Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #85559 |
| commutator, prime radical, cosocle, hypocenter | #73292 | C2 |
| Frattini, socle | C23 | #1020 |
| Fitting | #2840 | D3 |
| LCS | G ▷ #73292 | |
| UCS | C1 ◁ C2 | |
182 conjugacy classes of subgroups; only the 11 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 11 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:
2,3,2^7|a^2=g,e^2=g,f^2=g,b^a=b^2,c^a=cd,e^a=ef,f^a=fg,h^a=hi,c^b=cd,d^b=c,e^b=efg,f^b=e,h^b=i,i^b=hi,e^c=ehi,f^c=fh,e^d=eh,f^d=fi,f^e=fg19 conjugacy classes — view the full character table.
| maximal subgroup of | #72209668, #72209879, C3×#11714256, … |
| maximal normal subgroup of | #72209879, C3×#11714256, #72231805, … |
| maximal quotient of | #72209875, C3×#11714256 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088637);