Finite Group DatabasebetaFinite Group Databasebeta| order | 400 |
| isoclinism | isoclinic class of #1035 |
| automorphism | Aut(G) = C4×AGL(1,ℤ/25) (order 2000) · Inn(G) = #1035 · Out(G) = C4×C5 |
| homology | Abelianization (H1) = C16 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C25 ⋊ C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #1035 |
| commutator, hypocenter | C25 | C16 |
| Frattini | C4×C5 | AGL(1,𝔽5) |
| Fitting | C4×C25 | C4 |
| socle, prime radical | C2×C5 | #170 |
| cosocle | #6014 | C2 |
| LCS | G ▷ C25 | |
| UCS | C1 ◁ C4 | |
15 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 400 |
A minimal faithful permutation representation — 2 generators on 41 points:
(2,4,5,10)(3,8,9,20)(6,14,15,19)(7,17,18,11)(12,22,23,25)(13,16,21,24)(26,27,28,31,29,32,34,37,30,33,35,38,36,39,40,41);(1,2,6,16,13,3,7,4,11,22,8,19,12,23,14,20,25,17,10,18,9,21,24,15,5)Polycyclic presentation:
2^4,5^2|a^2=b,b^2=c,c^2=d,e^5=f,e^a=e^2f,f^a=f^2,e^b=e^4f^4,f^b=f^440 conjugacy classes — view the full character table.
| maximal subgroup of | D25⋊C16, #11716406, C2×#87709, … |
| maximal normal subgroup of | D25⋊C16, #11716406, C2×#87709, … |
| maximal quotient of | #11716344, C2×#87709, C3×#87709, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(400, 3);