Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #3529 ⋊ D3 |
| derived | #5465 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #5934 |
| commutator, cosocle, hypocenter | #5465 | C22 |
| Frattini | C22 | C2×#1020 |
| Fitting | #3529 | D3 |
| socle | C2 | C2×#5934 |
| hypercenter | C2×C4 | #1020 |
| LCS | G ▷ #5465 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
492 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 19 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:
2^2,3,2^6|b^2=i,c^a=c^2,d^a=de,f^a=fg,h^a=hi,d^c=e,e^c=de,f^c=g,g^c=fg,f^d=fi,g^d=ghi,f^e=fh,g^e=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72209927, #72218487, #72234215, … |
| maximal normal subgroup of | #72209927, #72234215, C2×#11715752, … |
| maximal quotient of | #72227894, #72242868, C4×#85557, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090133);