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, Frattini, socle | C22 | C2×#1020 |
| commutator, prime radical, cosocle | C2×#997 | C22 |
| Fitting | C4×H(5,𝔽2) | D3 |
| hypercenter | C2×C4 | #1020 |
| hypocenter | #997 | Q8 |
| LCS | G ▷ C2×#997 ▷ #997 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
597 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 597 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
2^3,3,2^5|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=ef,g^a=gh,e^d=f,f^d=ef,g^d=h,h^d=gh,h^e=hi,g^f=gi37 conjugacy classes — view the full character table.
| maximal subgroup of | Q8×#5934, #72242986, #72242987 |
| maximal normal subgroup of | Q8×#5934, #72242986, #72242987 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090128);