Finite Group DatabasebetaFinite Group Databasebeta| order | 36162 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
| Fitting, socle, derived, Hall | C72×C41 ⋊ C2×C32 |
| Hall | ΓL(1,𝔽8)2×C41 ⋊ C2 |
| Hall | C7⋊D287 ⋊ C32 |
| Hall | ΓL(1,𝔽8)2 ⋊ D41 |
| Sylow | C72 ⋊ C32×D41 |
| Sylow | C41 ⋊ ΓL(1,𝔽8)⋊AGL(1,𝔽7) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C72×C41 | C2×C32 |
| LCS | G ▷ C72×C41 | |
| UCS | C1 | |
112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 861 |
A minimal faithful permutation representation — 2 generators on 55 points:
(2,5,6,3,4,7)(8,9)(10,11)(12,14)(16,17)(18,19)(20,21)(22,23)(24,25)(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);(1,2,4,7,6,5,3)(9,11,13)(10,12,14)(15,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17)Polycyclic presentation:
2,3^2,7^2,41|d^a=d^6,e^a=e^6,f^a=f^40,e^b=e^2,d^c=d^2526 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36162, 74);