Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C2×#1020 |
| commutator, cosocle, hypocenter | #5465 | C22 |
| Fitting | C2×H(3,𝔽4) | D3 |
| prime radical | C2×#5465 | C2 |
| hypercenter | C23 | #1020 |
| LCS | G ▷ #5465 | |
| UCS | C1 ◁ C22 ◁ C23 | |
382 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 382 conjugacy classes of subgroups — too many to draw. Only its 21 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:
2^2,3,2^6|a^2=i,d^2=i,e^2=i,f^2=hi,g^2=hi,b^a=bhi,c^a=c^2,d^a=de,e^a=ei,f^a=fg,g^a=ghi,d^c=e,e^c=dei,f^c=g,g^c=fghi,e^d=ei,g^d=gh,f^e=fh,g^f=ghi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72228061 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090025);