Finite Group DatabasebetaFinite Group Databasebeta| order | 29988 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C3×C17⋊(ΓL(1,𝔽8)⋊D7) |
| unambiguous semidirect product | (C2×C17)⋊(C3×ΓL(1,𝔽8)⋊D7) |
| unambiguous semidirect product | (C2×ΓL(1,𝔽8))⋊(C3×D119) |
| Fitting, socle | C2×C3×C72×C17 ⋊ C2×C3 |
| derived, Hall | C72×C17 ⋊ C22×C32 |
| Hall | C3×C7×C17×ΓL(1,𝔽8) ⋊ C22 |
| Hall | C2×C7⋊D119 ⋊ C32 |
| Hall | C3×C7×ΓL(1,𝔽8) ⋊ C2×D17 |
| Sylow | C72 ⋊ C2×C32×D17 |
| Sylow | C17 ⋊ C2×C3×ΓL(1,𝔽8)⋊D7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C17⋊(ΓL(1,𝔽8)⋊D7) |
| commutator, cosocle, hypocenter | C72×C17 | C22×C32 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C72×C17 | C2×C3 |
| LCS | G ▷ C72×C17 | |
| UCS | C1 ◁ C2×C3 | |
280 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 2142 |
A minimal faithful permutation representation — 2 generators on 36 points:
(7,10,11,8,9,12)(14,15)(16,17)(18,19)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36);(1,2)(3,4,5)(6,7,9,12,11,10,8)(13,14,16,18,19,17,15)(20,21,23,25,27,29,31,33,35,36,34,32,30,28,26,24,22)1812 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(29988, 513);