Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | H(7,𝔽2) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C22×#1020 |
| commutator, cosocle, hypocenter | #997 | C23 |
| Fitting | H(7,𝔽2) | D3 |
| hypercenter | D4 | #1020 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2 ◁ D4 | |
1498 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 1498 conjugacy classes of subgroups — too many to draw. Only its 42 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|b^2=i,g^2=i,h^2=i,c^a=ci,d^a=d^2,e^a=ef,g^a=gh,h^a=hi,c^b=ci,e^d=f,f^d=ef,g^d=h,h^d=ghi,h^e=hi,g^f=gi,h^g=hi43 conjugacy classes — view the full character table.
| maximal subgroup of | #72209902, #72209903, #72215710, … |
| maximal normal subgroup of | #72215710, #72215767, #72236766, … |
| maximal quotient of | #72227837, #72227857, #72228016, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090210);