Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C24×Q8 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×#1020 |
| commutator, cosocle | C2×C22⋊A4 | C23 |
| Fitting | C24×Q8 | D3 |
| socle | C25 | C22×D3 |
| hypercenter | Q8 | #1020 |
| hypocenter | C22⋊A4 | #41 |
| LCS | G ▷ C2×C22⋊A4 ▷ C22⋊A4 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
1698 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 1698 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^4,3,2^4|b^2=d,c^2=d,c^a=cd,e^a=e^2,f^a=fg,h^a=hi,c^b=cd,f^e=g,g^e=fg,h^e=i,i^e=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72228251, #72228285, #41×#1020, … |
| maximal normal subgroup of | #72228285, #41×#1020, #72341520, … |
| maximal quotient of | #72341586 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090186);