Finite Group DatabasebetaFinite Group Databasebeta| order | 32634 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C37⋊C3)⋊#64418 |
| Fitting, socle, Hall | C72×C37 ⋊ C3×D3 |
| derived | C37×#3862 ⋊ C2×C3 |
| Hall | C37⋊C3×#3862 ⋊ C2 |
| Sylow | C72 ⋊ C3⋊(D37⋊C3) |
| Sylow | C37 ⋊ C3×#64418 |
| 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:
(1,4,14,9,13,2,3,12,11,10,8,6,7,5)(16,20,21,17,18,22)(19,28,29,23,24,37)(25,40,41,31,32,48)(26,39,35,33,30,27)(34,50,51,43,42,46)(36,38,49,45,44,47);(3,7,8)(4,9,6)(5,10,12)(11,14,13)(15,16,19,25,32,42,47,48,45,35,21,20,26,34,44,24,30,37,46,51,29,39,28,38,43,33,18,22,27,36,41,49,50,40,31,23,17)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^26,e^c=e^2f^6,f^c=f^4174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32634, 162);