Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22×C42 |
| direct product | #127×C42⋊C3 |
| Fitting, Sylow | C42×#127 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×C42⋊C3 |
| commutator, cosocle | C22×C42 | C23×C3 |
| Frattini, socle | C24 | C23×A4 |
| Fitting | C42×#127 | C3 |
| hypercenter | #127 | C42⋊C3 |
| hypocenter | C42 | C3×#127 |
| LCS | G ▷ C22×C42 ▷ C42 | |
| UCS | C1 ◁ C22 ◁ #127 | |
2480 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 2480 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 96 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,7,8)(9,10,18)(11,13,14)(12,17,20)(15,19,16);(1,2,3,4)(9,16,11,12)(10,17,13,15);(2,4)(6,8)112 conjugacy classes — view the full character table.
| maximal subgroup of | C42⋊C3×#127⋊C3, #127×#127⋊C3, #72258879 |
| maximal normal subgroup of | C42⋊C3×#127⋊C3, #127×#127⋊C3, #72258879 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,6,7,8)(9,10,18)(11,13,14)(12,17,20)(15,19,16), (1,2,3,4)(9,16,11,12)(10,17,13,15), (2,4)(6,8));