Finite Group DatabasebetaFinite Group Databasebeta| order | 20124 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D39⋊D129 |
| Fitting, socle, derived, Hall | C32×C13×C43 ⋊ C22 |
| Hall | C13×C43 ⋊ D32 |
| Hall | C32×C43 ⋊ C2×D13 |
| Hall | C32×C13 ⋊ C2×D43 |
| Sylow | C43 ⋊ (C3⋊D3)⋊D13 |
| Sylow | C13 ⋊ D3×D129 |
| Sylow | C32 ⋊ D13×D43 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C32×C13×C43 | C22 |
| LCS | G ▷ C32×C13×C43 | |
| UCS | C1 | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 62 |
| transitive degree | μt(G) = 3354 |
A minimal faithful permutation representation — 2 generators on 62 points:
(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58,59)(60,61);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45,47,49,51,53,55,56,54,52,50,48,46)(58,59)(61,62)Polycyclic presentation:
2^2,3^2,13,43|c^a=c^2,d^a=d^2,f^a=f^42,d^b=d^2,e^b=e^121368 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20124, 238);