Finite Group DatabasebetaFinite Group Databasebeta| order | 5760 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | C2×D5×Ω+(4,𝔽3) |
| Fitting | C2×C5×H(5,𝔽2) ⋊ C2×C32 |
| derived | C5×H(5,𝔽2) ⋊ C22×C32 |
| Hall | C2×D5×H(5,𝔽2) ⋊ C32 |
| Sylow | C5 ⋊ C22×Ω+(4,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D5×A42 |
| commutator, cosocle, hypocenter | C5×H(5,𝔽2) | C22×C32 |
| Frattini | C2 | C2×D5×A42 |
| Fitting | C2×C5×H(5,𝔽2) | C2×C32 |
| socle | C22×C5 | C2×A42 |
| LCS | G ▷ C5×H(5,𝔽2) | |
| UCS | C1 ◁ C22 | |
1454 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 1454 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 2 generators on 31 points:
(4,5)(6,7)(8,22,23,12,20,11)(9,28,16,18,19,15)(10,21)(13,30,31,27,26,25)(14,17)(24,29);(1,2)(3,4,6,7,5)(8,17,27,11,9,30,12,14,13,23,18,26)(10,15,31,22,19,24,21,16,25,20,28,29)200 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D5×SO+(4,𝔽3), C2×AGL(1,𝔽5)×Ω+(4,𝔽3), C2×#72229366 |
| maximal normal subgroup of | C2×D5×SO+(4,𝔽3), C2×AGL(1,𝔽5)×Ω+(4,𝔽3), C2×#72229366 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,5)(6,7)(8,22,23,12,20,11)(9,28,16,18,19,15)(10,21)(13,30,31,27,26,25)(14,17)(24,29), (1,2)(3,4,6,7,5)(8,17,27,11,9,30,12,14,13,23,18,26)(10,15,31,22,19,24,21,16,25,20,28,29));