Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #11711670 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #1655 ⋊ D3 |
| derived | #5464 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #5464 | C22 |
| Frattini | C42 | C2×S4 |
| Fitting | #1655 | D3 |
| socle | C22 | #1020⋊C2 |
| LCS | G ▷ #5464 | |
| UCS | C1 | |
200 conjugacy classes of subgroups; only the 11 normal subgroups are shown.
This group has 200 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.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2^2,3,2^6|f^2=h,g^2=i,c^a=c^2,d^a=de,f^a=fgh,h^a=hi,d^b=dfi,e^b=egi,f^b=fh,g^b=gi,d^c=e,e^c=de,f^c=gh,g^c=fg,h^c=i,i^c=hi,f^d=fh,g^d=ghi,f^e=fhi,g^e=gi18 conjugacy classes — view the full character table.
| maximal subgroup of | #72340786, #72340788 |
| maximal normal subgroup of | #72340786 |
| maximal quotient of | #72344170 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1086051);