Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C33 · Schur multiplier (H2) = C2×C34 |
| direct product | C2×C3×#92391 |
| Fitting | C2×C3×H(5,𝔽3) ⋊ C2 |
| derived | C33 ⋊ C22×C33 |
| Sylow | C3×H(5,𝔽3) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #92391 |
| commutator, cosocle, hypocenter | C33 | C22×C33 |
| Frattini | C3 | C2×C33×C3⋊D3 |
| Fitting | C2×C3×H(5,𝔽3) | C2 |
| socle | C2×C32 | C32×C3⋊D3 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C2×C3 | |
4276 conjugacy classes of subgroups; only the 424 normal subgroups are shown.
This group has 4276 conjugacy classes of subgroups — too many to draw. Only its 424 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 3 generators on 32 points:
(5,14,17,8,9,16)(6,22,24,11,12,23)(7,13)(10,29,18,19,20,27)(15,21)(25,28)(26,30);(4,6,11)(5,18,30)(7,12,23)(8,25,29)(9,20,21)(10,28,17)(13,24,22)(14,26,19)(15,27,16)(31,32);(1,2,3)(4,5,8)(6,10,15)(7,9,16)(11,21,19)(12,18,26)(13,17,14)(20,25,24)(22,30,27)(23,28,29)330 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×#72221916, C2×#72221892, C2×C3×#13091108, … |
| maximal quotient of | C2×C3×#13091503 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,14,17,8,9,16)(6,22,24,11,12,23)(7,13)(10,29,18,19,20,27)(15,21)(25,28)(26,30), (4,6,11)(5,18,30)(7,12,23)(8,25,29)(9,20,21)(10,28,17)(13,24,22)(14,26,19)(15,27,16)(31,32), (1,2,3)(4,5,8)(6,10,15)(7,9,16)(11,21,19)(12,18,26)(13,17,14)(20,25,24)(22,30,27)(23,28,29));