Finite Group DatabasebetaFinite Group Databasebeta| order | 1000 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | H(3,𝔽5) ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | ((C5⋊D5)⋊C5)⋊C2 |
| commutator, Fitting, prime radical, cosocle | C2×H(3,𝔽5) | C22 |
| Frattini, socle | C2×C5 | D52 |
| hypocenter | H(3,𝔽5) | Q8 |
| LCS | G ▷ C2×H(3,𝔽5) ▷ H(3,𝔽5) | |
| UCS | C1 ◁ C2 | |
54 conjugacy classes of subgroups.
| transitive degree | μt(G) = 200 |
Polycyclic presentation:
2^3,5^3|a^2=c,b^2=c,b^a=bc,e^a=e^4,f^a=f^4,d^b=d^4,f^b=f^4,e^d=ef37 conjugacy classes — view the full character table.
| maximal subgroup of | #14522305, #14522307, ((C5⋊D5)⋊C5)⋊Q8, … |
| maximal normal subgroup of | #14522305, #14522307, ((C5⋊D5)⋊C5)⋊Q8, … |
| maximal quotient of | #14522147, #14522148, C2×#11758657 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1000, 42);