Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| direct product | GL(2,𝔽3)×SL(2,ℤ/4) |
| derived | A4×SL(2,𝔽3) ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | S42 |
| commutator, hypocenter | A4×SL(2,𝔽3) | C2×C4 |
| Fitting | C23×Q8 | D32 |
| socle | C24 | D3×S4 |
| prime radical | C2×A4×GL(2,𝔽3) | C2 |
| cosocle | C2×A4×SL(2,𝔽3) | C22 |
| LCS | G ▷ A4×SL(2,𝔽3) | |
| UCS | C1 ◁ C22 | |
941 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 941 conjugacy classes of subgroups — too many to draw. Only its 43 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) = 96 |
A minimal faithful permutation representation — 2 generators on 16 points:
(5,7,6)(9,14,10,11,12,15,13,16);(1,2,3,4)(7,8)(10,11,15)(13,16,14)80 conjugacy classes — view the full character table.
| maximal subgroup of | GL(2,𝔽3)×GL(2,ℤ/4) |
| maximal normal subgroup of | GL(2,𝔽3)×GL(2,ℤ/4) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,7,6)(9,14,10,11,12,15,13,16), (1,2,3,4)(7,8)(10,11,15)(13,16,14));