Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
| direct product | C22×#87520 |
| Fitting | C22×#585 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | #1020 |
| commutator, cosocle, hypocenter | #997 | C24 |
| Frattini | C2 | C23×#1020 |
| Fitting | C22×#585 | D3 |
| socle | C23 | C2×#1020 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C22×C4 | |
12857 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 12857 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(3,4)(7,16,13)(8,15,14)(10,20,17)(11,19,18);(1,2)(3,4)(5,19)(6,7)(8,18)(9,10)(11,16)(12,15)(13,20)(14,17);(1,2);(7,13)(8,15)(10,17)(11,19)(14,16)(18,20)104 conjugacy classes — view the full character table.
| maximal subgroup of | #72341494, C2×#72215767, C2×#72236738, … |
| maximal normal subgroup of | #72341494, C2×#72215767, C2×#72236738, … |
| maximal quotient of | C2×C4×#87520 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4)(7,16,13)(8,15,14)(10,20,17)(11,19,18), (1,2)(3,4)(5,19)(6,7)(8,18)(9,10)(11,16)(12,15)(13,20)(14,17), (1,2), (7,13)(8,15)(10,17)(11,19)(14,16)(18,20));