Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| direct product | C2×#11710964 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #11710964 |
| commutator, cosocle, hypocenter | #1304⋊C3 | C22 |
| Frattini | C25 | C2×S4 |
| Fitting | C2×#1304 | D3 |
| socle | C23 | #4621 |
| prime radical | C2×#1304⋊C3 | C2 |
| LCS | G ▷ #1304⋊C3 | |
| UCS | C1 ◁ C2 | |
587 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 587 conjugacy classes of subgroups — too many to draw. Only its 19 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 18 points:
(3,18,9,10,8,17,12,5)(4,16,7,11,13,15,6,14);(1,2)(4,5,18)(6,8,7)(9,16,17)(10,14,13)(11,12,15)38 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72210571 |
| maximal normal subgroup of | C2×#72210571 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,18,9,10,8,17,12,5)(4,16,7,11,13,15,6,14), (1,2)(4,5,18)(6,8,7)(9,16,17)(10,14,13)(11,12,15));