Finite Group DatabasebetaFinite Group Databasebeta| order | 36414 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3×C17 |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊((C17⋊D17)⋊C3) |
| unambiguous semidirect product | (C17⋊(C17⋊AGL(1,𝔽7)))⋊C3 |
| unambiguous semidirect product | #27131466⋊C3 |
| unambiguous semidirect product | #27131472⋊C3 |
| Fitting, socle, derived, Hall | C7×C172 ⋊ C2×C32 |
| Hall | ΓL(1,𝔽8)×C172⋊C3 ⋊ C2 |
| Hall | C7⋊(C17⋊D17) ⋊ C32 |
| Sylow | C172 ⋊ C3×AGL(1,𝔽7) |
| Sylow | C7 ⋊ C3×(C17⋊D17)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C7×C172 | C2×C32 |
| LCS | G ▷ C7×C172 | |
| UCS | C1 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 58 |
| transitive degree | μt(G) = 357 |
A minimal faithful permutation representation — 2 generators on 58 points:
(2,5,6,3,4,7)(8,10)(9,12)(11,17)(13,43)(14,16)(15,46)(18,35)(19,21)(20,24)(22,23)(25,47)(26,28)(29,37)(30,31)(32,33)(34,56)(36,39)(38,50)(40,45)(41,53)(44,51)(48,49)(52,55)(57,58);(1,2,4,7,6,5,3)(8,9,11)(10,15,27)(12,20,38)(13,24,44)(14,26,39)(16,22,17)(18,34,23)(19,36,31)(21,41,55)(25,45,57)(28,32,30)(29,49,56)(33,50,47)(35,52,42)(37,53,58)(40,54,43)(46,48,51)Polycyclic presentation:
2,3^2,7,17^2|d^a=d^6,e^a=e^16,f^a=f^16,e^b=e^2f^15,f^b=e^12f^14,d^c=d^2261 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36414, 66);