Finite Group DatabasebetaFinite Group Databasebeta| order | 3600 |
| isoclinism | isoclinic class of C3⋊#1043 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C22×C3×C3⋊#1043 |
| Fitting, socle | C22×C32×C52 ⋊ C4 |
| derived | C3×C52 ⋊ C22×C3×C4 |
| Hall | C32×C52 ⋊ C22×C4 |
| Sylow | C52 ⋊ C22×C3×Dic3 |
| Sylow | C32 ⋊ C22×#1043 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | C3⋊#1043 |
| commutator, hypocenter | C3×C52 | C22×C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C22×C32×C52 | C4 |
| prime radical | C22×C32×C5⋊D5 | C2 |
| cosocle | C3×C5⋊D5 | C23×C3 |
| LCS | G ▷ C3×C52 | |
| UCS | C1 ◁ C22×C3 | |
992 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 992 conjugacy classes of subgroups — too many to draw. Only its 226 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) = 600 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3)(4,5,6)(12,13,16,14,15)(19,20);(7,8,11,9,10)(17,18);(5,6)(8,9,10,11)(13,14,15,16)288 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(4,5,6)(12,13,16,14,15)(19,20), (7,8,11,9,10)(17,18), (5,6)(8,9,10,11)(13,14,15,16));