Finite Group DatabasebetaFinite Group Databasebeta| order | 9216 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C27 |
| direct product | #10202 |
| Fitting, socle | C28 ⋊ D32 |
| derived | C22⋊A42 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22⋊A42 | C22 |
| Fitting, socle | C28 | D32 |
| LCS | G ▷ C22⋊A42 | |
| UCS | C1 | |
58083 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 58083 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 36 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,4,3)(6,8)(9,12,10,11)(13,16);(2,4,3)(5,6,8,7)(10,12)(14,15,16)100 conjugacy classes — view the full character table.
| maximal subgroup of | #72226610, #72258869 |
| maximal normal subgroup of | #72226610, #72258869 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,4,3)(6,8)(9,12,10,11)(13,16), (2,4,3)(5,6,8,7)(10,12)(14,15,16));