Finite Group DatabasebetaFinite Group Databasebeta| order | 2000 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | H(3,𝔽5) ⋊ D8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | (((C5⋊D5)⋊C5)⋊C2)⋊C2 |
| commutator, cosocle | #92525 | C22 |
| Frattini, socle | C2×C5 | D52⋊C2 |
| Fitting | C2×H(3,𝔽5) | D4 |
| hypocenter | H(3,𝔽5) | D8 |
| LCS | G ▷ #92525 ▷ C2×H(3,𝔽5) ▷ H(3,𝔽5) | |
| UCS | C1 ◁ C2 | |
73 conjugacy classes of subgroups.
| transitive degree | μt(G) = 200 |
Polycyclic presentation:
2^4,5^3|a^2=c,c^2=d,b^a=bc,e^a=e^2,f^a=f^3g^3,c^b=cd,e^b=f^4g,f^b=e^4g^4,g^b=g^4,e^c=e^4,f^c=f^4g^2,f^e=fg41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2000, 269);