Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C214 |
| direct product | D3×#15947 |
| Fitting | C3×#15947 ⋊ C2 |
| Sylow | C3 ⋊ C2×#15947 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C24×D3 |
| commutator, socle, cosocle | C24×C3 | C25 |
| Fitting | C3×#15947 | C2 |
| hypercenter | #15947 | D3 |
| hypocenter | C3 | C2×#15947 |
| LCS | G ▷ C24×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ #15947 | |
55540 conjugacy classes of subgroups; only the 1569 normal subgroups are shown.
This group has 55540 conjugacy classes of subgroups — too many to draw. Only its 1569 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 19 points:
(2,3);(1,2,3)(9,11)(17,19);(5,7)(13,15);(12,13)(14,15)(16,17)(18,19);(4,5)(6,7)(8,9)(10,11)174 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((2,3), (1,2,3)(9,11)(17,19), (5,7)(13,15), (12,13)(14,15)(16,17)(18,19), (4,5)(6,7)(8,9)(10,11));