Finite Group DatabasebetaFinite Group Databasebeta| order | 4608 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | GL(2,𝔽3)×GL(2,ℤ/4) |
| Fitting | C24×Q8 ⋊ D32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×S42 |
| commutator, cosocle | C2×A4×SL(2,𝔽3) | C23 |
| Fitting | C24×Q8 | D32 |
| socle | C24 | C2×D3×S4 |
| hypercenter | C23 | S42 |
| hypocenter | A4×SL(2,𝔽3) | C2×D4 |
| LCS | G ▷ C2×A4×SL(2,𝔽3) ▷ A4×SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C23 | |
5359 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 5359 conjugacy classes of subgroups — too many to draw. Only its 82 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 — 3 generators on 16 points:
(5,7,6)(9,14,10,11,12,15,13,16);(2,4)(10,11,15)(13,16,14);(1,2,3,4)(7,8)112 conjugacy classes — view the full character table.
| automorphism group of | C32×C42 |
| outer automorphism group of | C32×C42 |
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), (2,4)(10,11,15)(13,16,14), (1,2,3,4)(7,8));