Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1020⋊C2 |
| commutator, cosocle, hypocenter | #997 | C23 |
| Frattini | C23 | C22×S4 |
| Fitting | #3025 | D3 |
| prime radical | #3025⋊C3 | C2 |
| hypercenter | C4 | #1020⋊C2 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2 ◁ C4 | |
502 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 502 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
Polycyclic presentation:
2^3,3,2^5|a^2=i,b^2=i,b^a=bi,d^a=d^2,e^a=ef,g^a=gh,e^c=egi,f^c=fhi,e^d=f,f^d=ef,g^d=h,h^d=gh,h^e=hi,g^f=gi34 conjugacy classes — view the full character table.
| maximal subgroup of | #72236838, #72236883, #72236902, … |
| maximal normal subgroup of | #72236838, #72236883, #72236902, … |
| maximal quotient of | #72341389, #72341416, #72341767, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088558);