(3,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | AGL(3,𝔽2) |
| commutator, hypocenter | AGL(3,𝔽2) | C2×C5 |
| Frattini | C1 | G |
| Fitting, socle | C24×C5 | GL(3,𝔽2) |
| cosocle | C23 | C2×C5×GL(3,𝔽2) |
| LCS | G ▷ AGL(3,𝔽2) | |
| UCS | C1 ◁ C2×C5 | |
752 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 752 conjugacy classes of subgroups — too many to draw. Only its 12 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) = 70 |
A minimal faithful permutation representation — 2 generators on 15 points:
(3,4,6,5,7,8,9)(11,12,13,14,15);(1,2)(3,5,8,10,4,7,9)110 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D5×AGL(3,𝔽2) |
| maximal normal subgroup of | C2×D5×AGL(3,𝔽2) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4,6,5,7,8,9)(11,12,13,14,15), (1,2)(3,5,8,10,4,7,9));