Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22 | #1020⋊C2 |
| commutator, cosocle | C2×#997 | C22 |
| Frattini | C24 | C2×S4 |
| Fitting | C2×B(4,𝔽2) | D3 |
| prime radical | C2×B(4,𝔽2)⋊C3 | C2 |
| hypocenter | #997 | D4 |
| LCS | G ▷ C2×#997 ▷ #997 | |
| UCS | C1 ◁ C22 | |
350 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 32 |
Polycyclic presentation:
2^3,3,2^5|a^2=i,e^2=i,f^2=i,b^a=bc,d^a=d^2,e^a=ef,f^a=fi,g^a=gh,e^b=egi,f^b=fhi,e^d=f,f^d=efi,g^d=h,h^d=gh,f^e=fi,h^e=hi,g^f=gi31 conjugacy classes — view the full character table.
| maximal subgroup of | #72227711, #72227771, #72227868, … |
| maximal normal subgroup of | #72227711, #72227771, #72228017, … |
| maximal quotient of | #72209935, #72342005, C2×#11711634 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1086015);