Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C24×C32 ⋊ C2×D3 |
1176 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 1176 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2^4,3^3,2^2|f^a=f^2,g^a=g^2,c^b=cd,e^b=e^2,g^b=g^2,h^b=hi,f^e=fg,h^e=i,i^e=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72213328, C2×#13164261, #72220640, … |
| maximal normal subgroup of | C2×#13164261, D4×((C3⋊S4)⋊C3)⋊C2 |
| maximal quotient of | #72220372, C2×#13164261, #72220640, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30324);