Finite Group DatabasebetaFinite Group Databasebeta| order | 33516 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×D19×ΓL(1,𝔽8)⋊D7 |
| Fitting, socle | C3×C72×C19 ⋊ C22×C3 |
| derived, Hall | C72×C19 ⋊ C22×C32 |
| Hall | C3×C7×C19×ΓL(1,𝔽8) ⋊ C22 |
| Hall | D19×C7⋊D7 ⋊ C32 |
| Hall | C3×C7×ΓL(1,𝔽8) ⋊ C2×D19 |
| Sylow | C72 ⋊ C2×C32×D19 |
| Sylow | C19 ⋊ C2×C3×ΓL(1,𝔽8)⋊D7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | D19×ΓL(1,𝔽8)⋊D7 |
| commutator, cosocle, hypocenter | C72×C19 | C22×C32 |
| Frattini | C1 | G |
| Fitting, socle | C3×C72×C19 | C22×C3 |
| LCS | G ▷ C72×C19 | |
| UCS | C1 ◁ C3 | |
280 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 1197 |
A minimal faithful permutation representation — 2 generators on 36 points:
(5,6)(7,8)(9,10)(12,15,16,13,14,17)(18,19,21,23,25,27,29,31,33,35,36,34,32,30,28,26,24,22,20);(1,2,3)(4,5,7,9,10,8,6)(11,12,14,17,16,15,13)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)726 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33516, 1012);