Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C25 |
| direct product | C2×#11714279 |
| Fitting | C2×#2679 ⋊ D3 |
| derived | #2679⋊C3 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #85559 |
| commutator, cosocle, hypocenter | #2679⋊C3 | C22 |
| Frattini | C23 | C2×#1020 |
| Fitting | C2×#2679 | D3 |
| socle | C24 | #1020 |
| LCS | G ▷ #2679⋊C3 | |
| UCS | C1 ◁ C22 | |
2599 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 2599 conjugacy classes of subgroups — too many to draw. Only its 23 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) = 24 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,2)(3,9,12)(4,7,6)(5,13,11)(8,14,10);(3,5)(4,11,13,12)(6,10)(7,9,14,8)46 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((1,2)(3,9,12)(4,7,6)(5,13,11)(8,14,10), (3,5)(4,11,13,12)(6,10)(7,9,14,8));