Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C22×C4⋊C8 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×C4⋊S4 |
| commutator, cosocle | C2×C4×A4 | C23 |
| Frattini | C2×C4 | C22×S4 |
| Fitting | C22×C4⋊C8 | D3 |
| socle | C24 | C2×D12 |
| hypercenter | C4⋊C8 | S4 |
| hypocenter | A4 | #459 |
| LCS | G ▷ C2×C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4⋊C8 | |
654 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 654 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3,2^2|b^2=ef,c^2=e,e^2=f,b^a=be,c^a=cef,e^a=ef,g^a=g^2,h^a=hi,c^b=cd,h^g=i,i^g=hi58 conjugacy classes — view the full character table.
| maximal subgroup of | #72342098 |
| maximal normal subgroup of | #72342098 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1087852);