Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23×C3 |
No aliases for this group.
| Fitting | C32×H(5,𝔽2) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×(C3⋊D3)⋊A4 |
| commutator, cosocle, hypocenter | C32×Q8 | C23×C3 |
| Frattini, socle | C2×C3 | C22×D3×A4 |
| Fitting | C32×H(5,𝔽2) | C2×C3 |
| hypercenter | Q8 | (C3⋊D3)⋊A4 |
| LCS | G ▷ C32×Q8 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
1095 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 1095 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^3,3,2^2,3^2,2|b^2=i,c^2=i,e^2=i,f^2=i,g^a=g^2,h^a=h^2,c^b=ci,e^d=ef,f^d=e,g^d=gh,f^e=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72220370, #72220372, C2×#13164635 |
| maximal normal subgroup of | #72220370, #72220372, C2×#13164635 |
| maximal quotient of | C2×#13164635 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30698);