Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C24×C3 |
| direct product | A42×#41 |
| Fitting, Sylow | C24×#41 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×A42 |
| commutator, socle, cosocle | C25 | C23×C32 |
| Frattini | C2 | C23×A42 |
| Fitting | C24×#41 | C32 |
| hypercenter | #41 | A42 |
| hypocenter | C24 | C32×#41 |
| LCS | G ▷ C25 ▷ C24 | |
| UCS | C1 ◁ C4 ◁ #41 | |
5075 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 5075 conjugacy classes of subgroups — too many to draw. Only its 187 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)(2,3)(5,8,6)(9,11,12,16)(10,14,13,15);(2,4,3)(6,8,7)(10,13)(14,15);(9,10,12,13)(11,15,16,14)160 conjugacy classes — view the full character table.
| maximal subgroup of | A42×AGL(1,ℤ/8), #41×A4⋊S4, #72258962, … |
| maximal normal subgroup of | A42×AGL(1,ℤ/8), #41×A4⋊S4, #72258962, … |
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)(9,11,12,16)(10,14,13,15), (2,4,3)(6,8,7)(10,13)(14,15), (9,10,12,13)(11,15,16,14));