Finite Group DatabasebetaFinite Group Databasebeta| order | 36822 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C17⋊((C19⋊C3)⋊D19) |
| unambiguous semidirect product | (C19⋊C3)⋊D323 |
| Fitting, socle, derived, Hall | C17×C192 ⋊ C2×C3 |
| Hall | C17×C19×C19⋊C3 ⋊ C2 |
| Hall | C17⋊(C19⋊D19) ⋊ C3 |
| Hall | C19×C19⋊C3 ⋊ D17 |
| Sylow | C192 ⋊ C3×D17 |
| Sylow | C17 ⋊ (C19⋊C3)⋊D19 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C17×C192 | C2×C3 |
| LCS | G ▷ C17×C192 | |
| UCS | C1 | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 969 |
A minimal faithful permutation representation — 2 generators on 55 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(19,23,24,20,21,25)(22,31,28,26,27,33)(29,34,36,35,32,30)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55);(1,2,4,6,8,10,12,14,16,17,15,13,11,9,7,5,3)(18,19,22,28,34,31,35,21,25,30,36,24,23,29,27,32,33,26,20)(37,38,40,42,44,46,48,50,52,54,55,53,51,49,47,45,43,41,39)Polycyclic presentation:
2,3,17,19^2|c^a=c^16,d^a=d^18,e^a=e^18,d^b=d^71458 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36822, 25);