Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
| direct product | C2×#11710960 |
| Fitting | C2×#1304 ⋊ D3 |
| derived | #1304⋊C3 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #11710960 |
| commutator, cosocle, hypocenter | #1304⋊C3 | C22 |
| Frattini | C25 | C2×S4 |
| Fitting | C2×#1304 | D3 |
| socle | C23 | #4622 |
| LCS | G ▷ #1304⋊C3 | |
| UCS | C1 ◁ C2 | |
919 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 919 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:
(3,11,10,14,8,4,16,13)(5,7,9,17,12,6,15,18);(1,2)(4,12,10,9,5,17)(6,7,8)(11,14,16,15,13,18)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((3,11,10,14,8,4,16,13)(5,7,9,17,12,6,15,18), (1,2)(4,12,10,9,5,17)(6,7,8)(11,14,16,15,13,18));