Finite Group DatabasebetaFinite Group Databasebeta| order | 1000 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, Sylow | H(3,𝔽5) ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C5 | #6057 |
| commutator, prime radical, cosocle | #6953 | C22 |
| Fitting, hypocenter | H(3,𝔽5) | Q8 |
| LCS | G ▷ #6953 ▷ H(3,𝔽5) | |
| UCS | C1 ◁ C5 | |
36 conjugacy classes of subgroups.
| transitive degree | μt(G) = 50 |
Polycyclic presentation:
2^3,5^3|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^3f^3,d^b=ef^4,e^b=d^4f,d^c=d^4,e^c=e^4f^2,e^d=ef28 conjugacy classes — view the full character table.
| maximal subgroup of | (#92537⋊C2)⋊C2, C2×#11758708, #72209686 |
| maximal normal subgroup of | (#92537⋊C2)⋊C2, C2×#11758708, #72209686 |
| maximal quotient of | #14522310, C2×#11758708 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1000, 93);