Finite Group DatabasebetaFinite Group Databasebeta| order | 1200 |
| isoclinism | isoclinic class of #226 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C25 · Schur multiplier (H2) = C1 |
| direct product | C25×#226 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C25 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C2×C25 |
| Frattini, socle | C2×C5 | C5×S4 |
| Fitting | Q8×C25 | D3 |
| prime radical, cosocle | C5×SL(2,𝔽3) | C2×C5 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C25 | |
39 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 400 |
A minimal faithful permutation representation — 2 generators on 41 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25)(26,31,35,38,30,40,37,32)(27,36,39,29,33,28,41,34);(28,34,37)(29,35,36)(31,38,41)(32,39,40)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1200, 96);