Finite Group DatabasebetaFinite Group Databasebeta| order | 1000 |
| isoclinism | isoclinic class of (((C5⋊D5)⋊C5)⋊C2)⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (((C5⋊D5)⋊C5)⋊C2)⋊C2 |
| Fitting, Sylow | H(3,𝔽5) ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle | #6953 | C22 |
| Frattini, socle | C5 | D52⋊C2 |
| Fitting, hypocenter | H(3,𝔽5) | D4 |
| LCS | G ▷ #6953 ▷ H(3,𝔽5) | |
| UCS | C1 | |
50 conjugacy classes of subgroups.
| transitive degree | μt(G) = 50 |
Polycyclic presentation:
2^3,5^3|b^a=bc,d^a=d^4,e^a=def,f^a=f^4,d^b=d^2e^4,e^b=d^3e^3f^3,f^b=f^4,d^c=d^4,e^c=e^4f^2,e^d=ef22 conjugacy classes — view the full character table.
| maximal subgroup of | (#92537⋊C2)⋊C2, C2×(((C5⋊D5)⋊C5)⋊C2)⋊C2 |
| maximal normal subgroup of | (#92537⋊C2)⋊C2, C2×(((C5⋊D5)⋊C5)⋊C2)⋊C2 |
| maximal quotient of | #14522305, #14522306, #14522307, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1000, 92);