Finite Group DatabasebetaFinite Group Databasebeta| order | 4032 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C215 |
| direct product | C24×D3×AGL(1,𝔽7) |
| Fitting, socle | C24×C3×C7 ⋊ C22×C3 |
| derived | C3×C7 ⋊ C26×C3 |
| Hall | C3×ΓL(1,𝔽8) ⋊ C26 |
| Sylow | C7 ⋊ C25×C3×D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | D3×AGL(1,𝔽7) |
| commutator, cosocle, hypocenter | C3×C7 | C26×C3 |
| Frattini | C1 | G |
| Fitting, socle | C24×C3×C7 | C22×C3 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C24 | |
23348 conjugacy classes of subgroups; only the 6839 normal subgroups are shown.
This group has 23348 conjugacy classes of subgroups — too many to draw. Only its 6839 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) = 336 |
A minimal faithful permutation representation — 6 generators on 18 points:
(7,8)(9,10,11)(12,13,18)(14,15,17);(5,6);(3,4);(13,14)(15,17)(16,18);(1,2)(3,4)(5,6)(7,8)(10,11);(1,2)336 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D3×#120×AGL(1,𝔽7), C24×C3⋊D3×AGL(1,𝔽7) |
| maximal normal subgroup of | D3×#120×AGL(1,𝔽7) |
| maximal quotient of | D3×H(5,𝔽2)×AGL(1,𝔽7), C24×C3⋊D3×AGL(1,𝔽7) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((7,8)(9,10,11)(12,13,18)(14,15,17), (5,6), (3,4), (13,14)(15,17)(16,18), (1,2)(3,4)(5,6)(7,8)(10,11), (1,2));