Finite Group DatabasebetaFinite Group Databasebeta| order | 33516 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×ΓL(1,𝔽8)⋊#11716328 |
| unambiguous semidirect product | (C2×ΓL(1,𝔽8))⋊#11716328 |
| Fitting, socle | C2×C72×C19 ⋊ C2×C32 |
| derived, Hall | C72×C19 ⋊ C22×C32 |
| Hall | ΓL(1,𝔽8)×#87705 ⋊ C22 |
| Hall | C2×C7⋊D133 ⋊ C32 |
| Sylow | C72 ⋊ C2×C3×D19⋊C3 |
| Sylow | C19 ⋊ C2×ΓL(1,𝔽8)⋊AGL(1,𝔽7) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | ΓL(1,𝔽8)⋊#11716328 |
| commutator, cosocle, hypocenter | C72×C19 | C22×C32 |
| Frattini | C1 | G |
| Fitting, socle | C2×C72×C19 | C2×C32 |
| LCS | G ▷ C72×C19 | |
| UCS | C1 ◁ C2 | |
280 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 798 |
A minimal faithful permutation representation — 2 generators on 35 points:
(4,7,8,5,6,9)(10,11)(12,13)(14,16)(17,18)(19,21)(20,27)(22,24)(23,28)(25,26)(29,31)(30,33)(34,35);(1,2)(3,4,6,9,8,7,5)(11,13,15)(12,14,16)(18,20,23)(19,22,24)(21,26,30)(25,29,34)(27,31,33)(28,32,35)262 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33516, 946);