Finite Group DatabasebetaFinite Group Databasebeta| order | 3840 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
| direct product | D5×#72991 |
| Fitting | C5×#379 ⋊ C2×D3 |
| Sylow | C5 ⋊ C2×#72991 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D5×#1020⋊C2 |
| commutator, cosocle | C2×C5×C22⋊A4 | C23 |
| Frattini | C23 | C2×D5×S4 |
| Fitting | C5×#379 | C2×D3 |
| socle | C23×C5 | C22×S4 |
| hypocenter | C5×C22⋊A4 | C2×D4 |
| LCS | G ▷ C2×C5×C22⋊A4 ▷ C5×C22⋊A4 | |
| UCS | C1 ◁ C2 | |
3080 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 3080 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 3 generators on 17 points:
(2,3)(4,5)(11,15,14)(12,13,16);(1,2,4,5,3)(7,9)(13,16)(15,17);(6,7,8,9)(10,11)(12,14)(13,15)(16,17)88 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#11713200, #72243964, AGL(1,𝔽5)×#72991 |
| maximal normal subgroup of | D5×#11713200, #72243964, AGL(1,𝔽5)×#72991 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3)(4,5)(11,15,14)(12,13,16), (1,2,4,5,3)(7,9)(13,16)(15,17), (6,7,8,9)(10,11)(12,14)(13,15)(16,17));