Finite Group DatabasebetaFinite Group Databasebeta| order | 2880 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | D5×Ω+(4,𝔽3) |
| Fitting, derived | C5×H(5,𝔽2) ⋊ C2×C32 |
| Hall | D5×H(5,𝔽2) ⋊ C32 |
| Sylow | C5 ⋊ C2×Ω+(4,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D5×A42 |
| commutator, Fitting, cosocle, hypocenter | C5×H(5,𝔽2) | C2×C32 |
| socle | C2×C5 | C2×A42 |
| LCS | G ▷ C5×H(5,𝔽2) | |
| UCS | C1 ◁ C2 | |
316 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,21,22,15,10,20,5,7,17,12,13,6)(2,18,14,9,24,4,11,23,3,8,19,16)(26,27)(28,29);(3,12,4)(7,13,8)(9,21,10)(14,15,16)(17,19,18)(22,24,23)(25,26,28,29,27)100 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×Ω+(4,𝔽3), D5×SO+(4,𝔽3), #72229366, … |
| maximal normal subgroup of | AGL(1,𝔽5)×Ω+(4,𝔽3), D5×SO+(4,𝔽3), #72229366, … |
| maximal quotient of | C2×D5×Ω+(4,𝔽3) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,21,22,15,10,20,5,7,17,12,13,6)(2,18,14,9,24,4,11,23,3,8,19,16)(26,27)(28,29), (3,12,4)(7,13,8)(9,21,10)(14,15,16)(17,19,18)(22,24,23)(25,26,28,29,27));