Finite Group DatabasebetaFinite Group Databasebeta| order | 1200 |
| isoclinism | isoclinic class of #6037 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×#87780 |
| Hall | C3×C52 ⋊ C4⋊C4 |
| Sylow | C52 ⋊ C3×C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | D52 |
| commutator | C2×C52 | C2×C3×C4 |
| Frattini | C22 | C3×D52 |
| Fitting, socle, prime radical | C22×C3×C52 | C22 |
| cosocle | C22×C52 | C22×C3 |
| hypocenter | C52 | C3×C4⋊C4 |
| LCS | G ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C22×C3 | |
136 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3)(4,5,6,7)(9,11)(13,14)(15,16)(17,18,20,21,19);(8,9,10,11)(12,13,15,16,14)(18,19)(20,21)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1200, 282);