Finite Group DatabasebetaFinite Group Databasebeta| order | 3600 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C3×C5 |
| direct product | C5⋊D5×#638 |
| Fitting | C22×C32×C52 ⋊ C22 |
| Hall | C32×C52 ⋊ C2×D4 |
| Sylow | C52 ⋊ C2×#638 |
| Sylow | C32 ⋊ D4×C5⋊D5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×C3⋊D3×C5⋊D5 |
| commutator, socle, cosocle | C2×C32×C52 | C23 |
| Fitting | C22×C32×C52 | C22 |
| hypercenter | C22 | C3⋊D3×C5⋊D5 |
| hypocenter | C32×C52 | C2×D4 |
| LCS | G ▷ C2×C32×C52 ▷ C32×C52 | |
| UCS | C1 ◁ C2 ◁ C22 | |
1296 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 1296 conjugacy classes of subgroups — too many to draw. Only its 206 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 900 |
A minimal faithful permutation representation — 3 generators on 20 points:
(6,7)(8,9)(11,12)(13,14)(18,19,20);(2,4)(10,11,13,14,12)(16,17)(19,20);(1,2)(3,4)(5,6,8,9,7)(15,16,17)294 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D4×C3⋊D3×C5⋊D5, #638×#1043, #72230625, … |
| maximal normal subgroup of | D4×C3⋊D3×C5⋊D5, #638×#1043, #72230625, … |
| maximal quotient of | C2×C5⋊D5×#638 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((6,7)(8,9)(11,12)(13,14)(18,19,20), (2,4)(10,11,13,14,12)(16,17)(19,20), (1,2)(3,4)(5,6,8,9,7)(15,16,17));