Finite Group DatabasebetaFinite Group Databasebeta| order | 4608 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
| direct product | C2×#72215293 |
| Fitting | C27 ⋊ D32 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×S42 |
| commutator, cosocle | C2×A42 | C24 |
| Frattini | C2 | C22×S42 |
| Fitting | C27 | D32 |
| socle | C26 | C2×D32 |
| hypercenter | C23 | S42 |
| hypocenter | A42 | C22×D4 |
| LCS | G ▷ C2×A42 ▷ A42 | |
| UCS | C1 ◁ C22 ◁ C23 | |
19553 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 19553 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 4 generators on 14 points:
(7,8,10,9)(11,14,13,12);(1,2)(8,9,10);(4,6);(3,4)(5,6)(13,14)128 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#72215293 |
| maximal quotient of | #72258669, D3×#72215293 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((7,8,10,9)(11,14,13,12), (1,2)(8,9,10), (4,6), (3,4)(5,6)(13,14));