Finite Group DatabasebetaFinite Group Databasebeta| order | 12342 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C11 |
| unambiguous semidirect product | C17⋊((C11⋊D11)⋊C3) |
| unambiguous semidirect product | (C11⋊D187)⋊C3 |
| Fitting, socle, derived, Hall | C112×C17 ⋊ C2×C3 |
| Hall | C17×C112⋊C3 ⋊ C2 |
| Hall | C11⋊D187 ⋊ C3 |
| Hall | C112⋊C3 ⋊ D17 |
| Sylow | C112 ⋊ C3×D17 |
| Sylow | C17 ⋊ (C11⋊D11)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C112×C17 | C2×C3 |
| LCS | G ▷ C112×C17 | |
| UCS | C1 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 561 |
A minimal faithful permutation representation — 2 generators on 50 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19,21)(20,32,34,24,25,41)(22,40,37,29,30,47)(23,28,38,33,27,39)(26,44,46,35,36,50)(31,45,49,42,43,48);(1,2,4,6,8,10,12,14,16,17,15,13,11,9,7,5,3)(18,20,26,37,39,48,49,38,47,35,24)(19,22,31,44,25,33,23,32,36,42,29)(21,27,30,41,43,46,50,45,34,40,28)Polycyclic presentation:
2,3,11^2,17|c^a=c^10,d^a=d^10,e^a=e^16,c^b=c^7d^5,d^b=c^4d^3370 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12342, 13);