Finite Group DatabasebetaFinite Group Databasebeta| order | 23322 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C23⋊((C13⋊C3)⋊D13) |
| unambiguous semidirect product | (C13⋊C3)⋊D299 |
| Fitting, socle, derived, Hall | C132×C23 ⋊ C2×C3 |
| Hall | C13×C23×C13⋊C3 ⋊ C2 |
| Hall | C13⋊D299 ⋊ C3 |
| Hall | C13×C13⋊C3 ⋊ D23 |
| Sylow | C132 ⋊ C3×D23 |
| Sylow | C23 ⋊ (C13⋊C3)⋊D13 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C132×C23 | C2×C3 |
| LCS | G ▷ C132×C23 | |
| UCS | C1 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 897 |
A minimal faithful permutation representation — 2 generators on 49 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(25,29,30,26,27,31)(28,35,34,32,33,36)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49);(1,2,4,6,8,10,12,14,16,18,20,22,23,21,19,17,15,13,11,9,7,5,3)(24,25,28,27,31,34,33,35,36,30,29,32,26)(37,38,40,42,44,46,48,49,47,45,43,41,39)Polycyclic presentation:
2,3,13^2,23|c^a=c^12,d^a=d^12,e^a=e^22,c^b=c^31051 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(23322, 20);