Finite Group DatabasebetaFinite Group Databasebeta| order | 32634 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, Hall | C72×C37 ⋊ C3×D3 |
| derived | C37×#3862 ⋊ C2×C3 |
| Hall | #30654588 ⋊ C2 |
| Sylow | C72 ⋊ C3⋊(D37⋊C3) |
| Sylow | C37 ⋊ #64418⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C37×#3862 | C2×C3 |
| Fitting, socle | C72×C37 | C3×D3 |
| LCS | G ▷ C37×#3862 | |
| UCS | C1 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 518 |
A minimal faithful permutation representation — 2 generators on 51 points:
(1,4,7,6,13,10)(2,3,5,11,9,14)(8,12)(16,20,21,17,18,22)(19,28,29,23,24,38)(25,41,33,31,32,43)(26,40,36,34,30,27)(35,51,48,45,42,49)(37,39,44,46,47,50);(3,7,8)(4,9,6)(5,10,12)(11,14,13)(15,16,19,25,33,44,51,41,45,34,18,22,27,37,48,29,40,28,39,47,24,30,38,49,46,36,21,20,26,35,32,42,50,43,31,23,17)Polycyclic presentation:
2,3^2,37,7^2|c^a=c^2,d^a=d^36,e^a=e^4f,f^a=e^6f^3,d^b=d^10,e^b=e^2,f^b=f^2,e^c=e^2f^6,f^c=f^4134 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32634, 166);