Finite Group DatabasebetaFinite Group Databasebeta| order | 3744 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | GL(2,𝔽3)×D13⋊C3 |
| Fitting | Q8×C13 ⋊ C2×C3×D3 |
| derived | C13×SL(2,𝔽3) ⋊ C22×C3 |
| Sylow | C13 ⋊ C2×C3×GL(2,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | S4×D13⋊C3 |
| commutator, cosocle, hypocenter | C13×SL(2,𝔽3) | C22×C3 |
| Fitting | Q8×C13 | C2×C3×D3 |
| socle | C2×C13 | C2×C3×S4 |
| LCS | G ▷ C13×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
218 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 218 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 104 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,5,3)(4,8,7)(10,14,15,11,12,16)(13,20,19,17,18,21);(2,5)(3,7)(6,8)(9,10,13,12,16,19,18,20,21,15,14,17,11)64 conjugacy classes — view the full character table.
| maximal subgroup of | D13⋊C3×#5935, GL(2,𝔽3)×AGL(1,𝔽13) |
| maximal normal subgroup of | GL(2,𝔽3)×AGL(1,𝔽13) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,5,3)(4,8,7)(10,14,15,11,12,16)(13,20,19,17,18,21), (2,5)(3,7)(6,8)(9,10,13,12,16,19,18,20,21,15,14,17,11));