Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | C22×#3025⋊C3 |
| Fitting, Sylow | C22×#3025 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | #120⋊C3 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C24×C3 |
| Frattini | C23 | C24×A4 |
| Fitting | C22×#3025 | C3 |
| socle | C23 | C2×#120⋊C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C22×C4 | |
8336 conjugacy classes of subgroups; only the 228 normal subgroups are shown.
This group has 8336 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:
(3,4)(5,7,15)(6,11,18)(9,10,16)(13,14,19);(1,2)(3,4)(5,19)(6,7)(8,18)(9,12)(10,11)(13,17)(14,16)(15,20);(8,17)(12,20);(1,2)128 conjugacy classes — view the full character table.
| maximal subgroup of | #72341860, C22×#11714162, #72341885, … |
| maximal normal subgroup of | #72341860, C22×#11714162, #72341885, … |
| maximal quotient of | C2×#72245692 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4)(5,7,15)(6,11,18)(9,10,16)(13,14,19), (1,2)(3,4)(5,19)(6,7)(8,18)(9,12)(10,11)(13,17)(14,16)(15,20), (8,17)(12,20), (1,2));