Finite Group DatabasebetaFinite Group Databasebeta| order | 12900 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C43⋊#64470 |
| derived, Hall | C43×C52⋊C3 ⋊ C4 |
| Hall | C52×C43 ⋊ Dic3 |
| Hall | C52⋊C3 ⋊ Dic43 |
| Sylow | C43 ⋊ #64470 |
| Sylow | C52 ⋊ Dic129 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C43⋊#3874 |
| commutator, hypocenter | C43×C52⋊C3 | C4 |
| Fitting, socle | C2×C52×C43 | D3 |
| prime radical, cosocle | C2×C43×C52⋊C3 | C2 |
| LCS | G ▷ C43×C52⋊C3 | |
| UCS | C1 ◁ C2 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 62 |
| transitive degree | μt(G) = 2580 |
A minimal faithful permutation representation — 2 generators on 62 points:
(1,3,7,9,4)(2,10,8,15,14,11,12,5,6,13)(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)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60,61,62);(1,2,5)(3,6,11)(4,8,13)(7,12,15)(9,14,10)(16,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,58,56,54,52,50,48,46,44,42,40,38,36,34,32,30,28,26,24,22,20,18)Polycyclic presentation:
2,3,43,5^2,2|a^2=f,b^a=b^2,c^a=c^42,d^a=d^2e,e^a=d^2e^3,d^b=d^2e^4,e^b=d^2e^2488 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12900, 48);