Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26×C4 |
| direct product | C22×#3027⋊C3 |
| Fitting, Sylow | C22×#3027 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120⋊C3 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C24×C3 |
| Frattini | C23 | C24×A4 |
| Fitting | C22×#3027 | C3 |
| hypercenter | C22×C4 | #120⋊C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C23 ◁ C22×C4 | |
8356 conjugacy classes of subgroups; only the 228 normal subgroups are shown.
This group has 8356 conjugacy classes of subgroups — too many to draw. Only its 228 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:
(1,2)(3,4)(5,12,9,11,7,20,10,18,15,6,16,14)(8,13,17,19);(3,4)(5,16,10,7)(6,18,11,12)(8,15,17,9)(13,20,19,14);(6,11)(8,17)(12,18)(14,20);(1,2)92 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72236755, C22×#11714184, C2×#72236759, … |
| maximal normal subgroup of | C2×#72236755, C22×#11714184, C2×#72236759, … |
| maximal quotient of | C2×#72245692 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,4)(5,12,9,11,7,20,10,18,15,6,16,14)(8,13,17,19), (3,4)(5,16,10,7)(6,18,11,12)(8,15,17,9)(13,20,19,14), (6,11)(8,17)(12,18)(14,20), (1,2));