Finite Group DatabasebetaFinite Group Databasebeta| order | 12900 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(C43⋊#1035) |
| unambiguous semidirect product | C43⋊(C3⋊#1035) |
| Fitting, derived, Hall | C3×C25×C43 ⋊ C4 |
| Hall | C25×C43 ⋊ Dic3 |
| Hall | C3×C43 ⋊ #1035 |
| Hall | C3×C25 ⋊ Dic43 |
| Sylow | C43 ⋊ C3⋊#1035 |
| Sylow | C25 ⋊ Dic129 |
| Sylow | C3 ⋊ C43⋊#1035 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C3×C25×C43 | C4 |
| Frattini | C5 | C3⋊(C43⋊AGL(1,𝔽5)) |
| socle | C3×C5×C43 | AGL(1,𝔽5) |
| prime radical, cosocle | C3×C43×D25 | C2 |
| LCS | G ▷ C3×C25×C43 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 3225 |
A minimal faithful permutation representation — 2 generators on 71 points:
(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)(45,47,48,53)(46,51,52,63)(49,57,58,62)(50,60,61,54)(55,65,66,68)(56,59,64,67)(70,71);(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)(44,50,55,53,48,46,62,60,58,56,51,68,67,59,65,63,64,49,54,57,52,45,47,66,61)(69,70,71)Polycyclic presentation:
2^2,3,5,43,5|a^2=b,d^5=f,c^a=c^2,d^a=d^2f,e^a=e^42,f^a=f^2,d^b=d^4f^4,f^b=f^4906 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12900, 25);