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 ⋊ D3×C37⋊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:
(1,10,8,4,7,9)(2,3,13,6,11,14)(5,12)(16,17,19)(18,21,26)(20,24,23)(22,28,29)(25,31,33)(27,34,30)(32,39,37)(35,42,41)(36,40,38)(43,47,45)(44,48,46)(49,51,50);(3,7,6)(4,9,11)(5,10,13)(8,14,12)(15,16,18,22,29,37,45,50,47,31,17,20,25,32,40,26,33,41,46,39,21,27,35,43,48,34,19,23,30,38,28,36,44,49,51,42,24)Polycyclic presentation:
2,3^2,37,7^2|c^a=c^2,e^a=f^3,f^a=e^5,d^b=d^10,e^b=e^2,f^b=f^2,e^c=e^4,f^c=f^2260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32634, 168);