Finite Group DatabasebetaFinite Group Databasebeta| order | 3888 |
| isoclinism | isoclinic class of ((C3⋊D3)⋊C3)⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C26×C3 |
| direct product | C22×C32×((C3⋊D3)⋊C3)⋊C2 |
| Fitting | C22×C32×H(3,𝔽3) ⋊ C22 |
| derived | H(3,𝔽3) ⋊ C24×C32 |
| Sylow | C32×H(3,𝔽3) ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C32 | ((C3⋊D3)⋊C3)⋊C2 |
| commutator, cosocle, hypocenter | H(3,𝔽3) | C24×C32 |
| Frattini | C3 | C22×C32×D32 |
| Fitting | C22×C32×H(3,𝔽3) | C22 |
| socle | C22×C33 | D32 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C22×C32 | |
4674 conjugacy classes of subgroups; only the 654 normal subgroups are shown.
This group has 4674 conjugacy classes of subgroups — too many to draw. Only its 654 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) = 216 |
A minimal faithful permutation representation — 4 generators on 19 points:
(4,5,6)(7,8,11)(9,12,15,10,13,14);(1,2,3)(7,9,10)(8,12,13)(18,19);(8,11)(9,10)(12,15)(13,14);(16,17)396 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C22×C32×AGL(2,𝔽3) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,5,6)(7,8,11)(9,12,15,10,13,14), (1,2,3)(7,9,10)(8,12,13)(18,19), (8,11)(9,10)(12,15)(13,14), (16,17));