Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | GL(2,𝔽3)×GO+(4,𝔽2) |
| Fitting | C32×Q8 ⋊ D3×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | S4×GO+(4,𝔽2) |
| commutator, cosocle | C3⋊D3×SL(2,𝔽3) | C23 |
| Fitting | C32×Q8 | D3×D4 |
| socle | C2×C32 | D4×S4 |
| hypocenter | C32×SL(2,𝔽3) | C2×D4 |
| LCS | G ▷ C3⋊D3×SL(2,𝔽3) ▷ C32×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
1276 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 1276 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2,8,4,7,6)(3,5)(9,12,13,10)(11,14);(12,14);(2,5)(3,7)(6,8)72 conjugacy classes — view the full character table.
| maximal subgroup of | GL(2,𝔽3)×AΓL(1,𝔽9) |
| maximal normal subgroup of | GL(2,𝔽3)×AΓL(1,𝔽9) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,8,4,7,6)(3,5)(9,12,13,10)(11,14), (12,14), (2,5)(3,7)(6,8));