Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C42 |
| unambiguous semidirect product | #17281⋊C3 |
| Fitting, derived, Sylow | #17281 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #5464 |
| commutator, Fitting, cosocle, hypocenter | #17281 | C3 |
| Frattini, socle | C24 | C22⋊A4 |
| LCS | G ▷ #17281 | |
| UCS | C1 ◁ C22 | |
252 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 19 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:
3,2^8|b^2=h,c^2=fhi,d^2=f,e^2=f,b^a=c,c^a=bcfi,d^a=dei,e^a=dfgi,g^a=fgh,h^a=fhi,i^a=h,c^b=cf,d^b=dg,e^b=ei,d^c=dh,e^c=efg,e^d=ef26 conjugacy classes — view the full character table.
| maximal subgroup of | #72210258, #72213093, #72240999, … |
| maximal normal subgroup of | #72266640, C2×#17281⋊C3, #72278691, … |
| maximal quotient of | #72210077, #72266471, #72266606, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083716);