Finite Group DatabasebetaFinite Group Databasebeta| order | 972 |
| isoclinism | isoclinic class of #92161⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #92161⋊C2 |
| unambiguous semidirect product | #92162⋊C2 |
| unambiguous semidirect product | #92199⋊C2 |
| Fitting, derived, Sylow | #6857 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | #6857 | C22 |
| Frattini | C33 | D32 |
| socle | C32 | ((C3⋊D3)⋊C3)⋊C2 |
| LCS | G ▷ #6857 | |
| UCS | C1 | |
142 conjugacy classes of subgroups; only the 14 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 14 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 18 |
Polycyclic presentation:
2^2,3^5|d^3=g,d^a=d^2g^2,e^a=e^2f,g^a=g^2,c^b=c^2,e^b=e^2g,f^b=f^2,d^c=de,e^c=eg,e^d=ef26 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#92161⋊C2 |
| maximal normal subgroup of | C2×#92161⋊C2 |
| maximal quotient of | #92199⋊C4, #92161⋊C4, #92162⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(972, 102);