Finite Group DatabasebetaFinite Group Databasebeta| order | 36162 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C41⋊((C7⋊C9)⋊D7) |
| unambiguous semidirect product | (C7⋊C9)⋊D287 |
| derived, Hall | C72×C41 ⋊ C2×C9 |
| Hall | C7×C41×C7⋊C9 ⋊ C2 |
| Hall | C7⋊D287 ⋊ C9 |
| Hall | C7×C7⋊C9 ⋊ D41 |
| Sylow | C72 ⋊ C9×D41 |
| Sylow | C41 ⋊ (C7⋊C9)⋊D7 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | ΓL(1,𝔽8)⋊D287 |
| commutator, hypocenter | C72×C41 | C2×C9 |
| Fitting, socle, cosocle | C3×C72×C41 | C2×C3 |
| prime radical | C3×C7⋊D287 | C3 |
| LCS | G ▷ C72×C41 | |
| UCS | C1 ◁ C3 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 2583 |
A minimal faithful permutation representation — 2 generators on 64 points:
(2,5,6,3,4,7)(9,10)(11,12)(13,14)(15,19,23,17,22,18,20,16,21)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)(63,64);(1,2,4,7,6,5,3)(8,9,11,13,14,12,10)(24,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,61,63,64,62,60,58,56,54,52,50,48,46,44,42,40,38,36,34,32,30,28,26)Polycyclic presentation:
2,3,7^2,41,3|b^3=f,c^a=c^6,d^a=d^6,e^a=e^40,c^b=c^22166 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36162, 46);