Finite Group DatabasebetaFinite Group Databasebeta| order | 400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | D5×C22⋊D5 |
| Fitting | C22×C52 ⋊ C22 |
| Sylow | C52 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D52 |
| commutator, socle, cosocle | C2×C52 | C23 |
| Fitting | C22×C52 | C22 |
| hypercenter | C22 | D52 |
| hypocenter | C52 | C2×D4 |
| LCS | G ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ C22 | |
124 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 3 generators on 14 points:
(2,4)(5,6,8,9,7)(10,11,13,14,12);(1,2)(3,4)(11,12)(13,14);(6,7)(8,9)52 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×C22⋊D5, #11717352, D5×#709, … |
| maximal normal subgroup of | AGL(1,𝔽5)×C22⋊D5, #11717352, D5×#709, … |
| maximal quotient of | #11716957, #11716968, #11716969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(400, 179);