Finite Group DatabasebetaFinite Group Databasebeta| order | 4608 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25×C3 |
| direct product | D8×A4⋊S4 |
| Fitting | C24×D8 ⋊ C3⋊D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×A4⋊S4 |
| commutator, cosocle | C4×A42 | C23 |
| Frattini | C4 | C22×A4⋊S4 |
| Fitting | C24×D8 | C3⋊D3 |
| socle | C25 | D4×C3⋊D3 |
| hypercenter | D8 | A4⋊S4 |
| hypocenter | A42 | C2×D8 |
| LCS | G ▷ C4×A42 ▷ C2×A42 ▷ A42 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D8 | |
4580 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 4580 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 72 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,4,3,2)(5,8,6,7);(6,7,8)(10,13)(11,16)(14,15);(2,3,4)(6,7,8)(9,10,11,14,12,15,16,13)98 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,3,2)(5,8,6,7), (6,7,8)(10,13)(11,16)(14,15), (2,3,4)(6,7,8)(9,10,11,14,12,15,16,13));