Finite Group DatabasebetaFinite Group Databasebeta| order | 1200 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#87886 |
| Fitting | C22×C3×C52 ⋊ C22 |
| Hall | C3×C52 ⋊ C2×D4 |
| Sylow | C52 ⋊ C2×C3×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C2×D52 |
| commutator, cosocle | C2×C52 | C23×C3 |
| Frattini | C2 | C2×C3×D52 |
| Fitting | C22×C3×C52 | C22 |
| socle | C2×C3×C52 | C23 |
| hypercenter | C22×C3 | D52 |
| hypocenter | C52 | C2×C3×D4 |
| LCS | G ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3 | |
280 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,3)(4,5)(6,7)(8,9,11,12,10)(13,14,16,17,15);(5,7)(9,10)(11,12);(5,7)(14,15)(16,17)138 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#11717375, D3×#87886, #72224574, … |
| maximal normal subgroup of | C3×#11717375, D3×#87886, #72224574, … |
| maximal quotient of | C3×#11925664, C32×#87886, C2×C3×#87886, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1200, 775);