Finite Group DatabasebetaFinite Group Databasebeta| order | 11664 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | ((C3⋊D3)⋊C3)⋊C22 |
| Fitting, derived, Sylow | H(3,𝔽3)2 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | H(3,𝔽3)2 | C24 |
| Frattini, socle | C32 | D34 |
| LCS | G ▷ H(3,𝔽3)2 | |
| UCS | C1 | |
5606 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 5606 conjugacy classes of subgroups — too many to draw. Only its 191 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) = 54 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,5,2)(3,7,4)(6,9,8)(10,11)(12,18)(13,17);(2,5)(4,7)(8,9)(11,15)(12,14)(16,17);(2,4)(3,6)(5,7)(11,14)(12,15)(13,18)(16,17);(11,14)(12,15)(13,18)(16,17)121 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((1,5,2)(3,7,4)(6,9,8)(10,11)(12,18)(13,17), (2,5)(4,7)(8,9)(11,15)(12,14)(16,17), (2,4)(3,6)(5,7)(11,14)(12,15)(13,18)(16,17), (11,14)(12,15)(13,18)(16,17));