Finite Group DatabasebetaFinite Group Databasebeta| order | 1200 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#87871 |
| Hall | C3×C52 ⋊ #41 |
| Sylow | C52 ⋊ C3×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C2×D52 |
| commutator, cosocle | C2×C52 | C23×C3 |
| Frattini | C2 | C2×C3×D52 |
| Fitting | C3×C4×C52 | C22 |
| socle | C2×C3×C52 | C23 |
| hypercenter | C3×C4 | D52 |
| hypocenter | C52 | C3×#41 |
| LCS | G ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2×C3 ◁ C3×C4 | |
192 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 120 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3)(4,6,7,10)(5,8,9,11)(12,13,15,16,14)(17,18,20,21,19);(4,5,7,9)(6,11,10,8)(13,14)(15,16);(6,10)(8,11)(18,19)(20,21)138 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#11717471, D3×#87871, #72230622, … |
| maximal normal subgroup of | C3×#11717471, D3×#87871, #72230622, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1200, 760);