Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C4 · Schur multiplier (H2) = C35 |
| direct product | H(3,𝔽3)×(C3⋊Dic3)⋊C3 |
| Sylow | H(3,𝔽3)2 ⋊ C4 |
1409 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 1409 conjugacy classes of subgroups — too many to draw. Only its 174 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3,4)(6,11,13,8,9,12)(7,10);(5,6,8)(7,9,12)(10,13,11)(15,18,19)(17,21,22);(6,9,13)(8,11,12)(14,15,17)(16,19,21)(18,22,20)220 conjugacy classes — view the full character table.
| maximal subgroup of | #72229104, #72229107, #72229111, … |
| maximal normal subgroup of | #72229104, #72229107, #72229111, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3,4)(6,11,13,8,9,12)(7,10), (5,6,8)(7,9,12)(10,13,11)(15,18,19)(17,21,22), (6,9,13)(8,11,12)(14,15,17)(16,19,21)(18,22,20));