Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #11710650 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting, Sylow | C22×H(3,𝔽4) ⋊ C3 |
| derived | C26 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C26 | C22×C3 |
| Frattini | C22 | C22×C22⋊A4 |
| Fitting | C22×H(3,𝔽4) | C3 |
| socle | C24 | C22×A4 |
| LCS | G ▷ C26 | |
| UCS | C1 | |
1821 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 1821 conjugacy classes of subgroups — too many to draw. Only its 45 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:
3,2^8|b^2=h,c^2=h,b^a=bcdehi,c^a=bdgi,d^a=dgh,f^a=fg,g^a=f,h^a=hi,i^a=h,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^c=ei36 conjugacy classes — view the full character table.
| maximal subgroup of | #72234656, #72234952, #72235026, … |
| maximal normal subgroup of | #72234656, A4×#5450, C2×#11710650, … |
| maximal quotient of | #72247738, C2×#11710650 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085031);