Finite Group DatabasebetaFinite Group Databasebeta| order | 300 |
| isoclinism | isoclinic class of C22⋊(C52⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C2×C5 |
| unambiguous semidirect product | C22⋊(C52⋊C3) |
| unambiguous semidirect product | C52⋊A4 |
| Fitting, socle, derived, Hall | C22×C52 ⋊ C3 |
| Sylow | C52 ⋊ A4 |
| Sylow | C22 ⋊ C52⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C22×C52 | C3 |
| LCS | G ▷ C22×C52 | |
| UCS | C1 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 30 |
A minimal faithful permutation representation — 2 generators on 19 points:
(2,3,4)(5,6,8)(7,10,12)(9,14,17)(11,16,18)(13,19,15);(1,4)(2,3)(5,7,11,17,19)(6,10,16,9,15)(8,14,12,13,18)Polycyclic presentation:
3,2^2,5^2|b^a=c,c^a=bc,d^a=de^3,e^a=d^4e^336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(300, 43);