Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C24×C3 |
| direct product | C3×#11709314 |
| Fitting | C3×#15947 ⋊ C3 |
| derived | C26 ⋊ C22×C32 |
| Sylow | #15947 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #11709314 |
| commutator, cosocle, hypocenter | C26 | C22×C32 |
| Frattini | C24 | C22×C3×A4 |
| Fitting | C3×#15947 | C3 |
| socle | C24×C3 | C22×A4 |
| LCS | G ▷ C26 | |
| UCS | C1 ◁ C3 | |
2358 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 2358 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 72 |
A minimal faithful permutation representation — 2 generators on 19 points:
(6,10,7)(8,11,9)(12,13)(14,18,19,16,15,17);(1,2,3)(4,11,10,5)(6,9,7,8)(13,18)(16,17)90 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#72215851, C2×C3×#11709314, C3×#72209642, … |
| maximal normal subgroup of | C3×#72215851, C2×C3×#11709314, C3×#72209642, … |
| maximal quotient of | C2×C3×#11709314 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((6,10,7)(8,11,9)(12,13)(14,18,19,16,15,17), (1,2,3)(4,11,10,5)(6,9,7,8)(13,18)(16,17));