Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| Fitting | C2×C3×H(3,𝔽4) ⋊ C2 |
| Sylow | C3 ⋊ #63045 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C25×D3 |
| commutator, socle, cosocle | C22×C3 | C26 |
| Fitting | C2×C3×H(3,𝔽4) | C2 |
| hypercenter | C2×H(3,𝔽4) | D3 |
| hypocenter | C3 | #63045 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×H(3,𝔽4) | |
7484 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^8,3|a^2=g,b^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh,i^f=i^2108 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#63045 |
| maximal normal subgroup of | D3×#63045 |
| maximal quotient of | #72270674, #72270678 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1082791);