Finite Group DatabasebetaFinite Group Databasebeta| order | 12900 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C5 |
No aliases for this group.
| derived, Hall | C52×C43 ⋊ C3×C4 |
| Hall | C43⋊#1039 ⋊ C3 |
| Hall | C43×C52⋊C3 ⋊ C4 |
| Hall | C52⋊C3 ⋊ Dic43 |
| Sylow | C43 ⋊ #1039⋊C3 |
| Sylow | C52 ⋊ C3×Dic43 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #20264543 |
| commutator, hypocenter | C52×C43 | C3×C4 |
| Fitting, socle, cosocle | C2×C52×C43 | C2×C3 |
| prime radical | C2×C43×C52⋊C3 | C2 |
| LCS | G ▷ C52×C43 | |
| 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:
(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)(46,56,52,49,50,58)(48,53,57,54,55,51)(59,60,61,62);(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)(44,46,51,57,49)(45,48,50,56,54)(47,52,53,55,58)Polycyclic presentation:
2,3,5^2,43,2|a^2=f,c^a=c^4,d^a=d^4,e^a=e^42,c^b=cd,d^b=c^2d^3482 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12900, 52);