Finite Group DatabasebetaFinite Group Databasebeta| order | 400 |
| isoclinism | isoclinic class of #87831 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C52 ⋊ C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D5⋊AGL(1,𝔽5) |
| commutator, Fitting, socle | C2×C52 | C2×C4 |
| prime radical | C2×D52 | C2 |
| cosocle | C2×C5⋊D5 | C22 |
| hypocenter | C52 | C22⋊C4 |
| LCS | G ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 20 |
A minimal faithful permutation representation — 2 generators on 14 points:
(2,4)(5,6)(8,9)(10,11,14,12,13);(1,2)(3,4)(6,7,8,9)(11,12,13,14)Polycyclic presentation:
2^4,5^2|a^2=c,b^a=bd,e^a=e^3,f^a=f^3,e^b=e^4,e^c=e^4,f^c=f^422 conjugacy classes — view the full character table.
| maximal subgroup of | #11717309, #11717311, #11717312, … |
| maximal normal subgroup of | #11717309, #11717311, #11717312, … |
| maximal quotient of | #11716765, #11716766, #11716767, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(400, 125);