Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C2×C4 |
| unambiguous semidirect product | #13834⋊C3 |
| Fitting, derived, Sylow | #13834 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C22⋊(C42⋊C3) |
| commutator, Fitting, cosocle, hypocenter | #13834 | C3 |
| Frattini, socle | C24 | C22⋊A4 |
| LCS | G ▷ #13834 | |
| UCS | C1 ◁ C22 | |
290 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 21 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=i,d^2=fh,e^2=fg,b^a=bcdefgh,c^a=bcefghi,d^a=bcdfh,e^a=bdefgh,h^a=i,i^a=hi,c^b=cf,d^b=dg,e^b=efg,d^c=dfg,e^c=ef36 conjugacy classes — view the full character table.
| maximal subgroup of | #72210258, #72210818, #72211982, … |
| maximal normal subgroup of | #72210818, C2×#13834⋊C3, #72287301 |
| maximal quotient of | #72238854, #72239993, #72267016, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083606);