Finite Group DatabasebetaFinite Group Databasebeta| order | 12900 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C43⋊C3)⋊#1035 |
| unambiguous semidirect product | (C43⋊#1035)⋊C3 |
| Fitting, derived, Hall | C25×C43 ⋊ C3×C4 |
| Hall | C43⋊#1035 ⋊ C3 |
| Hall | C25×C43⋊C3 ⋊ C4 |
| Hall | C43⋊C3 ⋊ #1035 |
| Sylow | C43 ⋊ C3×#1035 |
| Sylow | C25 ⋊ (C43⋊C3)⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C25×C43 | C3×C4 |
| Frattini | C5 | (C43⋊C3)⋊AGL(1,𝔽5) |
| socle | C5×C43 | C3×AGL(1,𝔽5) |
| prime radical | D25×C43⋊C3 | C2 |
| cosocle | C43×D25 | C2×C3 |
| LCS | G ▷ C25×C43 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 1075 |
A minimal faithful permutation representation — 2 generators on 68 points:
(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,36)(12,26,22,20,16,13)(19,32,34,23,25,37)(21,38,39,30,29,33)(31,41,42,40,35,43)(45,47,48,53)(46,51,52,63)(49,57,58,62)(50,60,61,54)(55,65,66,68)(56,59,64,67);(1,2,5,11,19,22,7,6,12,21,31,39,15,26,14,25,35,42,28,37,38,27,18,29,34,36,43,41,32,10,16,24,33,40,30,20,4,8,13,23,17,9,3)(44,45,49,59,56,46,50,47,54,65,51,62,55,66,57,63,68,60,53,61,52,64,67,58,48)Polycyclic presentation:
2,3,2,5,43,5|a^2=c,d^5=f,d^a=d^2f,e^a=e^42,f^a=f^2,e^b=e^36,d^c=d^4f^4,f^c=f^4128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12900, 14);