Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C22×#73031 |
| Fitting | C22×#458 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C23 | #1020⋊C2 |
| commutator, cosocle, hypocenter | #997 | C24 |
| Frattini | C23 | C23×S4 |
| Fitting | C22×#458 | D3 |
| prime radical | C22×#5935 | C2 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C23 | |
3688 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 3688 conjugacy classes of subgroups — too many to draw. Only its 115 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)(5,16,14)(6,15,19)(7,9,8)(10,12,11);(7,13)(8,16)(10,18)(11,15)(14,17)(19,20);(5,6,9,12)(7,10,14,19)(8,15,16,11)(13,18,17,20);(1,2)80 conjugacy classes — view the full character table.
| maximal subgroup of | #72341575, C22×#11714162, C2×#72236843, … |
| maximal normal subgroup of | #72341575, C22×#11714162, C2×#72236843, … |
| maximal quotient of | C2×#72245729 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4)(5,16,14)(6,15,19)(7,9,8)(10,12,11), (7,13)(8,16)(10,18)(11,15)(14,17)(19,20), (5,6,9,12)(7,10,14,19)(8,15,16,11)(13,18,17,20), (1,2));