Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | A42×D8 |
| Fitting, Sylow | C24×D8 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×A42 |
| commutator, cosocle | C24×C4 | C22×C32 |
| Frattini | C4 | C22×A42 |
| Fitting | C24×D8 | C32 |
| socle | C25 | C32×D4 |
| hypercenter | D8 | A42 |
| hypocenter | C24 | C32×D8 |
| LCS | G ▷ C24×C4 ▷ C25 ▷ C24 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D8 | |
1481 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 1481 conjugacy classes of subgroups — too many to draw. Only its 77 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 — 2 generators on 16 points:
(1,4)(2,3)(5,8,6)(10,13)(11,16)(14,15);(2,4,3)(6,8,7)(9,10,11,14,12,15,16,13)112 conjugacy classes — view the full character table.
| maximal subgroup of | A42×AGL(1,ℤ/8), C2×A42×D8, #72258808, … |
| maximal normal subgroup of | A42×AGL(1,ℤ/8), C2×A42×D8, #72258808, … |
| maximal quotient of | C2×A42×D8 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4)(2,3)(5,8,6)(10,13)(11,16)(14,15), (2,4,3)(6,8,7)(9,10,11,14,12,15,16,13));