Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| direct product | GL(2,𝔽3)2 |
| Fitting | Q82 ⋊ D32 |
| derived | SL(2,𝔽3)2 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C22 | S42 |
| commutator, cosocle, hypocenter | SL(2,𝔽3)2 | C22 |
| Fitting | Q82 | D32 |
| LCS | G ▷ SL(2,𝔽3)2 | |
| UCS | C1 ◁ C22 | |
729 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 729 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) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,7,8)(2,6,4)(9,15,14,11,12,10,16,13);(2,5)(3,7)(6,8)(10,11,16)(13,14,15)64 conjugacy classes — view the full character table.
| maximal subgroup of | GL(2,𝔽3)×AGL(2,𝔽3) |
| maximal quotient of | GL(2,𝔽3)×AGL(2,𝔽3) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,7,8)(2,6,4)(9,15,14,11,12,10,16,13), (2,5)(3,7)(6,8)(10,11,16)(13,14,15));