Finite Group DatabasebetaFinite Group Databasebeta| order | 13824 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×D3×#10596770⋊C2 |
| Fitting, socle | C25×C3 ⋊ C2×GO+(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D3×#10596770⋊C2 |
| commutator, cosocle | C3×A4⋊S4 | C24 |
| Frattini | C1 | G |
| Fitting, socle | C25×C3 | C2×GO+(4,𝔽2) |
| hypocenter | C3×A42 | C22×D4 |
| LCS | G ▷ C3×A4⋊S4 ▷ C3×A42 | |
| UCS | C1 ◁ C2 | |
At least 6764 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has at least 6764 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 4 generators on 13 points:
(3,6,10)(7,9,8)(12,13);(7,9)(11,12,13);(3,4)(5,7)(6,9)(8,10);(1,2)120 conjugacy classes — view the full character table.
| automorphism group of | D3×H(5,𝔽2), #5966 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,6,10)(7,9,8)(12,13), (7,9)(11,12,13), (3,4)(5,7)(6,9)(8,10), (1,2));